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Theorem psgnunilem5 27408
Description: Lemma for psgnuni 27413. It is impossible to shift a transposition off the end because if the active transposition is at the right end, it is the only transposition moving  A in contradiction to this being a representation of the identity. (Contributed by Stefan O'Rear, 25-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  )
)
Assertion
Ref Expression
psgnunilem5  |-  ( ph  ->  ( I  +  1 )  e.  ( 0..^ L ) )
Distinct variable groups:    A, k    k, G    k, I    k, W
Allowed substitution hints:    ph( k)    D( k)    T( k)    L( k)    V( k)

Proof of Theorem psgnunilem5
Dummy variables  j 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noel 3634 . . . 4  |-  -.  A  e.  (/)
2 psgnunilem2.id . . . . . . . 8  |-  ( ph  ->  ( G  gsumg  W )  =  (  _I  |`  D )
)
32difeq1d 3466 . . . . . . 7  |-  ( ph  ->  ( ( G  gsumg  W ) 
\  _I  )  =  ( (  _I  |`  D ) 
\  _I  ) )
43dmeqd 5075 . . . . . 6  |-  ( ph  ->  dom  ( ( G 
gsumg  W )  \  _I  )  =  dom  ( (  _I  |`  D )  \  _I  ) )
5 resss 5173 . . . . . . . . 9  |-  (  _I  |`  D )  C_  _I
6 ssdif0 3688 . . . . . . . . 9  |-  ( (  _I  |`  D )  C_  _I  <->  ( (  _I  |`  D )  \  _I  )  =  (/) )
75, 6mpbi 201 . . . . . . . 8  |-  ( (  _I  |`  D )  \  _I  )  =  (/)
87dmeqi 5074 . . . . . . 7  |-  dom  (
(  _I  |`  D ) 
\  _I  )  =  dom  (/)
9 dm0 5086 . . . . . . 7  |-  dom  (/)  =  (/)
108, 9eqtri 2458 . . . . . 6  |-  dom  (
(  _I  |`  D ) 
\  _I  )  =  (/)
114, 10syl6eq 2486 . . . . 5  |-  ( ph  ->  dom  ( ( G 
gsumg  W )  \  _I  )  =  (/) )
1211eleq2d 2505 . . . 4  |-  ( ph  ->  ( A  e.  dom  ( ( G  gsumg  W ) 
\  _I  )  <->  A  e.  (/) ) )
131, 12mtbiri 296 . . 3  |-  ( ph  ->  -.  A  e.  dom  ( ( G  gsumg  W ) 
\  _I  ) )
14 psgnunilem2.d . . . . . . . . 9  |-  ( ph  ->  D  e.  V )
15 psgnunilem2.g . . . . . . . . . 10  |-  G  =  ( SymGrp `  D )
1615symggrp 15108 . . . . . . . . 9  |-  ( D  e.  V  ->  G  e.  Grp )
17 grpmnd 14822 . . . . . . . . 9  |-  ( G  e.  Grp  ->  G  e.  Mnd )
1814, 16, 173syl 19 . . . . . . . 8  |-  ( ph  ->  G  e.  Mnd )
19 psgnunilem2.t . . . . . . . . . . . 12  |-  T  =  ran  (pmTrsp `  D
)
20 eqid 2438 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
2119, 15, 20symgtrf 27401 . . . . . . . . . . 11  |-  T  C_  ( Base `  G )
22 sswrd 11742 . . . . . . . . . . 11  |-  ( T 
C_  ( Base `  G
)  -> Word  T  C_ Word  ( Base `  G ) )
2321, 22mp1i 12 . . . . . . . . . 10  |-  ( ph  -> Word  T  C_ Word  ( Base `  G
) )
24 psgnunilem2.w . . . . . . . . . 10  |-  ( ph  ->  W  e. Word  T )
2523, 24sseldd 3351 . . . . . . . . 9  |-  ( ph  ->  W  e. Word  ( Base `  G ) )
26 swrdcl 11771 . . . . . . . . 9  |-  ( W  e. Word  ( Base `  G
)  ->  ( W substr  <.
