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Theorem psgnunilem5 27285
Description: Lemma for psgnuni 27290. 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 3592 . . . 4  |-  -.  A  e.  (/)
2 psgnunilem2.id . . . . . . . 8  |-  ( ph  ->  ( G  gsumg  W )  =  (  _I  |`  D )
)
32difeq1d 3424 . . . . . . 7  |-  ( ph  ->  ( ( G  gsumg  W ) 
\  _I  )  =  ( (  _I  |`  D ) 
\  _I  ) )
43dmeqd 5031 . . . . . 6  |-  ( ph  ->  dom  ( ( G 
gsumg  W )  \  _I  )  =  dom  ( (  _I  |`  D )  \  _I  ) )
5 resss 5129 . . . . . . . . 9  |-  (  _I  |`  D )  C_  _I
6 ssdif0 3646 . . . . . . . . 9  |-  ( (  _I  |`  D )  C_  _I  <->  ( (  _I  |`  D )  \  _I  )  =  (/) )
75, 6mpbi 200 . . . . . . . 8  |-  ( (  _I  |`  D )  \  _I  )  =  (/)
87dmeqi 5030 . . . . . . 7  |-  dom  (
(  _I  |`  D ) 
\  _I  )  =  dom  (/)
9 dm0 5042 . . . . . . 7  |-  dom  (/)  =  (/)
108, 9eqtri 2424 . . . . . 6  |-  dom  (
(  _I  |`  D ) 
\  _I  )  =  (/)
114, 10syl6eq 2452 . . . . 5  |-  ( ph  ->  dom  ( ( G 
gsumg  W )  \  _I  )  =  (/) )
1211eleq2d 2471 . . . 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 15058 . . . . . . . . 9  |-  ( D  e.  V  ->  G  e.  Grp )
17 grpmnd 14772 . . . . . . . . 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 2404 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
2119, 15, 20symgtrf 27278 . . . . . . . . . . 11  |-  T  C_  ( Base `  G )
22 sswrd 11692 . . . . . . . . . . 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 3309 . . . . . . . . 9  |-  ( ph  ->  W  e. Word  ( Base `  G ) )
26 swrdcl 11721 . . . . . . . . 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 14741 . . . . . . . 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 15051 . . . . . . . 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 11688 . . . . . . . . . 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 6056 . . . . . . . . . 10  |-  ( ph  ->  ( 0..^ ( # `  W ) )  =  ( 0..^ L ) )
3936, 38eleqtrrd 2481 . . . . . . . . 9  |-  ( ph  ->  I  e.  ( 0..^ ( # `  W
) ) )
4035, 39ffvelrnd 5830 . . . . . . . 8  |-  ( ph  ->  ( W `  I
)  e.  T )
4121, 40sseldi 3306 . . . . . . 7  |-  ( ph  ->  ( W `  I
)  e.  ( Base `  G ) )
4215, 20elsymgbas2 15051 . . . . . . . 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 27276 . . . . . . . . . . . 12  |-  ( D  e.  V  ->  { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) }  e.  (SubGrp `  G ) )
47 subgsubm 14917 . . . . . . . . . . . 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 11112 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0..^ L )  C_  (
0 ... L )
5150, 36sseldi 3306 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  I  e.  ( 0 ... L ) )
52 elfzuz3 11012 . . . . . . . . . . . . . . . . . . . 20  |-  ( I  e.  ( 0 ... L )  ->  L  e.  ( ZZ>= `  I )
)
5351, 52syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  L  e.  ( ZZ>= `  I ) )
5437, 53eqeltrd 2478 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( # `  W
)  e.  ( ZZ>= `  I ) )
55 fzoss2 11118 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  W )  e.  ( ZZ>= `  I )  ->  ( 0..^ I ) 
C_  ( 0..^ (
# `  W )
) )
5654, 55syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 0..^ I ) 
C_  ( 0..^ (
# `  W )
) )
5756sselda 3308 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0..^ I ) )  ->  s  e.  ( 0..^ ( # `  W
) ) )
5835ffvelrnda 5829 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  s
)  e.  T )
5921, 58sseldi 3306 . . . . . . . . . . . . . . . 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 5687 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  s  ->  ( W `  k )  =  ( W `  s ) )
6362difeq1d 3424 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  s  ->  (
( W `  k
)  \  _I  )  =  ( ( W `
 s )  \  _I  ) )
6463dmeqd 5031 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  s  ->  dom  ( ( W `  k )  \  _I  )  =  dom  ( ( W `  s ) 
\  _I  ) )
6564eleq2d 2471 . . . . . . . . . . . . . . . . . . 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 2892 . . . . . . . . . . . . . . . . 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 2764 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  s  e.  ( 0..^ I ) )  ->  -.  A  e.  dom  ( ( W `  s )  \  _I  ) )
