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Theorem psgnunilem2 27397
Description: Lemma for psgnuni 27401. Induction step for moving a transposition as far to the right as possible. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
Hypotheses
Ref Expression
psgnunilem2.g  |-  G  =  ( SymGrp `  D )
psgnunilem2.t  |-  T  =  ran  (pmTrsp `  D
)
psgnunilem2.d  |-  ( ph  ->  D  e.  V )
psgnunilem2.w  |-  ( ph  ->  W  e. Word  T )
psgnunilem2.id  |-  ( ph  ->  ( G  gsumg  W )  =  (  _I  |`  D )
)
psgnunilem2.l  |-  ( ph  ->  ( # `  W
)  =  L )
psgnunilem2.ix  |-  ( ph  ->  I  e.  ( 0..^ L ) )
psgnunilem2.a  |-  ( ph  ->  A  e.  dom  (
( W `  I
)  \  _I  )
)
psgnunilem2.al  |-  ( ph  ->  A. k  e.  ( 0..^ I )  -.  A  e.  dom  (
( W `  k
)  \  _I  )
)
psgnunilem2.in  |-  ( ph  ->  -.  E. x  e. Word  T ( ( # `  x )  =  ( L  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D ) ) )
Assertion
Ref Expression
psgnunilem2  |-  ( ph  ->  E. w  e. Word  T
( ( ( G 
gsumg  w )  =  (  _I  |`  D )  /\  ( # `  w
)  =  L )  /\  ( ( I  +  1 )  e.  ( 0..^ L )  /\  A  e.  dom  ( ( w `  ( I  +  1
) )  \  _I  )  /\  A. j  e.  ( 0..^ ( I  +  1 ) )  -.  A  e.  dom  ( ( w `  j )  \  _I  ) ) ) )
Distinct variable groups:    j, k, w, A    x, j, D, w    ph, j    j, G   
x, k, G, w   
j, I, k, w, x    T, j, w, x   
j, W, k, w, x    w, L, x
Allowed substitution hints:    ph( x, w, k)    A( x)    D( k)    T( k)    L( j, k)    V( x, w, j, k)

Proof of Theorem psgnunilem2
Dummy variables  r 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psgnunilem2.w . . . . . . 7  |-  ( ph  ->  W  e. Word  T )
2 wrd0 11734 . . . . . . 7  |-  (/)  e. Word  T
3 splcl 11783 . . . . . . 7  |-  ( ( W  e. Word  T  /\  (/) 
e. Word  T )  ->  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )  e. Word  T
)
41, 2, 3sylancl 645 . . . . . 6  |-  ( ph  ->  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )  e. Word  T )
54adantr 453 . . . . 5  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( W splice  <.
I ,  ( I  +  2 ) ,  (/) >. )  e. Word  T
)
6 fzossfz 11159 . . . . . . . . . . 11  |-  ( 0..^ L )  C_  (
0 ... L )
7 psgnunilem2.ix . . . . . . . . . . 11  |-  ( ph  ->  I  e.  ( 0..^ L ) )
86, 7sseldi 3348 . . . . . . . . . 10  |-  ( ph  ->  I  e.  ( 0 ... L ) )
9 elfznn0 11085 . . . . . . . . . 10  |-  ( I  e.  ( 0 ... L )  ->  I  e.  NN0 )
108, 9syl 16 . . . . . . . . 9  |-  ( ph  ->  I  e.  NN0 )
11 2nn0 10240 . . . . . . . . . 10  |-  2  e.  NN0
12 nn0addcl 10257 . . . . . . . . . 10  |-  ( ( I  e.  NN0  /\  2  e.  NN0 )  -> 
( I  +  2 )  e.  NN0 )
1310, 11, 12sylancl 645 . . . . . . . . 9  |-  ( ph  ->  ( I  +  2 )  e.  NN0 )
1410nn0red 10277 . . . . . . . . . 10  |-  ( ph  ->  I  e.  RR )
15 nn0addge1 10268 . . . . . . . . . 10  |-  ( ( I  e.  RR  /\  2  e.  NN0 )  ->  I  <_  ( I  + 
2 ) )
1614, 11, 15sylancl 645 . . . . . . . . 9  |-  ( ph  ->  I  <_  ( I  +  2 ) )
17 elfz2nn0 11084 . . . . . . . . 9  |-  ( I  e.  ( 0 ... ( I  +  2 ) )  <->  ( I  e.  NN0  /\  ( I  +  2 )  e. 
