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Theorem psgnunilem5 27079
Description: Lemma for psgnuni 27084. 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 3568 . . . 4  |-  -.  A  e.  (/)
2 psgnunilem2.id . . . . . . . 8  |-  ( ph  ->  ( G  gsumg  W )  =  (  _I  |`  D )
)
32difeq1d 3400 . . . . . . 7  |-  ( ph  ->  ( ( G  gsumg  W ) 
\  _I  )  =  ( (  _I  |`  D ) 
\  _I  ) )
43dmeqd 5005 . . . . . 6  |-  ( ph  ->  dom  ( ( G 
gsumg  W )  \  _I  )  =  dom  ( (  _I  |`  D )  \  _I  ) )
5 resss 5103 . . . . . . . . 9  |-  (  _I  |`  D )  C_  _I
6 ssdif0 3622 . . . . . . . . 9  |-  ( (  _I  |`  D )  C_  _I  <->  ( (  _I  |`  D )  \  _I  )  =  (/) )
75, 6mpbi 200 . . . . . . . 8  |-  ( (  _I  |`  D )  \  _I  )  =  (/)
87dmeqi 5004 . . . . . . 7  |-  dom  (
(  _I  |`  D ) 
\  _I  )  =  dom  (/)
9 dm0 5016 . . . . . . 7  |-  dom  (/)  =  (/)
108, 9eqtri 2400 . . . . . 6  |-  dom  (
(  _I  |`  D ) 
\  _I  )  =  (/)
114, 10syl6eq 2428 . . . . 5  |-  ( ph  ->  dom  ( ( G 
gsumg  W )  \  _I  )  =  (/) )
1211eleq2d 2447 . . . 4  |-  ( ph  ->  ( A  e.  dom  ( ( G  gsumg  W ) 
\  _I  )  <->  A  e.  (/) ) )
131, 12mtbiri 295 . . 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 15023 . . . . . . . . 9  |-  ( D  e.  V  ->  G  e.  Grp )
17 grpmnd 14737 . . . . . . . . 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 2380 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
2119, 15, 20symgtrf 27072 . . . . . . . . . . 11  |-  T  C_  ( Base `  G )
22 sswrd 11657 . . . . . . . . . . 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 3285 . . . . . . . . 9  |-  ( ph  ->  W  e. Word  ( Base `  G ) )
26 swrdcl 11686 . . . . . . . . 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 14706 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( W substr  <. 0 ,  I >. )  e. Word  ( Base `  G ) )  ->  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  e.  ( Base `  G ) )
2918, 27, 28syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  e.  ( Base `  G
) )
3015, 20elsymgbas2 15016 . . . . . . . 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 233 . . . . . . 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 452 . . . . 5  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) : D -1-1-onto-> D )
34 wrdf 11653 . . . . . . . . . 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 6029 . . . . . . . . . 10  |-  ( ph  ->  ( 0..^ ( # `  W ) )  =  ( 0..^ L ) )
3936, 38eleqtrrd 2457 . . . . . . . . 9  |-  ( ph  ->  I  e.  ( 0..^ ( # `  W
) ) )
4035, 39ffvelrnd 5803 . . . . . . . 8  |-  ( ph  ->  ( W `  I
)  e.  T )
4121, 40sseldi 3282 . . . . . . 7  |-  ( ph  ->  ( W `  I
)  e.  ( Base `  G ) )
4215, 20elsymgbas2 15016 . . . . . . . 8  |-  ( ( W `  I )  e.  ( Base `  G
)  ->  ( ( W `  I )  e.  ( Base `  G
)  <->  ( W `  I ) : D -1-1-onto-> D
) )
4342ibi 233 . . . . . . 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 452 . . . . 5  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( W `  I ) : D -1-1-onto-> D )
4615, 20symgsssg 27070 . . . . . . . . . . . 12  |-  ( D  e.  V  ->  { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) }  e.  (SubGrp `  G ) )
47 subgsubm 14882 . . . . . . . . . . . 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 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) }  e.  (SubMnd `  G ) )
50 fzossfz 11080 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0..^ L )  C_  (
0 ... L )
5150, 36sseldi 3282 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  I  e.  ( 0 ... L ) )
52 elfzuz3 10981 . . . . . . . . . . . . . . . . . . . 20  |-  ( I  e.  ( 0 ... L )  ->  L  e.  ( ZZ>= `  I )
)
5351, 52syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  L  e.  ( ZZ>= `  I ) )