0 ,  I >. )  e. Word  ( Base `  G
) )
2725, 26syl 16 . . . . . . . 8  |-  ( ph  ->  ( W substr  <. 0 ,  I >. )  e. Word  ( Base `  G ) )
2820gsumwcl 14791 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( W substr  <. 0 ,  I >. )  e. Word  ( Base `  G ) )  ->  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  e.  ( Base `  G ) )
2918, 27, 28syl2anc 644 . . . . . . 7  |-  ( ph  ->  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  e.  ( Base `  G
) )
3015, 20elsymgbas2 15101 . . . . . . . 8  |-  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  e.  ( Base `  G
)  ->  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  e.  ( Base `  G
)  <->  ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) : D -1-1-onto-> D ) )
3130ibi 234 . . . . . . 7  |-  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  e.  ( Base `  G
)  ->  ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) : D -1-1-onto-> D
)
3229, 31syl 16 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) : D -1-1-onto-> D )
3332adantr 453 . . . . 5  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) : D -1-1-onto-> D )
34 wrdf 11738 . . . . . . . . . 10  |-  ( W  e. Word  T  ->  W : ( 0..^ (
# `  W )
) --> T )
3524, 34syl 16 . . . . . . . . 9  |-  ( ph  ->  W : ( 0..^ ( # `  W
) ) --> T )
36 psgnunilem2.ix . . . . . . . . . 10  |-  ( ph  ->  I  e.  ( 0..^ L ) )
37 psgnunilem2.l . . . . . . . . . . 11  |-  ( ph  ->  ( # `  W
)  =  L )
3837oveq2d 6100 . . . . . . . . . 10  |-  ( ph  ->  ( 0..^ ( # `  W ) )  =  ( 0..^ L ) )
3936, 38eleqtrrd 2515 . . . . . . . . 9  |-  ( ph  ->  I  e.  ( 0..^ ( # `  W
) ) )
4035, 39ffvelrnd 5874 . . . . . . . 8  |-  ( ph  ->  ( W `  I
)  e.  T )
4121, 40sseldi 3348 . . . . . . 7  |-  ( ph  ->  ( W `  I
)  e.  ( Base `  G ) )
4215, 20elsymgbas2 15101 . . . . . . . 8  |-  ( ( W `  I )  e.  ( Base `  G
)  ->  ( ( W `  I )  e.  ( Base `  G
)  <->  ( W `  I ) : D -1-1-onto-> D
) )
4342ibi 234 . . . . . . 7  |-  ( ( W `  I )  e.  ( Base `  G
)  ->  ( W `  I ) : D -1-1-onto-> D
)
4441, 43syl 16 . . . . . 6  |-  ( ph  ->  ( W `  I
) : D -1-1-onto-> D )
4544adantr 453 . . . . 5  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( W `  I ) : D -1-1-onto-> D )
4615, 20symgsssg 27399 . . . . . . . . . . . 12  |-  ( D  e.  V  ->  { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) }  e.  (SubGrp `  G ) )
47 subgsubm 14967 . . . . . . . . . . . 12  |-  ( { j  e.  ( Base `  G )  |  dom  ( j  \  _I  )  C_  ( _V  \  { A } ) }  e.  (SubGrp `  G
)  ->  { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) }  e.  (SubMnd `  G ) )
4814, 46, 473syl 19 . . . . . . . . . . 11  |-  ( ph  ->  { j  e.  (
Base `  G )  |  dom  ( j  \  _I  )  C_  ( _V 
\  { A }
) }  e.  (SubMnd `  G ) )
4948adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) }  e.  (SubMnd `  G ) )
50 fzossfz 11162 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0..^ L )  C_  (
0 ... L )
5150, 36sseldi 3348 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  I  e.  ( 0 ... L ) )
52 elfzuz3 11061 . . . . . . . . . . . . . . . . . . . 20  |-  ( I  e.  ( 0 ... L )  ->  L  e.  ( ZZ>= `  I )
)
5351, 52syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  L  e.  ( ZZ>= `  I ) )
5437, 53eqeltrd 2512 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( # `  W
)  e.  ( ZZ>= `  I ) )
55 fzoss2 11168 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  W )  e.  ( ZZ>= `  I )  ->  ( 0..^ I ) 
C_  ( 0..^ (
# `  W )
) )
5654, 55syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 0..^ I ) 
C_  ( 0..^ (
# `  W )
) )