NN0  /\  I  <_  ( I  +  2 ) ) )
1810, 13, 16, 17syl3anbrc 1139 . . . . . . . 8  |-  ( ph  ->  I  e.  ( 0 ... ( I  + 
2 ) ) )
19 psgnunilem2.g . . . . . . . . . . 11  |-  G  =  ( SymGrp `  D )
20 psgnunilem2.t . . . . . . . . . . 11  |-  T  =  ran  (pmTrsp `  D
)
21 psgnunilem2.d . . . . . . . . . . 11  |-  ( ph  ->  D  e.  V )
22 psgnunilem2.id . . . . . . . . . . 11  |-  ( ph  ->  ( G  gsumg  W )  =  (  _I  |`  D )
)
23 psgnunilem2.l . . . . . . . . . . 11  |-  ( ph  ->  ( # `  W
)  =  L )
24 psgnunilem2.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  dom  (
( W `  I
)  \  _I  )
)
25 psgnunilem2.al . . . . . . . . . . 11  |-  ( ph  ->  A. k  e.  ( 0..^ I )  -.  A  e.  dom  (
( W `  k
)  \  _I  )
)
2619, 20, 21, 1, 22, 23, 7, 24, 25psgnunilem5 27396 . . . . . . . . . 10  |-  ( ph  ->  ( I  +  1 )  e.  ( 0..^ L ) )
27 fzofzp1 11191 . . . . . . . . . 10  |-  ( ( I  +  1 )  e.  ( 0..^ L )  ->  ( (
I  +  1 )  +  1 )  e.  ( 0 ... L
) )
2826, 27syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( I  + 
1 )  +  1 )  e.  ( 0 ... L ) )
2910nn0cnd 10278 . . . . . . . . . . 11  |-  ( ph  ->  I  e.  CC )
30 ax-1cn 9050 . . . . . . . . . . . 12  |-  1  e.  CC
3130a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  1  e.  CC )
3229, 31, 31addassd 9112 . . . . . . . . . 10  |-  ( ph  ->  ( ( I  + 
1 )  +  1 )  =  ( I  +  ( 1  +  1 ) ) )
33 df-2 10060 . . . . . . . . . . 11  |-  2  =  ( 1  +  1 )
3433oveq2i 6094 . . . . . . . . . 10  |-  ( I  +  2 )  =  ( I  +  ( 1  +  1 ) )
3532, 34syl6reqr 2489 . . . . . . . . 9  |-  ( ph  ->  ( I  +  2 )  =  ( ( I  +  1 )  +  1 ) )
3623oveq2d 6099 . . . . . . . . 9  |-  ( ph  ->  ( 0 ... ( # `
 W ) )  =  ( 0 ... L ) )
3728, 35, 363eltr4d 2519 . . . . . . . 8  |-  ( ph  ->  ( I  +  2 )  e.  ( 0 ... ( # `  W
) ) )
382a1i 11 . . . . . . . 8  |-  ( ph  -> 
(/)  e. Word  T )
391, 18, 37, 38spllen 11785 . . . . . . 7  |-  ( ph  ->  ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. ) )  =  ( ( # `  W
)  +  ( (
# `  (/) )  -  ( ( I  + 
2 )  -  I
) ) ) )
40 hash0 11648 . . . . . . . . . . 11  |-  ( # `  (/) )  =  0
4140oveq1i 6093 . . . . . . . . . 10  |-  ( (
# `  (/) )  -  ( ( I  + 
2 )  -  I
) )  =  ( 0  -  ( ( I  +  2 )  -  I ) )
42 df-neg 9296 . . . . . . . . . 10  |-  -u (
( I  +  2 )  -  I )  =  ( 0  -  ( ( I  + 
2 )  -  I
) )
4341, 42eqtr4i 2461 . . . . . . . . 9  |-  ( (
# `  (/) )  -  ( ( I  + 
2 )  -  I
) )  =  -u ( ( I  + 
2 )  -  I
)
44 2cn 10072 . . . . . . . . . . 11  |-  2  e.  CC
45 pncan2 9314 . . . . . . . . . . 11  |-  ( ( I  e.  CC  /\  2  e.  CC )  ->  ( ( I  + 
2 )  -  I
)  =  2 )
4629, 44, 45sylancl 645 . . . . . . . . . 10  |-  ( ph  ->  ( ( I  + 
2 )  -  I
)  =  2 )
4746negeqd 9302 . . . . . . . . 9  |-  ( ph  -> 
-u ( ( I  +  2 )  -  I )  =  -u
2 )
4843, 47syl5eq 2482 . . . . . . . 8  |-  ( ph  ->  ( ( # `  (/) )  -  ( ( I  + 
2 )  -  I
) )  =  -u
2 )
4923, 48oveq12d 6101 . . . . . . 7  |-  ( ph  ->  ( ( # `  W
)  +  ( (
# `  (/) )  -  ( ( I  + 
2 )  -  I
) ) )  =  ( L  +  -u
2 ) )
50 elfzel2 11059 . . . . . . . . . 10  |-  ( I  e.  ( 0 ... L )  ->  L  e.  ZZ )
518, 50syl 16 . . . . . . . . 9  |-  ( ph  ->  L  e.  ZZ )
5251zcnd 10378 . . . . . . . 8  |-  ( ph  ->  L  e.  CC )
53 negsub 9351 . . . . . . . 8  |-  ( ( L  e.  CC  /\  2  e.  CC )  ->  ( L  +  -u
2 )  =  ( L  -  2 ) )
5452, 44, 53sylancl 645 . . . . . . 7  |-  ( ph  ->  ( L  +  -u
2 )  =  ( L  -  2 ) )
5539, 49, 543eqtrd 2474 . . . . . 6  |-  ( ph  ->  ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. ) )  =  ( L  -  2 ) )
5655adantr 453 . . . . 5  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. ) )  =  ( L  -  2 ) )
57 splid 11784 . . . . . . . . 9  |-  ( ( W  e. Word  T  /\  ( I  e.  (
0 ... ( I  + 
2 ) )  /\  ( I  +  2
)  e.  ( 0 ... ( # `  W
) ) ) )  ->  ( W splice  <. I ,  ( I  +  2 ) ,  ( W substr  <. I ,  ( I  +  2 ) >.
) >. )  =  W )
581, 18, 37, 57syl12anc 1183 . . . . . . . 8  |-  ( ph  ->  ( W splice  <. I ,  ( I  +  2 ) ,  ( W substr  <. I ,  ( I  +  2 ) >.
) >. )  =  W )
5958oveq2d 6099 . . . . . . 7  |-  ( ph  ->  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  ( W substr  <. I ,  ( I  +  2 ) >.