5437, 53eqeltrd 2454 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( # `  W
)  e.  ( ZZ>= `  I ) )
55 fzoss2 11086 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  W )  e.  ( ZZ>= `  I )  ->  ( 0..^ I ) 
C_  ( 0..^ (
# `  W )
) )
5654, 55syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 0..^ I ) 
C_  ( 0..^ (
# `  W )
) )
5756sselda 3284 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0..^ I ) )  ->  s  e.  ( 0..^ ( # `  W
) ) )
5835ffvelrnda 5802 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  s
)  e.  T )
5921, 58sseldi 3282 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  s
)  e.  ( Base `  G ) )
6057, 59syldan 457 . . . . . . . . . . . . . . 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 5661 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  s  ->  ( W `  k )  =  ( W `  s ) )
6362difeq1d 3400 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  s  ->  (
( W `  k
)  \  _I  )  =  ( ( W `
 s )  \  _I  ) )
6463dmeqd 5005 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  s  ->  dom  ( ( W `  k )  \  _I  )  =  dom  ( ( W `  s ) 
\  _I  ) )
6564eleq2d 2447 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  s  ->  ( A  e.  dom  ( ( W `  k ) 
\  _I  )  <->  A  e.  dom  ( ( W `  s )  \  _I  ) ) )
6665notbid 286 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  s  ->  ( -.  A  e.  dom  ( ( W `  k )  \  _I  ) 
<->  -.  A  e.  dom  ( ( W `  s )  \  _I  ) ) )
6766cbvralv 2868 . . . . . . . . . . . . . . . . 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 189 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. s  e.  ( 0..^ I )  -.  A  e.  dom  (
( W `  s
)  \  _I  )
)
6968r19.21bi 2740 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  s  e.  ( 0..^ I ) )  ->  -.  A  e.  dom  ( ( W `  s )  \  _I  ) )
70 difeq1 3394 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =  ( W `  s )  ->  (
j  \  _I  )  =  ( ( W `
 s )  \  _I  ) )
7170dmeqd 5005 . . . . . . . . . . . . . . . . . 18  |-  ( j  =  ( W `  s )  ->  dom  ( j  \  _I  )  =  dom  ( ( W `  s ) 
\  _I  ) )
7271sseq1d 3311 . . . . . . . . . . . . . . . . 17  |-  ( j  =  ( W `  s )  ->  ( dom  ( j  \  _I  )  C_  ( _V  \  { A } )  <->  dom  ( ( W `  s ) 
\  _I  )  C_  ( _V  \  { A } ) ) )
73 disj2 3611 . . . . . . . . . . . . . . . . . 18  |-  ( ( dom  ( ( W `
 s )  \  _I  )  i^i  { A } )  =  (/)  <->  dom  ( ( W `  s )  \  _I  )  C_  ( _V  \  { A } ) )
74 disjsn 3804 . . . . . . . . . . . . . . . . . 18  |-  ( ( dom  ( ( W `
 s )  \  _I  )  i^i  { A } )  =  (/)  <->  -.  A  e.  dom  ( ( W `  s ) 
\  _I  ) )
7573, 74bitr3i 243 . . . . . . . . . . . . . . . . 17  |-  ( dom  ( ( W `  s )  \  _I  )  C_  ( _V  \  { A } )  <->  -.  A  e.  dom  ( ( W `
 s )  \  _I  ) )
7672, 75syl6bb 253 . . . . . . . . . . . . . . . 16  |-  ( j  =  ( W `  s )  ->  ( dom  ( j  \  _I  )  C_  ( _V  \  { A } )  <->  -.  A  e.  dom  ( ( W `
 s )  \  _I  ) ) )
7776elrab 3028 . . . . . . . . . . . . . . 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 646 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  s  e.  ( 0..^ I ) )  ->  ( W `  s )  e.  {
j  e.  ( Base `  G )  |  dom  ( j  \  _I  )  C_  ( _V  \  { A } ) } )
79 eqid 2380 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 0..^ I )  |->  ( W `  s ) )  =  ( s  e.  ( 0..^ I )  |->  ( W `  s ) )
8078, 79fmptd 5825 . . . . . . . . . . . . 13  |-  ( ph  ->  ( s  e.  ( 0..^ I )  |->  ( W `  s ) ) : ( 0..^ I ) --> { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) } )
8137oveq2d 6029 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 0 ... ( # `
 W ) )  =  ( 0 ... L ) )
8251, 81eleqtrrd 2457 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  I  e.  ( 0 ... ( # `  W
) ) )
83 swrd0val 11688 . . . . . . . . . . . . . . . 16  |-  ( ( W  e. Word  T  /\  I  e.  ( 0 ... ( # `  W
