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Theorem psgnunilem5 16315
Description: Lemma for psgnuni 16320. 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 3789 . . . 4  |-  -.  A  e.  (/)
2 psgnunilem2.id . . . . . . . 8  |-  ( ph  ->  ( G  gsumg  W )  =  (  _I  |`  D )
)
32difeq1d 3621 . . . . . . 7  |-  ( ph  ->  ( ( G  gsumg  W ) 
\  _I  )  =  ( (  _I  |`  D ) 
\  _I  ) )
43dmeqd 5203 . . . . . 6  |-  ( ph  ->  dom  ( ( G 
gsumg  W )  \  _I  )  =  dom  ( (  _I  |`  D )  \  _I  ) )
5 resss 5295 . . . . . . . . 9  |-  (  _I  |`  D )  C_  _I
6 ssdif0 3885 . . . . . . . . 9  |-  ( (  _I  |`  D )  C_  _I  <->  ( (  _I  |`  D )  \  _I  )  =  (/) )
75, 6mpbi 208 . . . . . . . 8  |-  ( (  _I  |`  D )  \  _I  )  =  (/)
87dmeqi 5202 . . . . . . 7  |-  dom  (
(  _I  |`  D ) 
\  _I  )  =  dom  (/)
9 dm0 5214 . . . . . . 7  |-  dom  (/)  =  (/)
108, 9eqtri 2496 . . . . . 6  |-  dom  (
(  _I  |`  D ) 
\  _I  )  =  (/)
114, 10syl6eq 2524 . . . . 5  |-  ( ph  ->  dom  ( ( G 
gsumg  W )  \  _I  )  =  (/) )
1211eleq2d 2537 . . . 4  |-  ( ph  ->  ( A  e.  dom  ( ( G  gsumg  W ) 
\  _I  )  <->  A  e.  (/) ) )
131, 12mtbiri 303 . . 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 16220 . . . . . . . . 9  |-  ( D  e.  V  ->  G  e.  Grp )
17 grpmnd 15863 . . . . . . . . 9  |-  ( G  e.  Grp  ->  G  e.  Mnd )
1814, 16, 173syl 20 . . . . . . . 8  |-  ( ph  ->  G  e.  Mnd )
19 psgnunilem2.t . . . . . . . . . . . 12  |-  T  =  ran  (pmTrsp `  D
)
20 eqid 2467 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
2119, 15, 20symgtrf 16290 . . . . . . . . . . 11  |-  T  C_  ( Base `  G )
22 sswrd 12517 . . . . . . . . . . 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 3505 . . . . . . . . 9  |-  ( ph  ->  W  e. Word  ( Base `  G ) )
26 swrdcl 12605 . . . . . . . . 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 15831 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( W substr  <. 0 ,  I >. )  e. Word  ( Base `  G ) )  ->  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  e.  ( Base `  G ) )
2918, 27, 28syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  e.  ( Base `  G
) )
3015, 20elsymgbas2 16201 . . . . . . . 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 241 . . . . . . 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 465 . . . . 5  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) : D -1-1-onto-> D )
34 wrdf 12515 . . . . . . . . . 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 6298 . . . . . . . . . 10  |-  ( ph  ->  ( 0..^ ( # `  W ) )  =  ( 0..^ L ) )
3936, 38eleqtrrd 2558 . . . . . . . . 9  |-  ( ph  ->  I  e.  ( 0..^ ( # `  W
) ) )
4035, 39ffvelrnd 6020 . . . . . . . 8  |-  ( ph  ->  ( W `  I
)  e.  T )
4121, 40sseldi 3502 . . . . . . 7  |-  ( ph  ->  ( W `  I
)  e.  ( Base `  G ) )
4215, 20elsymgbas2 16201 . . . . . . . 8  |-  ( ( W `  I )  e.  ( Base `  G
)  ->  ( ( W `  I )  e.  ( Base `  G
)  <->  ( W `  I ) : D -1-1-onto-> D
) )
4342ibi 241 . . . . . . 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 465 . . . . 5  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( W `  I ) : D -1-1-onto-> D )
4615, 20symgsssg 16288 . . . . . . . . . . . 12  |-  ( D  e.  V  ->  { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) }  e.  (SubGrp `  G ) )
47 subgsubm 16018 . . . . . . . . . . . 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 20 . . . . . . . . . . 11  |-  ( ph  ->  { j  e.  (
Base `  G )  |  dom  ( j  \  _I  )  C_  ( _V 
\  { A }
) }  e.  (SubMnd `  G ) )
4948adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) }  e.  (SubMnd `  G ) )
50 fzossfz 11810 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0..^ L )  C_  (
0 ... L )
5150, 36sseldi 3502 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  I  e.  ( 0 ... L ) )
52 elfzuz3 11681 . . . . . . . . . . . . . . . . . . . 20  |-  ( I  e.  ( 0 ... L )  ->  L  e.  ( ZZ>= `  I )
)
5351, 52syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  L  e.  ( ZZ>= `  I ) )
5437, 53eqeltrd 2555 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( # `  W