) >. ) )  =  ( G  gsumg  W ) )
6059adantr 453 . . . . . 6  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  ( W substr  <. I ,  ( I  +  2 ) >. ) >. )
)  =  ( G 
gsumg  W ) )
61 eqid 2438 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
6219symggrp 15105 . . . . . . . . . 10  |-  ( D  e.  V  ->  G  e.  Grp )
6321, 62syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
64 grpmnd 14819 . . . . . . . . 9  |-  ( G  e.  Grp  ->  G  e.  Mnd )
6563, 64syl 16 . . . . . . . 8  |-  ( ph  ->  G  e.  Mnd )
6665adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  G  e.  Mnd )
6720, 19, 61symgtrf 27389 . . . . . . . . . 10  |-  T  C_  ( Base `  G )
68 sswrd 11739 . . . . . . . . . 10  |-  ( T 
C_  ( Base `  G
)  -> Word  T  C_ Word  ( Base `  G ) )
6967, 68ax-mp 8 . . . . . . . . 9  |- Word  T  C_ Word  (
Base `  G )
7069, 1sseldi 3348 . . . . . . . 8  |-  ( ph  ->  W  e. Word  ( Base `  G ) )
7170adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  W  e. Word  (
Base `  G )
)
7218adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  I  e.  ( 0 ... (
I  +  2 ) ) )
7337adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( I  +  2 )  e.  ( 0 ... ( # `
 W ) ) )
74 swrdcl 11768 . . . . . . . . 9  |-  ( W  e. Word  ( Base `  G
)  ->  ( W substr  <.
I ,  ( I  +  2 ) >.
)  e. Word  ( Base `  G ) )
7570, 74syl 16 . . . . . . . 8  |-  ( ph  ->  ( W substr  <. I ,  ( I  +  2 ) >. )  e. Word  ( Base `  G ) )
7675adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( W substr  <.
I ,  ( I  +  2 ) >.
)  e. Word  ( Base `  G ) )
77 wrd0 11734 . . . . . . . 8  |-  (/)  e. Word  ( Base `  G )
7877a1i 11 . . . . . . 7  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  (/)  e. Word  ( Base `  G ) )
7923oveq2d 6099 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 0..^ ( # `  W ) )  =  ( 0..^ L ) )
8026, 79eleqtrrd 2515 . . . . . . . . . . . 12  |-  ( ph  ->  ( I  +  1 )  e.  ( 0..^ ( # `  W
) ) )
81 swrds2 11882 . . . . . . . . . . . 12  |-  ( ( W  e. Word  T  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  ( W substr  <. I ,  ( I  +  2 )
>. )  =  <" ( W `  I
) ( W `  ( I  +  1
) ) "> )
821, 10, 80, 81syl3anc 1185 . . . . . . . . . . 11  |-  ( ph  ->  ( W substr  <. I ,  ( I  +  2 ) >. )  =  <" ( W `  I
) ( W `  ( I  +  1
) ) "> )
8382oveq2d 6099 . . . . . . . . . 10  |-  ( ph  ->  ( G  gsumg  ( W substr  <. I ,  ( I  +  2 ) >. ) )  =  ( G  gsumg 
<" ( W `  I ) ( W `
 ( I  + 
1 ) ) "> ) )
84 wrdf 11735 . . . . . . . . . . . . . . 15  |-  ( W  e. Word  T  ->  W : ( 0..^ (
# `  W )
) --> T )
851, 84syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  W : ( 0..^ ( # `  W
) ) --> T )
8679feq2d 5583 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( W : ( 0..^ ( # `  W
) ) --> T  <->  W :
( 0..^ L ) --> T ) )
8785, 86mpbid 203 . . . . . . . . . . . . 13  |-  ( ph  ->  W : ( 0..^ L ) --> T )
8887, 7ffvelrnd 5873 . . . . . . . . . . . 12  |-  ( ph  ->  ( W `  I
)  e.  T )
8967, 88sseldi 3348 . . . . . . . . . . 11  |-  ( ph  ->  ( W `  I
)  e.  ( Base `  G ) )
9087, 26ffvelrnd 5873 . . . . . . . . . . . 12  |-  ( ph  ->  ( W `  (
I  +  1 ) )  e.  T )
9167, 90sseldi 3348 . . . . . . . . . . 11  |-  ( ph  ->  ( W `  (
I  +  1 ) )  e.  ( Base `  G ) )
92 eqid 2438 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
9361, 92gsumws2 14790 . . . . . . . . . . 11  |-  ( ( G  e.  Mnd  /\  ( W `  I )  e.  ( Base `  G
)  /\  ( W `  ( I  +  1 ) )  e.  (
Base `  G )
)  ->  ( G  gsumg  <" ( W `  I ) ( W `
 ( I  + 
1 ) ) "> )  =  ( ( W `  I
) ( +g  `  G
) ( W `  ( I  +  1
) ) ) )
9465, 89, 91, 93syl3anc 1185 . . . . . . . . . 10  |-  ( ph  ->  ( G  gsumg 
<" ( W `  I ) ( W `
 ( I  + 
1 ) ) "> )  =  ( ( W `  I
) ( +g  `  G
) ( W `  ( I  +  1
) ) ) )
9519, 61, 92symgov 15102 . . . . . . . . . . 11  |-  ( ( ( W `  I
)  e.  ( Base `  G )  /\  ( W `  ( I  +  1 ) )  e.  ( Base `  G
) )  ->  (
( W `  I
) ( +g  `  G
) ( W `  ( I  +  1
) ) )  =  ( ( W `  I )  o.  ( W `  ( I  +  1 ) ) ) )
9689, 91, 95syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  ( ( W `  I ) ( +g  `  G ) ( W `
 ( I  + 
1 ) ) )  =  ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) ) )
9783, 94, 963eqtrd 2474 . . . . . . . . 9  |-  ( ph  ->  ( G  gsumg  ( W substr  <. I ,  ( I  +  2 ) >. ) )  =  ( ( W `  I )  o.  ( W `  ( I  +  1 ) ) ) )
9897adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( G  gsumg  ( W substr  <. I ,  ( I  +  2 )
>. ) )  =  ( ( W `  I
)  o.  ( W `
 ( I  + 
1 ) ) ) )
99 simpr 449 . . . . . . . 8  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)
10019symgid 15106 . . . . . . . . . . 11  |-  ( D  e.  V  ->  (  _I  |`  D )  =  ( 0g `  G
) )
10121, 100syl 16 . . . . . . . . . 10  |-  ( ph  ->  (  _I  |`  D )  =  ( 0g `  G ) )
102 eqid 2438 . . . . . . . . . . 11  |-  ( 0g
`  G )  =  ( 0g `  G
)
103102gsum0 14782 . . . . . . . . . 10  |-  ( G 
gsumg  (/) )  =  ( 0g
`  G )
104101, 103syl6eqr 2488 . . . . . . . . 9  |-  ( ph  ->  (  _I  |`  D )  =  ( G  gsumg  (/) ) )
105104adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  (  _I  |`  D )  =  ( G  gsumg  (/) ) )