) ) )  -> 
( W substr  <. 0 ,  I >. )  =  ( W  |`  ( 0..^ I ) ) )
8424, 82, 83syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( W substr  <. 0 ,  I >. )  =  ( W  |`  ( 0..^ I ) ) )
8535feqmptd 5711 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  W  =  ( s  e.  ( 0..^ (
# `  W )
)  |->  ( W `  s ) ) )
8685reseq1d 5078 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( W  |`  (
0..^ I ) )  =  ( ( s  e.  ( 0..^ (
# `  W )
)  |->  ( W `  s ) )  |`  ( 0..^ I ) ) )
87 resmpt 5124 . . . . . . . . . . . . . . . 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 2416 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( W substr  <. 0 ,  I >. )  =  ( s  e.  ( 0..^ I )  |->  ( W `
 s ) ) )
9089feq1d 5513 . . . . . . . . . . . . 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 224 . . . . . . . . . . . 12  |-  ( ph  ->  ( W substr  <. 0 ,  I >. ) : ( 0..^ I ) --> { j  e.  ( Base `  G )  |  dom  ( j  \  _I  )  C_  ( _V  \  { A } ) } )
9291adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( W substr  <. 0 ,  I >. ) : ( 0..^ I ) --> { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) } )
93 iswrdi 11651 . . . . . . . . . . 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 14704 . . . . . . . . . 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 643 . . . . . . . . 9  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  e. 
{ j  e.  (
Base `  G )  |  dom  ( j  \  _I  )  C_  ( _V 
\  { A }
) } )
97 difeq1 3394 . . . . . . . . . . . . . 14  |-  ( j  =  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  ->  ( j  \  _I  )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )
)
9897dmeqd 5005 . . . . . . . . . . . . 13  |-  ( j  =  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  ->  dom  ( j 
\  _I  )  =  dom  ( ( G 
gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  ) )
9998sseq1d 3311 . . . . . . . . . . . 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 3028 . . . . . . . . . . 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 451 . . . . . . . . . 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 3611 . . . . . . . . . . 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 3804 . . . . . . . . . . 11  |-  ( ( dom  ( ( G 
gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  i^i  { A } )  =  (/)  <->  -.  A  e.  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  ) )
104102, 103bitr3i 243 . . . . . . . . . 10  |-  ( dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  C_  ( _V  \  { A } )  <->  -.  A  e.  dom  ( ( G 
gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  ) )
105101, 104sylib 189 . . . . . . . . 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 452 . . . . . . . 8  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  A  e.  dom  ( ( W `
 I )  \  _I  ) )
109106, 108jca 519 . . . . . . 7  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( -.  A  e.  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  /\  A  e.  dom  ( ( W `  I )  \  _I  ) ) )
110109olcd 383 . . . . . 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 1315 . . . . . 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 204 . . . . 5  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( A  e.  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  \/_  A  e. 
dom  ( ( W `
 I )  \  _I  ) ) )
113 f1omvdco3 27054 . . . . 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 1184 . . . 4  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  A  e.  dom  ( ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) )  \  _I  ) )
11524adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  W  e. Word  T )
116 elfzo0 11094 . . . . . . . . . . . . . . 15  |-  ( I  e.  ( 0..^ L )  <->  ( I  e. 