5756sselda 3350 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0..^ I ) )  ->  s  e.  ( 0..^ ( # `  W
) ) )
5835ffvelrnda 5873 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  s
)  e.  T )
5921, 58sseldi 3348 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  s
)  e.  ( Base `  G ) )
6057, 59syldan 458 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  s  e.  ( 0..^ I ) )  ->  ( W `  s )  e.  (
Base `  G )
)
61 psgnunilem2.al . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A. k  e.  ( 0..^ I )  -.  A  e.  dom  (
( W `  k
)  \  _I  )
)
62 fveq2 5731 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  s  ->  ( W `  k )  =  ( W `  s ) )
6362difeq1d 3466 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  s  ->  (
( W `  k
)  \  _I  )  =  ( ( W `
 s )  \  _I  ) )
6463dmeqd 5075 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  s  ->  dom  ( ( W `  k )  \  _I  )  =  dom  ( ( W `  s ) 
\  _I  ) )
6564eleq2d 2505 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  s  ->  ( A  e.  dom  ( ( W `  k ) 
\  _I  )  <->  A  e.  dom  ( ( W `  s )  \  _I  ) ) )
6665notbid 287 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  s  ->  ( -.  A  e.  dom  ( ( W `  k )  \  _I  ) 
<->  -.  A  e.  dom  ( ( W `  s )  \  _I  ) ) )
6766cbvralv 2934 . . . . . . . . . . . . . . . . 17  |-  ( A. k  e.  ( 0..^ I )  -.  A  e.  dom  ( ( W `
 k )  \  _I  )  <->  A. s  e.  ( 0..^ I )  -.  A  e.  dom  (
( W `  s
)  \  _I  )
)
6861, 67sylib 190 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. s  e.  ( 0..^ I )  -.  A  e.  dom  (
( W `  s
)  \  _I  )
)
6968r19.21bi 2806 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  s  e.  ( 0..^ I ) )  ->  -.  A  e.  dom  ( ( W `  s )  \  _I  ) )
70 difeq1 3460 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =  ( W `  s )  ->  (
j  \  _I  )  =  ( ( W `
 s )  \  _I  ) )
7170dmeqd 5075 . . . . . . . . . . . . . . . . . 18  |-  ( j  =  ( W `  s )  ->  dom  ( j  \  _I  )  =  dom  ( ( W `  s ) 
\  _I  ) )
7271sseq1d 3377 . . . . . . . . . . . . . . . . 17  |-  ( j  =  ( W `  s )  ->  ( dom  ( j  \  _I  )  C_  ( _V  \  { A } )  <->  dom  ( ( W `  s ) 
\  _I  )  C_  ( _V  \  { A } ) ) )
73 disj2 3677 . . . . . . . . . . . . . . . . . 18  |-  ( ( dom  ( ( W `
 s )  \  _I  )  i^i  { A } )  =  (/)  <->  dom  ( ( W `  s )  \  _I  )  C_  ( _V  \  { A } ) )
74 disjsn 3870 . . . . . . . . . . . . . . . . . 18  |-  ( ( dom  ( ( W `
 s )  \  _I  )  i^i  { A } )  =  (/)  <->  -.  A  e.  dom  ( ( W `  s ) 
\  _I  ) )
7573, 74bitr3i 244 . . . . . . . . . . . . . . . . 17  |-  ( dom  ( ( W `  s )  \  _I  )  C_  ( _V  \  { A } )  <->  -.  A  e.  dom  ( ( W `
 s )  \  _I  ) )
7672, 75syl6bb 254 . . . . . . . . . . . . . . . 16  |-  ( j  =  ( W `  s )  ->  ( dom  ( j  \  _I  )  C_  ( _V  \  { A } )  <->  -.  A  e.  dom  ( ( W `
 s )  \  _I  ) ) )
7776elrab 3094 . . . . . . . . . . . . . . 15  |-  ( ( W `  s )  e.  { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) }  <->  ( ( W `  s )  e.  ( Base `  G
)  /\  -.  A  e.  dom  ( ( W `
 s )  \  _I  ) ) )
7860, 69, 77sylanbrc 647 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  s  e.  ( 0..^ I ) )  ->  ( W `  s )  e.  {
j  e.  ( Base `  G )  |  dom  ( j  \  _I  )  C_  ( _V  \  { A } ) } )
79 eqid 2438 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 0..^ I )  |->  ( W `  s ) )  =  ( s  e.  ( 0..^ I )  |->  ( W `  s ) )
8078, 79fmptd 5896 . . . . . . . . . . . . 13  |-  ( ph  ->  ( s  e.  ( 0..^ I )  |->  ( W `  s ) ) : ( 0..^ I ) --> { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) } )
8137oveq2d 6100 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 0 ... ( # `
 W ) )  =  ( 0 ... L ) )