10698, 99, 1053eqtrd 2474 . . . . . . 7  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( G  gsumg  ( W substr  <. I ,  ( I  +  2 )
>. ) )  =  ( G  gsumg  (/) ) )
10761, 66, 71, 72, 73, 76, 78, 106gsumspl 14791 . . . . . 6  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  ( W substr  <. I ,  ( I  +  2 ) >. ) >. )
)  =  ( G 
gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )
) )
10822adantr 453 . . . . . 6  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( G  gsumg  W )  =  (  _I  |`  D ) )
10960, 107, 1083eqtr3d 2478 . . . . 5  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. ) )  =  (  _I  |`  D ) )
110 fveq2 5730 . . . . . . . 8  |-  ( x  =  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )  ->  ( # `  x
)  =  ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )
) )
111110eqeq1d 2446 . . . . . . 7  |-  ( x  =  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )  ->  ( ( # `  x
)  =  ( L  -  2 )  <->  ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. ) )  =  ( L  -  2 ) ) )
112 oveq2 6091 . . . . . . . 8  |-  ( x  =  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )  ->  ( G  gsumg  x )  =  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )
) )
113112eqeq1d 2446 . . . . . . 7  |-  ( x  =  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )  ->  ( ( G  gsumg  x )  =  (  _I  |`  D )  <-> 
( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )
)  =  (  _I  |`  D ) ) )
114111, 113anbi12d 693 . . . . . 6  |-  ( x  =  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )  ->  ( ( ( # `  x )  =  ( L  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D ) )  <->  ( ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )
)  =  ( L  -  2 )  /\  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )
)  =  (  _I  |`  D ) ) ) )
115114rspcev 3054 . . . . 5  |-  ( ( ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )  e. Word  T  /\  ( (
# `  ( W splice  <.
I ,  ( I  +  2 ) ,  (/) >. ) )  =  ( L  -  2 )  /\  ( G 
gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )
)  =  (  _I  |`  D ) ) )  ->  E. x  e. Word  T
( ( # `  x
)  =  ( L  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) )
1165, 56, 109, 115syl12anc 1183 . . . 4  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  E. x  e. Word  T ( ( # `  x )  =  ( L  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D ) ) )
117 psgnunilem2.in . . . . 5  |-  ( ph  ->  -.  E. x  e. Word  T ( ( # `  x )  =  ( L  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D ) ) )
118117adantr 453 . . . 4  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  -.  E. x  e. Word  T ( ( # `  x )  =  ( L  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D ) ) )
119116, 118pm2.21dd 102 . . 3  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  E. w  e. Word  T ( ( ( G  gsumg  w )  =  (  _I  |`  D )  /\  ( # `  w
)  =  L )  /\  ( ( I  +  1 )  e.  ( 0..^ L )  /\  A  e.  dom  ( ( w `  ( I  +  1
) )  \  _I  )  /\  A. j  e.  ( 0..^ ( I  +  1 ) )  -.  A  e.  dom  ( ( w `  j )  \  _I  ) ) ) )
120119ex 425 . 2  |-  ( ph  ->  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )  ->  E. w  e. Word  T
( ( ( G 
gsumg  w )  =  (  _I  |`  D )  /\  ( # `  w
)  =  L )  /\  ( ( I  +  1 )  e.  ( 0..^ L )  /\  A  e.  dom  ( ( w `  ( I  +  1
) )  \  _I  )  /\  A. j  e.  ( 0..^ ( I  +  1 ) )  -.  A  e.  dom  ( ( w `  j )  \  _I  ) ) ) ) )
1211adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  ->  W  e. Word  T )
122 simprl 734 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
r  e.  T )
123 simprr 735 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
s  e.  T )
124122, 123s2cld 11835 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  ->  <" r s ">  e. Word  T )
125 splcl 11783 . . . . . . 7  |-  ( ( W  e. Word  T  /\  <" r s ">  e. Word  T )  ->  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  e. Word  T )
126121, 124, 125syl2anc 644 . . . . . 6  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  e. Word  T )
127126adantrr 699 . . . . 5  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  e. Word  T )
12865adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  G  e.  Mnd )
12970adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  W  e. Word  ( Base `  G ) )
13018adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  I  e.  ( 0 ... ( I  +  2 ) ) )
13137adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( I  + 
2 )  e.  ( 0 ... ( # `  W ) ) )
13269, 124sseldi 3348 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  ->  <" r s ">  e. Word  ( Base `  G ) )
133132adantrr 699 . . . . . . . 8  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  <" r s ">  e. Word  ( Base `  G ) )
13475adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( W substr  <. I ,  ( I  +  2 ) >. )  e. Word  ( Base `  G ) )
135 simprr1 1006 . . . . . . . . 9  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s ) )
13697adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( G  gsumg  ( W substr  <. I ,  ( I  +  2 ) >.