70 difeq1 3418 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =  ( W `  s )  ->  (
j  \  _I  )  =  ( ( W `
 s )  \  _I  ) )
7170dmeqd 5031 . . . . . . . . . . . . . . . . . 18  |-  ( j  =  ( W `  s )  ->  dom  ( j  \  _I  )  =  dom  ( ( W `  s ) 
\  _I  ) )
7271sseq1d 3335 . . . . . . . . . . . . . . . . 17  |-  ( j  =  ( W `  s )  ->  ( dom  ( j  \  _I  )  C_  ( _V  \  { A } )  <->  dom  ( ( W `  s ) 
\  _I  )  C_  ( _V  \  { A } ) ) )
73 disj2 3635 . . . . . . . . . . . . . . . . . 18  |-  ( ( dom  ( ( W `
 s )  \  _I  )  i^i  { A } )  =  (/)  <->  dom  ( ( W `  s )  \  _I  )  C_  ( _V  \  { A } ) )
74 disjsn 3828 . . . . . . . . . . . . . . . . . 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 3052 . . . . . . . . . . . . . . 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 2404 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 0..^ I )  |->  ( W `  s ) )  =  ( s  e.  ( 0..^ I )  |->  ( W `  s ) )
8078, 79fmptd 5852 . . . . . . . . . . . . 13  |-  ( ph  ->  ( s  e.  ( 0..^ I )  |->  ( W `  s ) ) : ( 0..^ I ) --> { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) } )
8137oveq2d 6056 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 0 ... ( # `
 W ) )  =  ( 0 ... L ) )
8251, 81eleqtrrd 2481 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  I  e.  ( 0 ... ( # `  W
) ) )
83 swrd0val 11723 . . . . . . . . . . . . . . . 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 5738 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  W  =  ( s  e.  ( 0..^ (
# `  W )
)  |->  ( W `  s ) ) )
8685reseq1d 5104 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( W  |`  (
0..^ I ) )  =  ( ( s  e.  ( 0..^ (
# `  W )
)  |->  ( W `  s ) )  |`  ( 0..^ I ) ) )
87 resmpt 5150 . . . . . . . . . . . . . . . 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 2440 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( W substr  <. 0 ,  I >. )  =  ( s  e.  ( 0..^ I )  |->  ( W `
 s ) ) )
9089feq1d 5539 . . . . . . . . . . . . 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 11686 . . . . . . . . . . 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 14739 . . . . . . . . . 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 3418 . . . . . . . . . . . . . 14  |-  ( j  =  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  ->  ( j  \  _I  )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )
)
9897dmeqd 5031 . . . . . . . . . . . . 13  |-  ( j  =  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  ->  dom  ( j 
\  _I  )  =  dom  ( ( G 
gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  ) )
9998sseq1d 3335 . . . . . . . . . . . 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 3052 . . . . . . . . . . 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 3635 . . . . . . . . . . 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 3828 . . . . . . . . . . 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 27260 . . . . 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 11126 . . . . . . . . . . . . . . 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 2478 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  W
)  e.  NN )
120 wrdfin 11689 . . . . . . . . . . . . 13  |-  ( W  e. Word  T  ->  W  e.  Fin )
121 hashnncl 11600 . . . . . . . . . . . . 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 11743 . . . . . . . . . 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 6055 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  W
)  -  1 )  =  ( L  - 
1 ) )
128127adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  (
( # `  W )  -  1 )  =  ( L  -  1 ) )
129118nncnd 9972 . . . . . . . . . . . . 13  |-  ( ph  ->  L  e.  CC )
130 ax-1cn 9004 . . . . . . . . . . . . . 14  |-  1  e.  CC
131130a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  1  e.  CC )
132 elfzoelz 11095 . . . . . . . . . . . . . . 15  |-  ( I  e.  ( 0..^ L )  ->  I  e.  ZZ )
13336, 132syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  I  e.  ZZ )