8251, 81eleqtrrd 2515 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  I  e.  ( 0 ... ( # `  W
) ) )
83 swrd0val 11773 . . . . . . . . . . . . . . . 16  |-  ( ( W  e. Word  T  /\  I  e.  ( 0 ... ( # `  W
) ) )  -> 
( W substr  <. 0 ,  I >. )  =  ( W  |`  ( 0..^ I ) ) )
8424, 82, 83syl2anc 644 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( W substr  <. 0 ,  I >. )  =  ( W  |`  ( 0..^ I ) ) )
8535feqmptd 5782 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  W  =  ( s  e.  ( 0..^ (
# `  W )
)  |->  ( W `  s ) ) )
8685reseq1d 5148 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( W  |`  (
0..^ I ) )  =  ( ( s  e.  ( 0..^ (
# `  W )
)  |->  ( W `  s ) )  |`  ( 0..^ I ) ) )
87 resmpt 5194 . . . . . . . . . . . . . . . 16  |-  ( ( 0..^ I )  C_  ( 0..^ ( # `  W
) )  ->  (
( s  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 s ) )  |`  ( 0..^ I ) )  =  ( s  e.  ( 0..^ I )  |->  ( W `  s ) ) )
8854, 55, 873syl 19 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( s  e.  ( 0..^ ( # `  W ) )  |->  ( W `  s ) )  |`  ( 0..^ I ) )  =  ( s  e.  ( 0..^ I )  |->  ( W `  s ) ) )
8984, 86, 883eqtrd 2474 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( W substr  <. 0 ,  I >. )  =  ( s  e.  ( 0..^ I )  |->  ( W `
 s ) ) )
9089feq1d 5583 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( W substr  <. 0 ,  I >. ) : ( 0..^ I ) --> { j  e.  ( Base `  G )  |  dom  ( j  \  _I  )  C_  ( _V  \  { A } ) }  <-> 
( s  e.  ( 0..^ I )  |->  ( W `  s ) ) : ( 0..^ I ) --> { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) } ) )
9180, 90mpbird 225 . . . . . . . . . . . 12  |-  ( ph  ->  ( W substr  <. 0 ,  I >. ) : ( 0..^ I ) --> { j  e.  ( Base `  G )  |  dom  ( j  \  _I  )  C_  ( _V  \  { A } ) } )
9291adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( W substr  <. 0 ,  I >. ) : ( 0..^ I ) --> { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) } )
93 iswrdi 11736 . . . . . . . . . . 11  |-  ( ( W substr  <. 0 ,  I >. ) : ( 0..^ I ) --> { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) }  ->  ( W substr  <. 0 ,  I >. )  e. Word  { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) } )
9492, 93syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( W substr  <. 0 ,  I >. )  e. Word  { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) } )
95 gsumwsubmcl 14789 . . . . . . . . . 10  |-  ( ( { j  e.  (
Base `  G )  |  dom  ( j  \  _I  )  C_  ( _V 
\  { A }
) }  e.  (SubMnd `  G )  /\  ( W substr  <. 0 ,  I >. )  e. Word  { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) } )  ->  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  e.  { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) } )
9649, 94, 95syl2anc 644 . . . . . . . . 9  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  e. 
{ j  e.  (
Base `  G )  |  dom  ( j  \  _I  )  C_  ( _V 
\  { A }
) } )
97 difeq1 3460 . . . . . . . . . . . . . 14  |-  ( j  =  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  ->  ( j  \  _I  )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )
)
9897dmeqd 5075 . . . . . . . . . . . . 13  |-  ( j  =  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  ->  dom  ( j 
\  _I  )  =  dom  ( ( G 
gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  ) )
9998sseq1d 3377 . . . . . . . . . . . 12  |-  ( j  =  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  ->  ( dom  ( j  \  _I  )  C_  ( _V  \  { A } )  <->  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  C_  ( _V 
\  { A }
) ) )
10099elrab 3094 . . . . . . . . . . 11  |-  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  e. 