) )  =  ( ( W `  I
)  o.  ( W `
 ( I  + 
1 ) ) ) )
13765adantr 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  ->  G  e.  Mnd )
13867a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  T  C_  ( Base `  G ) )
139138sselda 3350 . . . . . . . . . . . . 13  |-  ( (
ph  /\  r  e.  T )  ->  r  e.  ( Base `  G
) )
140139adantrr 699 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
r  e.  ( Base `  G ) )
141138sselda 3350 . . . . . . . . . . . . 13  |-  ( (
ph  /\  s  e.  T )  ->  s  e.  ( Base `  G
) )
142141adantrl 698 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
s  e.  ( Base `  G ) )
14361, 92gsumws2 14790 . . . . . . . . . . . 12  |-  ( ( G  e.  Mnd  /\  r  e.  ( Base `  G )  /\  s  e.  ( Base `  G
) )  ->  ( G  gsumg 
<" r s "> )  =  ( r ( +g  `  G
) s ) )
144137, 140, 142, 143syl3anc 1185 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( G  gsumg 
<" r s "> )  =  ( r ( +g  `  G
) s ) )
14519, 61, 92symgov 15102 . . . . . . . . . . . 12  |-  ( ( r  e.  ( Base `  G )  /\  s  e.  ( Base `  G
) )  ->  (
r ( +g  `  G
) s )  =  ( r  o.  s
) )
146140, 142, 145syl2anc 644 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( r ( +g  `  G ) s )  =  ( r  o.  s ) )
147144, 146eqtrd 2470 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( G  gsumg 
<" r s "> )  =  ( r  o.  s ) )
148147adantrr 699 . . . . . . . . 9  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( G  gsumg  <" r
s "> )  =  ( r  o.  s ) )
149135, 136, 1483eqtr4rd 2481 . . . . . . . 8  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( G  gsumg  <" r
s "> )  =  ( G  gsumg  ( W substr  <. I ,  ( I  +  2 ) >.
) ) )
15061, 128, 129, 130, 131, 133, 134, 149gsumspl 14791 . . . . . . 7  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) )  =  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  ( W substr  <. I ,  ( I  +  2 ) >.
) >. ) ) )
15159adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  ( W substr  <. I ,  ( I  +  2 ) >. ) >. )
)  =  ( G 
gsumg  W ) )
15222adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( G  gsumg  W )  =  (  _I  |`  D ) )
153150, 151, 1523eqtrd 2474 . . . . . 6  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) )  =  (  _I  |`  D ) )
15418adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  ->  I  e.  ( 0 ... ( I  + 
2 ) ) )
15537adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( I  +  2 )  e.  ( 0 ... ( # `  W
) ) )
156121, 154, 155, 124spllen 11785 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( # `  ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) )  =  ( ( # `  W
)  +  ( (
# `  <" r
s "> )  -  ( ( I  +  2 )  -  I ) ) ) )
157 s2len 11853 . . . . . . . . . . . . 13  |-  ( # `  <" r s "> )  =  2
158157oveq1i 6093 . . . . . . . . . . . 12  |-  ( (
# `  <" r
s "> )  -  ( ( I  +  2 )  -  I ) )  =  ( 2  -  (
( I  +  2 )  -  I ) )
15946oveq2d 6099 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 2  -  (
( I  +  2 )  -  I ) )  =  ( 2  -  2 ) )
16044subidi 9373 . . . . . . . . . . . . 13  |-  ( 2  -  2 )  =  0
161159, 160syl6eq 2486 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2  -  (
( I  +  2 )  -  I ) )  =  0 )
162158, 161syl5eq 2482 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  <" r s "> )  -  ( (
I  +  2 )  -  I ) )  =  0 )
163162oveq2d 6099 . . . . . . . . . 10  |-  ( ph  ->  ( ( # `  W
)  +  ( (
# `  <" r
s "> )  -  ( ( I  +  2 )  -  I ) ) )  =  ( ( # `  W )  +  0 ) )
16423, 52eqeltrd 2512 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  W
)  e.  CC )
165164addid1d 9268 . . . . . . . . . 10  |-  ( ph  ->  ( ( # `  W
)  +  0 )  =  ( # `  W
) )
166163, 165, 233eqtrd 2474 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  W
)  +  ( (
# `  <" r
s "> )  -  ( ( I  +  2 )  -  I ) ) )  =  L )
167166adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( ( # `  W
)  +  ( (
# `  <" r
s "> )  -  ( ( I  +  2 )  -  I ) ) )  =  L )
168156, 167eqtrd 2470 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( # `  ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) )  =  L )
169168adantrr 699 . . . . . 6  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  <" r s "> >. )
)  =  L )
170153, 169jca 520 . . . . 5  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( ( G 
gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )
)  =  (  _I  |`  D )  /\  ( # `
 ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )
)  =  L ) )
17126adantr 453 . . . . . 6  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( I  + 
1 )  e.  ( 0..^ L ) )
172 simprr2 1007 . . . . . . 7  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  A  e.  dom  ( s  \  _I  ) )
173 1nn0 10239 . . . . . . . . . . . . . . 15  |-  1  e.  NN0
174 2nn 10135 . . . . . . . . . . . . . . 15  |-  2  e.  NN
175 1lt2 10144 . . . . . . . . . . . . . . 15  |-  1  <  2
176 elfzo0 11173 . . . . . . . . . . . . . . 15  |-  ( 1  e.  ( 0..^ 2 )  <->  ( 1  e. 