NN0  /\  L  e.  NN  /\  I  <  L
) )
117116simp2bi 973 . . . . . . . . . . . . . 14  |-  ( I  e.  ( 0..^ L )  ->  L  e.  NN )
11836, 117syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  L  e.  NN )
11937, 118eqeltrd 2454 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  W
)  e.  NN )
120 wrdfin 11654 . . . . . . . . . . . . 13  |-  ( W  e. Word  T  ->  W  e.  Fin )
121 hashnncl 11565 . . . . . . . . . . . . 13  |-  ( W  e.  Fin  ->  (
( # `  W )  e.  NN  <->  W  =/=  (/) ) )
12224, 120, 1213syl 19 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  W
)  e.  NN  <->  W  =/=  (/) ) )
123119, 122mpbid 202 . . . . . . . . . . 11  |-  ( ph  ->  W  =/=  (/) )
124123adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  W  =/=  (/) )
125 wrdeqcats1 11708 . . . . . . . . . 10  |-  ( ( W  e. Word  T  /\  W  =/=  (/) )  ->  W  =  ( ( W substr  <. 0 ,  ( (
# `  W )  -  1 ) >.
) concat  <" ( W `
 ( ( # `  W )  -  1 ) ) "> ) )
126115, 124, 125syl2anc 643 . . . . . . . . 9  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  W  =  ( ( W substr  <. 0 ,  ( (
# `  W )  -  1 ) >.
) concat  <" ( W `
 ( ( # `  W )  -  1 ) ) "> ) )
12737oveq1d 6028 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  W
)  -  1 )  =  ( L  - 
1 ) )
128127adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  (
( # `  W )  -  1 )  =  ( L  -  1 ) )
129118nncnd 9941 . . . . . . . . . . . . 13  |-  ( ph  ->  L  e.  CC )
130 ax-1cn 8974 . . . . . . . . . . . . . 14  |-  1  e.  CC
131130a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  1  e.  CC )
132 elfzoelz 11063 . . . . . . . . . . . . . . 15  |-  ( I  e.  ( 0..^ L )  ->  I  e.  ZZ )
13336, 132syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  I  e.  ZZ )
134133zcnd 10301 . . . . . . . . . . . . 13  |-  ( ph  ->  I  e.  CC )
135129, 131, 134subadd2d 9355 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( L  - 
1 )  =  I  <-> 
( I  +  1 )  =  L ) )
136135biimpar 472 . . . . . . . . . . 11  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( L  -  1 )  =  I )
137128, 136eqtrd 2412 . . . . . . . . . 10  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  (
( # `  W )  -  1 )  =  I )
138 opeq2 3920 . . . . . . . . . . . 12  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  <. 0 ,  ( ( # `  W )  -  1 ) >.  =  <. 0 ,  I >. )
139138oveq2d 6029 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  ( W substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  =  ( W substr  <. 0 ,  I >. ) )
140 fveq2 5661 . . . . . . . . . . . 12  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  ( W `  ( ( # `
 W )  - 
1 ) )  =  ( W `  I
) )
141140s1eqd 11674 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  <" ( W `  ( ( # `
 W )  - 
1 ) ) ">  =  <" ( W `  I ) "> )
142139, 141oveq12d 6031 . . . . . . . . . 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 2412 . . . . . . . 8  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  W  =  ( ( W substr  <. 0 ,  I >. ) concat  <" ( W `  I ) "> ) )
145144oveq2d 6029 . . . . . . 7  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( G  gsumg  W )  =  ( G  gsumg  ( ( W substr  <. 0 ,  I >. ) concat  <" ( W `  I ) "> ) ) )
14641s1cld 11676 . . . . . . . . 9  |-  ( ph  ->  <" ( W `
 I ) ">  e. Word  ( Base `  G ) )
147 eqid 2380 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
14820, 147gsumccat 14707 . . . . . . . . 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 1184 . . . . . . . 8  |-  ( ph  ->  ( G  gsumg  ( ( W substr  <. 0 ,  I >. ) concat  <" ( W `  I ) "> ) )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) ( +g  `  G