{ j  e.  (
Base `  G )  |  dom  ( j  \  _I  )  C_  ( _V 
\  { A }
) }  <->  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  e.  ( Base `  G
)  /\  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  C_  ( _V 
\  { A }
) ) )
101100simprbi 452 . . . . . . . . . 10  |-  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  e. 
{ j  e.  (
Base `  G )  |  dom  ( j  \  _I  )  C_  ( _V 
\  { A }
) }  ->  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  C_  ( _V  \  { A } ) )
102 disj2 3677 . . . . . . . . . . 11  |-  ( ( dom  ( ( G 
gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  i^i  { A } )  =  (/)  <->  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  C_  ( _V  \  { A } ) )
103 disjsn 3870 . . . . . . . . . . 11  |-  ( ( dom  ( ( G 
gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  i^i  { A } )  =  (/)  <->  -.  A  e.  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  ) )
104102, 103bitr3i 244 . . . . . . . . . 10  |-  ( dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  C_  ( _V  \  { A } )  <->  -.  A  e.  dom  ( ( G 
gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  ) )
105101, 104sylib 190 . . . . . . . . 9  |-  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  e. 
{ j  e.  (
Base `  G )  |  dom  ( j  \  _I  )  C_  ( _V 
\  { A }
) }  ->  -.  A  e.  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  ) )
10696, 105syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  -.  A  e.  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  ) )
107 psgnunilem2.a . . . . . . . . 9  |-  ( ph  ->  A  e.  dom  (
( W `  I
)  \  _I  )
)
108107adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  A  e.  dom  ( ( W `
 I )  \  _I  ) )
109106, 108jca 520 . . . . . . 7  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( -.  A  e.  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  /\  A  e.  dom  ( ( W `  I )  \  _I  ) ) )
110109olcd 384 . . . . . 6  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  (
( A  e.  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  /\  -.  A  e.  dom  ( ( W `  I )  \  _I  ) )  \/  ( -.  A  e.  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  /\  A  e.  dom  ( ( W `  I )  \  _I  ) ) ) )
111 excxor 1319 . . . . . 6  |-  ( ( A  e.  dom  (
( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  \/_  A  e. 
dom  ( ( W `
 I )  \  _I  ) )  <->  ( ( A  e.  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  /\  -.  A  e.  dom  ( ( W `
 I )  \  _I  ) )  \/  ( -.  A  e.  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  /\  A  e.  dom  ( ( W `  I )  \  _I  ) ) ) )
112110, 111sylibr 205 . . . . 5  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( A  e.  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  \/_  A  e. 
dom  ( ( W `
 I )  \  _I  ) ) )
113 f1omvdco3 27383 . . . . 5  |-  ( ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) : D -1-1-onto-> D  /\  ( W `
 I ) : D -1-1-onto-> D  /\  ( A  e.  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  \/_  A  e. 
dom  ( ( W `
 I )  \  _I  ) ) )  ->  A  e.  dom  ( ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) )  \  _I  ) )
11433, 45, 112, 113syl3anc 1185 . . . 4  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  A  e.  dom  ( ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) )  \  _I  ) )
11524adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  W  e. Word  T )
116 elfzo0 11176 . . . . . . . . . . . . . . 15  |-  ( I  e.  ( 0..^ L )  <->  ( I  e. 
NN0  /\  L  e.  NN  /\  I  <  L
) )
117116simp2bi 974 . . . . . . . . . . . . . 14  |-  ( I  e.  ( 0..^ L )  ->  L  e.  NN )
11836, 117syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  L  e.  NN )
11937, 118eqeltrd 2512 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  W
)  e.  NN )
120 wrdfin 11739 . . . . . . . . . . . . 13  |-  ( W  e. Word  T  ->  W  e.  Fin )
121 hashnncl 11650 . . . . . . . . . . . . 13  |-  ( W  e.  Fin  ->  (
( # `  W )  e.  NN  <->  W  =/=  (/) ) )
12224, 120, 1213syl 19 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  W
)  e.  NN  <->  W  =/=  (/) ) )
123119, 122mpbid 203 . . . . . . . . . . 11  |-  ( ph  ->  W  =/=  (/) )
124123adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  W  =/=  (/) )
125 wrdeqcats1 11793 . . . . . . . . . 10  |-  ( ( W  e. Word  T  /\  W  =/=  (/) )  ->  W  =  ( ( W substr  <. 0 ,  ( (