NN0  /\  2  e.  NN  /\  1  <  2
) )
177173, 174, 175, 176mpbir3an 1137 . . . . . . . . . . . . . 14  |-  1  e.  ( 0..^ 2 )
178157oveq2i 6094 . . . . . . . . . . . . . 14  |-  ( 0..^ ( # `  <" r s "> ) )  =  ( 0..^ 2 )
179177, 178eleqtrri 2511 . . . . . . . . . . . . 13  |-  1  e.  ( 0..^ ( # `  <" r s "> ) )
180179a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
1  e.  ( 0..^ ( # `  <" r s "> ) ) )
181121, 154, 155, 124, 180splfv2a 11787 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  ( I  +  1 ) )  =  (
<" r s "> `  1 )
)
182 s2fv1 11852 . . . . . . . . . . . 12  |-  ( s  e.  T  ->  ( <" r s "> `  1 )  =  s )
183182ad2antll 711 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( <" r s "> `  1
)  =  s )
184181, 183eqtrd 2470 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  ( I  +  1 ) )  =  s )
185184adantrr 699 . . . . . . . . 9  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  (
I  +  1 ) )  =  s )
186185difeq1d 3466 . . . . . . . 8  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r s "> >. ) `  ( I  +  1 ) )  \  _I  )  =  ( s  \  _I  ) )
187186dmeqd 5074 . . . . . . 7  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  dom  ( (
( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  ( I  +  1 ) )  \  _I  )  =  dom  ( s 
\  _I  ) )
188172, 187eleqtrrd 2515 . . . . . 6  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  (
I  +  1 ) )  \  _I  )
)
189 fzosplitsni 11198 . . . . . . . . . . 11  |-  ( I  e.  ( ZZ>= `  0
)  ->  ( j  e.  ( 0..^ ( I  +  1 ) )  <-> 
( j  e.  ( 0..^ I )  \/  j  =  I ) ) )
190 nn0uz 10522 . . . . . . . . . . 11  |-  NN0  =  ( ZZ>= `  0 )
191189, 190eleq2s 2530 . . . . . . . . . 10  |-  ( I  e.  NN0  ->  ( j  e.  ( 0..^ ( I  +  1 ) )  <->  ( j  e.  ( 0..^ I )  \/  j  =  I ) ) )
19210, 191syl 16 . . . . . . . . 9  |-  ( ph  ->  ( j  e.  ( 0..^ ( I  + 
1 ) )  <->  ( j  e.  ( 0..^ I )  \/  j  =  I ) ) )
193192adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( j  e.  ( 0..^ ( I  +  1 ) )  <-> 
( j  e.  ( 0..^ I )  \/  j  =  I ) ) )
194 fveq2 5730 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  j  ->  ( W `  k )  =  ( W `  j ) )
195194difeq1d 3466 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  j  ->  (
( W `  k
)  \  _I  )  =  ( ( W `
 j )  \  _I  ) )
196195dmeqd 5074 . . . . . . . . . . . . . . . . 17  |-  ( k  =  j  ->  dom  ( ( W `  k )  \  _I  )  =  dom  ( ( W `  j ) 
\  _I  ) )
197196eleq2d 2505 . . . . . . . . . . . . . . . 16  |-  ( k  =  j  ->  ( A  e.  dom  ( ( W `  k ) 
\  _I  )  <->  A  e.  dom  ( ( W `  j )  \  _I  ) ) )
198197notbid 287 . . . . . . . . . . . . . . 15  |-  ( k  =  j  ->  ( -.  A  e.  dom  ( ( W `  k )  \  _I  ) 
<->  -.  A  e.  dom  ( ( W `  j )  \  _I  ) ) )
199198rspccva 3053 . . . . . . . . . . . . . 14  |-  ( ( A. k  e.  ( 0..^ I )  -.  A  e.  dom  (
( W `  k
)  \  _I  )  /\  j  e.  (
0..^ I ) )  ->  -.  A  e.  dom  ( ( W `  j )  \  _I  ) )
20025, 199sylan 459 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  ( 0..^ I ) )  ->  -.  A  e.  dom  ( ( W `  j )  \  _I  ) )
201200adantlr 697 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  -.  A  e.  dom  ( ( W `  j )  \  _I  ) )
2021ad2antrr 708 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  W  e. Word  T
)
20318ad2antrr 708 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  I  e.  ( 0 ... ( I  +  2 ) ) )
20437ad2antrr 708 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  ( I  + 
2 )  e.  ( 0 ... ( # `  W ) ) )
205124adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  <" r s ">  e. Word  T
)
206 simpr 449 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  j  e.  ( 0..^ I ) )
207202, 203, 204, 205, 206splfv1 11786 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  j
)  =  ( W `
 j ) )
208207difeq1d 3466 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  ( ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r s "> >. ) `  j )  \  _I  )  =  ( ( W `  j )  \  _I  ) )
209208dmeqd 5074 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  dom  ( (
( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  j )  \  _I  )  =  dom  ( ( W `  j ) 
\  _I  ) )
210201, 209neleqtrrd 2534 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  -.  A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  j
)  \  _I  )
)
211210ex 425 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( j  e.  ( 0..^ I )  ->  -.  A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  j
)  \  _I  )
) )
212211adantrr 699 . . . . . . . . 9  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( j  e.  ( 0..^ I )  ->  -.  A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  j
)  \  _I  )
) )
213 simprr3 1008 . . . . . . . . . . 11  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  -.  A  e.  dom  ( r  \  _I  ) )
214 0nn0 10238 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  NN0
215 2pos 10084 . . . . . . . . . . . . . . . . . . . 20  |-  0  <  2
216 elfzo0 11173 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0  e.  ( 0..^ 2 )  <->  ( 0  e. 
NN0  /\  2  e.  NN  /\  0  <  2
) )
217214, 174, 215, 216mpbir3an 1137 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  ( 0..^ 2 )
218217, 178eleqtrri 2511 . . . . . . . . . . . . . . . . . 18  |-  0  e.  ( 0..^ ( # `  <" r s "> ) )
219218a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
0  e.  ( 0..^ ( # `  <" r s "> ) ) )
220121, 154, 155, 124, 219splfv2a 11787 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  ( I  +  0 ) )  =  (
<" r s "> `  0 )
)
22129addid1d 9268 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( I  +  0 )  =  I )
222221adantr 453 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( I  +  0 )  =  I )
223222fveq2d 5734 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  ( I  +  0 ) )  =  ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  I ) )
224 s2fv0 11851 . . . . . . . . . . . . . . . . 17  |-  ( r  e.  T  ->  ( <" r s "> `  0 )  =  r )
225224ad2antrl 710 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( <" r s "> `  0
)  =  r )
226220, 223, 2253eqtr3d 2478 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  I )  =  r )
227226difeq1d 3466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  I
)  \  _I  )  =  ( r  \  _I  ) )
228227dmeqd 5074 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  ->  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  I
)  \  _I  )  =  dom  ( r  \  _I  ) )
229228eleq2d 2505 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  I
)  \  _I  )  <->  A  e.  dom  ( r 
\  _I  ) ) )
230229adantrr 699 . . . . . . . . . . 11  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( A  e. 