) ( G  gsumg  <" ( W `  I ) "> ) ) )
150149adantr 452 . . . . . . 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 14705 . . . . . . . . . . 11  |-  ( ( W `  I )  e.  ( Base `  G
)  ->  ( G  gsumg  <" ( W `  I ) "> )  =  ( W `  I ) )
15241, 151syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( G  gsumg 
<" ( W `  I ) "> )  =  ( W `  I ) )
153152oveq2d 6029 . . . . . . . . 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 15020 . . . . . . . . . 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 643 . . . . . . . . 9  |-  ( ph  ->  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) ( +g  `  G
) ( W `  I ) )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `
 I ) ) )
156153, 155eqtrd 2412 . . . . . . . 8  |-  ( ph  ->  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) ( +g  `  G
) ( G  gsumg  <" ( W `  I ) "> ) )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `
 I ) ) )
157156adantr 452 . . . . . . 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 2416 . . . . . 6  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( G  gsumg  W )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) ) )
159158difeq1d 3400 . . . . 5  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  (
( G  gsumg  W )  \  _I  )  =  ( (
( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) )  \  _I  ) )
160159dmeqd 5005 . . . 4  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  dom  ( ( G  gsumg  W ) 
\  _I  )  =  dom  ( ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) )  \  _I  ) )
161114, 160eleqtrrd 2457 . . 3  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  A  e.  dom  ( ( G 
gsumg  W )  \  _I  ) )
16213, 161mtand 641 . 2  |-  ( ph  ->  -.  ( I  + 
1 )  =  L )
163 fzostep1 11117 . . . 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 367 . 2  |-  ( ph  ->  ( -.  ( I  +  1 )  e.  ( 0..^ L )  ->  ( I  + 
1 )  =  L ) )
166162, 165mt3d 119 1  |-  ( ph  ->  ( I  +  1 )  e.  ( 0..^ L ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    \/_ wxo 1310    = wceq 1649    e. wcel 1717    =/= wne 2543   A.wral 2642   {crab 2646   _Vcvv 2892    \ cdif 3253    i^i cin 3255    C_ wss 3256   (/)c0 3564   {csn 3750   <.cop 3753   class class class wbr 4146    e. cmpt 4200    _I cid 4427   dom cdm 4811   ran crn 4812    |` cres 4813    o. ccom 4815   -->wf 5383   -1-1-onto->wf1o 5386   ` cfv 5387  (class class class)co 6013   Fincfn 7038   CCcc 8914   0cc0 8916   1c1 8917    + caddc 8919    < clt 9046    - cmin 9216   NNcn 9925   NN0cn0 10146   ZZcz 10207   ZZ>=cuz 10413   ...cfz 10968  ..^cfzo 11058   #chash 11538  Word cword 11637   concat cconcat 11638   <"cs1 11639   substr csubstr 11640   Basecbs 13389   +g cplusg 13449    gsumg cgsu 13644   Mndcmnd 14604   Grpcgrp 14605  SubMndcsubmnd 14657  SubGrpcsubg 14858   SymGrpcsymg 15012  pmTrspcpmtr 27046
This theorem is referenced by:  psgnunilem2  27080
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-xor 1311  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-n0 10147  df-z 10208  df-uz 10414  df-fz 10969  df-fzo 11059  df-seq 11244  df-hash 11539  df-word 11643  df-concat 11644  df-s1 11645  df-substr 11646  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-tset 13468  df-0g 13647  df-gsum 13648  df-mnd 14610  df-submnd 14659  df-grp 14732  df-minusg 14733  df-subg 14861  df-symg 15013  df-pmtr 27047
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