# `  W )  -  1 ) >.
) concat  <" ( W `
 ( ( # `  W )  -  1 ) ) "> ) )
126115, 124, 125syl2anc 644 . . . . . . . . 9  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  W  =  ( ( W substr  <. 0 ,  ( (
# `  W )  -  1 ) >.
) concat  <" ( W `
 ( ( # `  W )  -  1 ) ) "> ) )
12737oveq1d 6099 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  W
)  -  1 )  =  ( L  - 
1 ) )
128127adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  (
( # `  W )  -  1 )  =  ( L  -  1 ) )
129118nncnd 10021 . . . . . . . . . . . . 13  |-  ( ph  ->  L  e.  CC )
130 ax-1cn 9053 . . . . . . . . . . . . . 14  |-  1  e.  CC
131130a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  1  e.  CC )
132 elfzoelz 11145 . . . . . . . . . . . . . . 15  |-  ( I  e.  ( 0..^ L )  ->  I  e.  ZZ )
13336, 132syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  I  e.  ZZ )
134133zcnd 10381 . . . . . . . . . . . . 13  |-  ( ph  ->  I  e.  CC )
135129, 131, 134subadd2d 9435 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( L  - 
1 )  =  I  <-> 
( I  +  1 )  =  L ) )
136135biimpar 473 . . . . . . . . . . 11  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( L  -  1 )  =  I )
137128, 136eqtrd 2470 . . . . . . . . . 10  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  (
( # `  W )  -  1 )  =  I )
138 opeq2 3987 . . . . . . . . . . . 12  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  <. 0 ,  ( ( # `  W )  -  1 ) >.  =  <. 0 ,  I >. )
139138oveq2d 6100 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  ( W substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  =  ( W substr  <. 0 ,  I >. ) )
140 fveq2 5731 . . . . . . . . . . . 12  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  ( W `  ( ( # `
 W )  - 
1 ) )  =  ( W `  I
) )
141140s1eqd 11759 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  <" ( W `  ( ( # `
 W )  - 
1 ) ) ">  =  <" ( W `  I ) "> )
142139, 141oveq12d 6102 . . . . . . . . . 10  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  (
( W substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. ) concat  <" ( W `  ( ( # `
 W )  - 
1 ) ) "> )  =  ( ( W substr  <. 0 ,  I >. ) concat  <" ( W `  I ) "> ) )
143137, 142syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  (
( W substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. ) concat  <" ( W `  ( ( # `
 W )  - 
1 ) ) "> )  =  ( ( W substr  <. 0 ,  I >. ) concat  <" ( W `  I ) "> ) )
144126, 143eqtrd 2470 . . . . . . . 8  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  W  =  ( ( W substr  <. 0 ,  I >. ) concat  <" ( W `  I ) "> ) )
145144oveq2d 6100 . . . . . . 7  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( G  gsumg  W )  =  ( G  gsumg  ( ( W substr  <. 0 ,  I >. ) concat  <" ( W `  I ) "> ) ) )
14641s1cld 11761 . . . . . . . . 9  |-  ( ph  ->  <" ( W `
 I ) ">  e. Word  ( Base `  G ) )
147 eqid 2438 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
14820, 147gsumccat 14792 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  ( W substr  <. 0 ,  I >. )  e. Word  ( Base `  G )  /\  <" ( W `  I ) ">  e. Word  ( Base `  G
) )  ->  ( G  gsumg  ( ( W substr  <. 0 ,  I >. ) concat  <" ( W `  I ) "> ) )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) ( +g  `  G
) ( G  gsumg  <" ( W `  I ) "> ) ) )
14918, 27, 146, 148syl3anc 1185 . . . . . . . 8  |-  ( ph  ->  ( G  gsumg  ( ( W substr  <. 0 ,  I >. ) concat  <" ( W `  I ) "> ) )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) ( +g  `  G
) ( G  gsumg  <" ( W `  I ) "> ) ) )
150149adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( G  gsumg  ( ( W substr  <. 0 ,  I >. ) concat  <" ( W `  I ) "> ) )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) ( +g  `  G
) ( G  gsumg  <" ( W `  I ) "> ) ) )
15120gsumws1 14790 . . . . . . . . . . 11  |-  ( ( W `  I )  e.  ( Base `  G