134133zcnd 10332 . . . . . . . . . . . . 13  |-  ( ph  ->  I  e.  CC )
135129, 131, 134subadd2d 9386 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( L  - 
1 )  =  I  <-> 
( I  +  1 )  =  L ) )
136135biimpar 472 . . . . . . . . . . 11  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( L  -  1 )  =  I )
137128, 136eqtrd 2436 . . . . . . . . . 10  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  (
( # `  W )  -  1 )  =  I )
138 opeq2 3945 . . . . . . . . . . . 12  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  <. 0 ,  ( ( # `  W )  -  1 ) >.  =  <. 0 ,  I >. )
139138oveq2d 6056 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  ( W substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  =  ( W substr  <. 0 ,  I >. ) )
140 fveq2 5687 . . . . . . . . . . . 12  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  ( W `  ( ( # `
 W )  - 
1 ) )  =  ( W `  I
) )
141140s1eqd 11709 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  <" ( W `  ( ( # `
 W )  - 
1 ) ) ">  =  <" ( W `  I ) "> )
142139, 141oveq12d 6058 . . . . . . . . . 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 2436 . . . . . . . 8  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  W  =  ( ( W substr  <. 0 ,  I >. ) concat  <" ( W `  I ) "> ) )
145144oveq2d 6056 . . . . . . 7  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( G  gsumg  W )  =  ( G  gsumg  ( ( W substr  <. 0 ,  I >. ) concat  <" ( W `  I ) "> ) ) )
14641s1cld 11711 . . . . . . . . 9  |-  ( ph  ->  <" ( W `
 I ) ">  e. Word  ( Base `  G ) )
147 eqid 2404 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
14820, 147gsumccat 14742 . . . . . . . . 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 14740 . . . . . . . . . . 11  |-  ( ( W `  I )  e.  ( Base `  G
)  ->  ( G  gsumg  <" ( W `  I ) "> )  =  ( W `  I ) )
15241, 151syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( G  gsumg 
<" ( W `  I ) "> )  =  ( W `  I ) )
153152oveq2d 6056 . . . . . . . . 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 15055 . . . . . . . . . 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 2436 . . . . . . . 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 2440 . . . . . 6  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( G  gsumg  W )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) ) )
159158difeq1d 3424 . . . . 5  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  (
( G  gsumg  W )  \  _I  )  =  ( (
( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) )  \  _I  ) )
160159dmeqd 5031 . . . 4  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  dom  ( ( G  gsumg  W ) 
\  _I  )  =  dom  ( ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) )  \  _I  ) )
161114, 160eleqtrrd 2481 . . 3  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  A  e.  dom  ( ( G 
gsumg  W )  \  _I  ) )
16213, 161mtand 641 . 2  |-  ( ph  ->  -.  ( I  + 
1 )  =  L )
163 fzostep1 11152 . . . 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 1721    =/= wne 2567   A.wral 2666   {crab 2670   _Vcvv 2916    \ cdif 3277    i^i cin 3279    C_ wss 3280   (/)c0 3588   {csn 3774   <.cop 3777   class class class wbr 4172    e. cmpt 4226    _I cid 4453   dom cdm 4837   ran crn 4838    |` cres 4839    o. ccom 4841   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    < clt 9076    - cmin 9247   NNcn 9956   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999  ..^cfzo 11090   #chash 11573  Word cword 11672   concat cconcat 11673   <"cs1 11674   substr csubstr 11675   Basecbs 13424   +g cplusg 13484    gsumg cgsu 13679   Mndcmnd 14639   Grpcgrp 14640  SubMndcsubmnd 14692  SubGrpcsubg 14893   SymGrpcsymg 15047  pmTrspcpmtr 27252
This theorem is referenced by:  psgnunilem2  27286
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-word 11678  df-concat 11679  df-s1 11680  df-substr 11681  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-tset 13503  df-0g 13682  df-gsum 13683  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-subg 14896  df-symg 15048  df-pmtr 27253
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