dom  ( ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r s "> >. ) `  I )  \  _I  ) 
<->  A  e.  dom  (
r  \  _I  )
) )
231213, 230mtbird 294 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  -.  A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  I
)  \  _I  )
)
232 fveq2 5730 . . . . . . . . . . . . . 14  |-  ( j  =  I  ->  (
( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  j )  =  ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  I ) )
233232difeq1d 3466 . . . . . . . . . . . . 13  |-  ( j  =  I  ->  (
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  j )  \  _I  )  =  ( (
( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  I )  \  _I  ) )
234233dmeqd 5074 . . . . . . . . . . . 12  |-  ( j  =  I  ->  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  j
)  \  _I  )  =  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r s "> >. ) `  I )  \  _I  ) )
235234eleq2d 2505 . . . . . . . . . . 11  |-  ( j  =  I  ->  ( A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  j )  \  _I  ) 
<->  A  e.  dom  (
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  I )  \  _I  ) ) )
236235notbid 287 . . . . . . . . . 10  |-  ( j  =  I  ->  ( -.  A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  j
)  \  _I  )  <->  -.  A  e.  dom  (
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  I )  \  _I  ) ) )
237231, 236syl5ibrcom 215 . . . . . . . . 9  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( j  =  I  ->  -.  A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r s "> >. ) `  j )  \  _I  ) ) )
238212, 237jaod 371 . . . . . . . 8  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( ( j  e.  ( 0..^ I )  \/  j  =  I )  ->  -.  A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  j )  \  _I  ) ) )
239193, 238sylbid 208 . . . . . . 7  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( j  e.  ( 0..^ ( I  +  1 ) )  ->  -.  A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  j
)  \  _I  )
) )
240239ralrimiv 2790 . . . . . 6  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  A. j  e.  ( 0..^ ( I  + 
1 ) )  -.  A  e.  dom  (
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  j )  \  _I  ) )
241171, 188, 2403jca 1135 . . . . 5  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  ( ( I  +  1 )  e.  ( 0..^ L )  /\  A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  (
I  +  1 ) )  \  _I  )  /\  A. j  e.  ( 0..^ ( I  + 
1 ) )  -.  A  e.  dom  (
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  j )  \  _I  ) ) )
242 oveq2 6091 . . . . . . . . 9  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( G  gsumg  w )  =  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )
) )
243242eqeq1d 2446 . . . . . . . 8  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( ( G  gsumg  w )  =  (  _I  |`  D )  <-> 
( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )
)  =  (  _I  |`  D ) ) )
244 fveq2 5730 . . . . . . . . 9  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( # `  w
)  =  ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )
) )
245244eqeq1d 2446 . . . . . . . 8  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( ( # `  w
)  =  L  <->  ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  <" r s "> >. )
)  =  L ) )
246243, 245anbi12d 693 . . . . . . 7  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( ( ( G 
gsumg  w )  =  (  _I  |`  D )  /\  ( # `  w
)  =  L )  <-> 
( ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) )  =  (  _I  |`  D )  /\  ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  <" r s "> >. )
)  =  L ) ) )
247 fveq1 5729 . . . . . . . . . . 11  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( w `  (
I  +  1 ) )  =  ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r s "> >. ) `  ( I  +  1 ) ) )
248247difeq1d 3466 . . . . . . . . . 10  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( ( w `  ( I  +  1
) )  \  _I  )  =  ( (
( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  ( I  +  1 ) )  \  _I  ) )
249248dmeqd 5074 . . . . . . . . 9  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  dom  ( ( w `
 ( I  + 
1 ) )  \  _I  )  =  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  (
I  +  1 ) )  \  _I  )
)
250249eleq2d 2505 . . . . . . . 8  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( A  e.  dom  ( ( w `  ( I  +  1
) )  \  _I  ) 
<->  A  e.  dom  (
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  ( I  +  1 ) )  \  _I  ) ) )
251 fveq1 5729 . . . . . . . . . . . . 13  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( w `  j
)  =  ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r s "> >. ) `  j ) )
252251difeq1d 3466 . . . . . . . . . . . 12  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( ( w `  j )  \  _I  )  =  ( (
( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  j )  \  _I  ) )
253252dmeqd 5074 . . . . . . . . . . 11  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  dom  ( ( w `
 j )  \  _I  )  =  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  j
)  \  _I  )
)
254253eleq2d 2505 . . . . . . . . . 10  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( A  e.  dom  ( ( w `  j )  \  _I  ) 
<->  A  e.  dom  (
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  j )  \  _I  ) ) )
255254notbid 287 . . . . . . . . 9  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( -.  A  e. 
dom  ( ( w `
 j )  \  _I  )  <->  -.  A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  j
)  \  _I  )
) )
256255ralbidv 2727 . . . . . . . 8  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( A. j  e.  ( 0..^ ( I  +  1 ) )  -.  A  e.  dom  ( ( w `  j )  \  _I  ) 
<-> 
A. j  e.  ( 0..^ ( I  + 
1 ) )  -.  A  e.  dom  (
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  j )  \  _I  ) ) )
257250, 2563anbi23d 1258 . . . . . . 7  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( ( ( I  +  1 )  e.  ( 0..^ L )  /\  A  e.  dom  ( ( w `  ( I  +  1
) )  \  _I  )  /\  A. j  e.  ( 0..^ ( I  +  1 ) )  -.  A  e.  dom  ( ( w `  j )  \  _I  ) )  <->  ( (
I  +  1 )  e.  ( 0..^ L )  /\  A  e. 