)  ->  ( G  gsumg  <" ( W `  I ) "> )  =  ( W `  I ) )
15241, 151syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( G  gsumg 
<" ( W `  I ) "> )  =  ( W `  I ) )
153152oveq2d 6100 . . . . . . . . 9  |-  ( ph  ->  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) ( +g  `  G
) ( G  gsumg  <" ( W `  I ) "> ) )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) ( +g  `  G
) ( W `  I ) ) )
15415, 20, 147symgov 15105 . . . . . . . . . 10  |-  ( ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  e.  ( Base `  G
)  /\  ( W `  I )  e.  (
Base `  G )
)  ->  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) ( +g  `  G ) ( W `  I
) )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) ) )
15529, 41, 154syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) ( +g  `  G
) ( W `  I ) )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `
 I ) ) )
156153, 155eqtrd 2470 . . . . . . . 8  |-  ( ph  ->  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) ( +g  `  G
) ( G  gsumg  <" ( W `  I ) "> ) )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `
 I ) ) )
157156adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  (
( G  gsumg  ( W substr  <. 0 ,  I >. ) ) ( +g  `  G ) ( G  gsumg 
<" ( W `  I ) "> ) )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) ) )
158145, 150, 1573eqtrd 2474 . . . . . 6  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( G  gsumg  W )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) ) )
159158difeq1d 3466 . . . . 5  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  (
( G  gsumg  W )  \  _I  )  =  ( (
( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) )  \  _I  ) )
160159dmeqd 5075 . . . 4  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  dom  ( ( G  gsumg  W ) 
\  _I  )  =  dom  ( ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) )  \  _I  ) )
161114, 160eleqtrrd 2515 . . 3  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  A  e.  dom  ( ( G 
gsumg  W )  \  _I  ) )
16213, 161mtand 642 . 2  |-  ( ph  ->  -.  ( I  + 
1 )  =  L )
163 fzostep1 11202 . . . 4  |-  ( I  e.  ( 0..^ L )  ->  ( (
I  +  1 )  e.  ( 0..^ L )  \/  ( I  +  1 )  =  L ) )
16436, 163syl 16 . . 3  |-  ( ph  ->  ( ( I  + 
1 )  e.  ( 0..^ L )  \/  ( I  +  1 )  =  L ) )
165164ord 368 . 2  |-  ( ph  ->  ( -.  ( I  +  1 )  e.  ( 0..^ L )  ->  ( I  + 
1 )  =  L ) )
166162, 165mt3d 120 1  |-  ( ph  ->  ( I  +  1 )  e.  ( 0..^ L ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    \/_ wxo 1314    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   {crab 2711   _Vcvv 2958    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630   {csn 3816   <.cop 3819   class class class wbr 4215    e. cmpt 4269    _I cid 4496   dom cdm 4881   ran crn 4882    |` cres 4883    o. ccom 4885   -->wf 5453   -1-1-onto->wf1o 5456   ` cfv 5457  (class class class)co 6084   Fincfn 7112   CCcc 8993   0cc0 8995   1c1 8996    + caddc 8998    < clt 9125    - cmin 9296   NNcn 10005   NN0cn0 10226   ZZcz 10287   ZZ>=cuz 10493   ...cfz 11048  ..^cfzo 11140   #chash 11623  Word cword 11722   concat cconcat 11723   <"cs1 11724   substr csubstr 11725   Basecbs 13474   +g cplusg 13534    gsumg cgsu 13729   Mndcmnd 14689   Grpcgrp 14690  SubMndcsubmnd 14742  SubGrpcsubg 14943   SymGrpcsymg 15097  pmTrspcpmtr 27375
This theorem is referenced by:  psgnunilem2  27409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
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-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-fzo 11141  df-seq 11329  df-hash 11624  df-word 11728  df-concat 11729  df-s1 11730  df-substr 11731  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-tset 13553  df-0g 13732  df-gsum 13733  df-mnd 14695  df-submnd 14744  df-grp 14817  df-minusg 14818  df-subg 14946  df-symg 15098  df-pmtr 27376
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