dom  ( ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r s "> >. ) `  ( I  +  1 ) )  \  _I  )  /\  A. j  e.  ( 0..^ ( I  +  1 ) )  -.  A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  j
)  \  _I  )
) ) )
258246, 257anbi12d 693 . . . . . 6  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( ( ( ( G  gsumg  w )  =  (  _I  |`  D )  /\  ( # `  w
)  =  L )  /\  ( ( I  +  1 )  e.  ( 0..^ L )  /\  A  e.  dom  ( ( w `  ( I  +  1
) )  \  _I  )  /\  A. j  e.  ( 0..^ ( I  +  1 ) )  -.  A  e.  dom  ( ( w `  j )  \  _I  ) ) )  <->  ( (
( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )
)  =  (  _I  |`  D )  /\  ( # `
 ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )
)  =  L )  /\  ( ( I  +  1 )  e.  ( 0..^ L )  /\  A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  (
I  +  1 ) )  \  _I  )  /\  A. j  e.  ( 0..^ ( I  + 
1 ) )  -.  A  e.  dom  (
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  j )  \  _I  ) ) ) ) )
259258rspcev 3054 . . . . 5  |-  ( ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  e. Word  T  /\  ( ( ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )
)  =  (  _I  |`  D )  /\  ( # `
 ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )
)  =  L )  /\  ( ( I  +  1 )  e.  ( 0..^ L )  /\  A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  (
I  +  1 ) )  \  _I  )  /\  A. j  e.  ( 0..^ ( I  + 
1 ) )  -.  A  e.  dom  (
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  j )  \  _I  ) ) ) )  ->  E. w  e. Word  T
( ( ( G 
gsumg  w )  =  (  _I  |`  D )  /\  ( # `  w
)  =  L )  /\  ( ( I  +  1 )  e.  ( 0..^ L )  /\  A  e.  dom  ( ( w `  ( I  +  1
) )  \  _I  )  /\  A. j  e.  ( 0..^ ( I  +  1 ) )  -.  A  e.  dom  ( ( w `  j )  \  _I  ) ) ) )
260127, 170, 241, 259syl12anc 1183 . . . 4  |-  ( (
ph  /\  ( (
r  e.  T  /\  s  e.  T )  /\  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )  ->  E. w  e. Word  T
( ( ( G 
gsumg  w )  =  (  _I  |`  D )  /\  ( # `  w
)  =  L )  /\  ( ( I  +  1 )  e.  ( 0..^ L )  /\  A  e.  dom  ( ( w `  ( I  +  1
) )  \  _I  )  /\  A. j  e.  ( 0..^ ( I  +  1 ) )  -.  A  e.  dom  ( ( w `  j )  \  _I  ) ) ) )
261260expr 600 . . 3  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( ( ( ( W `  I )  o.  ( W `  ( I  +  1
) ) )  =  ( r  o.  s
)  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) )  ->  E. w  e. Word  T ( ( ( G  gsumg  w )  =  (  _I  |`  D )  /\  ( # `  w
)  =  L )  /\  ( ( I  +  1 )  e.  ( 0..^ L )  /\  A  e.  dom  ( ( w `  ( I  +  1
) )  \  _I  )  /\  A. j  e.  ( 0..^ ( I  +  1 ) )  -.  A  e.  dom  ( ( w `  j )  \  _I  ) ) ) ) )
262261rexlimdvva 2839 . 2  |-  ( ph  ->  ( E. r  e.  T  E. s  e.  T  ( ( ( W `  I )  o.  ( W `  ( I  +  1
) ) )  =  ( r  o.  s
)  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) )  ->  E. w  e. Word  T ( ( ( G  gsumg  w )  =  (  _I  |`  D )  /\  ( # `  w
)  =  L )  /\  ( ( I  +  1 )  e.  ( 0..^ L )  /\  A  e.  dom  ( ( w `  ( I  +  1
) )  \  _I  )  /\  A. j  e.  ( 0..^ ( I  +  1 ) )  -.  A  e.  dom  ( ( w `  j )  \  _I  ) ) ) ) )
26320, 21, 88, 90, 24psgnunilem1 27395 . 2  |-  ( ph  ->  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )  \/  E. r  e.  T  E. s  e.  T  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  ( r  o.  s )  /\  A  e.  dom  ( s  \  _I  )  /\  -.  A  e. 
dom  ( r  \  _I  ) ) ) )
264120, 262, 263mpjaod 372 1  |-  ( ph  ->  E. w  e. Word  T
( ( ( G 
gsumg  w )  =  (  _I  |`  D )  /\  ( # `  w
)  =  L )  /\  ( ( I  +  1 )  e.  ( 0..^ L )  /\  A  e.  dom  ( ( w `  ( I  +  1
) )  \  _I  )  /\  A. j  e.  ( 0..^ ( I  +  1 ) )  -.  A  e.  dom  ( ( w `  j )  \  _I  ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708    \ cdif 3319    C_ wss 3322   (/)c0 3630   <.cop 3819   <.cotp 3820   class class class wbr 4214    _I cid 4495   dom cdm 4880   ran crn 4881    |` cres 4882    o. ccom 4884   -->wf 5452   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993    + caddc 8995    < clt 9122    <_ cle 9123    - cmin 9293   -ucneg 9294   NNcn 10002   2c2 10051   NN0cn0 10223   ZZcz 10284   ZZ>=cuz 10490   ...cfz 11045  ..^cfzo 11137   #chash 11620  Word cword 11719   substr csubstr 11722   splice csplice 11723   <"cs2 11807   Basecbs 13471   +g cplusg 13531   0gc0g 13725    gsumg cgsu 13726   Mndcmnd 14686   Grpcgrp 14687   SymGrpcsymg 15094  pmTrspcpmtr 27363
This theorem is referenced by:  psgnunilem3  27398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-xor 1315  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-ot 3826  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-fzo 11138  df-seq 11326  df-hash 11621  df-word 11725  df-concat 11726  df-s1 11727  df-substr 11728  df-splice 11729  df-s2 11814  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-tset 13550  df-0g 13729  df-gsum 13730  df-mnd 14692  df-submnd 14741  df-grp 14814  df-minusg 14815  df-subg 14943  df-symg 15095  df-pmtr 27364
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