)  e.  ( ZZ>= `  I ) )
55 fzoss2 11817 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  W )  e.  ( ZZ>= `  I )  ->  ( 0..^ I ) 
C_  ( 0..^ (
# `  W )
) )
5654, 55syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 0..^ I ) 
C_  ( 0..^ (
# `  W )
) )
5756sselda 3504 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0..^ I ) )  ->  s  e.  ( 0..^ ( # `  W
) ) )
5835ffvelrnda 6019 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  s
)  e.  T )
5921, 58sseldi 3502 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  s
)  e.  ( Base `  G ) )
6057, 59syldan 470 . . . . . . . . . . . . . . 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 5864 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  s  ->  ( W `  k )  =  ( W `  s ) )
6362difeq1d 3621 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  s  ->  (
( W `  k
)  \  _I  )  =  ( ( W `
 s )  \  _I  ) )
6463dmeqd 5203 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  s  ->  dom  ( ( W `  k )  \  _I  )  =  dom  ( ( W `  s ) 
\  _I  ) )
6564eleq2d 2537 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  s  ->  ( A  e.  dom  ( ( W `  k ) 
\  _I  )  <->  A  e.  dom  ( ( W `  s )  \  _I  ) ) )
6665notbid 294 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  s  ->  ( -.  A  e.  dom  ( ( W `  k )  \  _I  ) 
<->  -.  A  e.  dom  ( ( W `  s )  \  _I  ) ) )
6766cbvralv 3088 . . . . . . . . . . . . . . . . 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 196 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A. s  e.  ( 0..^ I )  -.  A  e.  dom  (
( W `  s
)  \  _I  )
)
6968r19.21bi 2833 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  s  e.  ( 0..^ I ) )  ->  -.  A  e.  dom  ( ( W `  s )  \  _I  ) )
70 difeq1 3615 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =  ( W `  s )  ->  (
j  \  _I  )  =  ( ( W `
 s )  \  _I  ) )
7170dmeqd 5203 . . . . . . . . . . . . . . . . . 18  |-  ( j  =  ( W `  s )  ->  dom  ( j  \  _I  )  =  dom  ( ( W `  s ) 
\  _I  ) )
7271sseq1d 3531 . . . . . . . . . . . . . . . . 17  |-  ( j  =  ( W `  s )  ->  ( dom  ( j  \  _I  )  C_  ( _V  \  { A } )  <->  dom  ( ( W `  s ) 
\  _I  )  C_  ( _V  \  { A } ) ) )
73 disj2 3874 . . . . . . . . . . . . . . . . . 18  |-  ( ( dom  ( ( W `
 s )  \  _I  )  i^i  { A } )  =  (/)  <->  dom  ( ( W `  s )  \  _I  )  C_  ( _V  \  { A } ) )
74 disjsn 4088 . . . . . . . . . . . . . . . . . 18  |-  ( ( dom  ( ( W `
 s )  \  _I  )  i^i  { A } )  =  (/)  <->  -.  A  e.  dom  ( ( W `  s ) 
\  _I  ) )
7573, 74bitr3i 251 . . . . . . . . . . . . . . . . 17  |-  ( dom  ( ( W `  s )  \  _I  )  C_  ( _V  \  { A } )  <->  -.  A  e.  dom  ( ( W `
 s )  \  _I  ) )
7672, 75syl6bb 261 . . . . . . . . . . . . . . . 16  |-  ( j  =  ( W `  s )  ->  ( dom  ( j  \  _I  )  C_  ( _V  \  { A } )  <->  -.  A  e.  dom  ( ( W `
 s )  \  _I  ) ) )
7776elrab 3261 . . . . . . . . . . . . . . 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 664 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  s  e.  ( 0..^ I ) )  ->  ( W `  s )  e.  {
j  e.  ( Base `  G )  |  dom  ( j  \  _I  )  C_  ( _V  \  { A } ) } )
79 eqid 2467 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 0..^ I )  |->  ( W `  s ) )  =  ( s  e.  ( 0..^ I )  |->  ( W `  s ) )
8078, 79fmptd 6043 . . . . . . . . . . . . 13  |-  ( ph  ->  ( s  e.  ( 0..^ I )  |->  ( W `  s ) ) : ( 0..^ I ) --> { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) } )
8137oveq2d 6298 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 0 ... ( # `
 W ) )  =  ( 0 ... L ) )
8251, 81eleqtrrd 2558 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  I  e.  ( 0 ... ( # `  W
) ) )
83 swrd0val 12607 . . . . . . . . . . . . . . . 16  |-  ( ( W  e. Word  T  /\  I  e.  ( 0 ... ( # `  W
) ) )  -> 
( W substr  <. 0 ,  I >. )  =  ( W  |`  ( 0..^ I ) ) )
8424, 82, 83syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( W substr  <. 0 ,  I >. )  =  ( W  |`  ( 0..^ I ) ) )
8535feqmptd 5918 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  W  =  ( s  e.  ( 0..^ (
# `  W )
)  |->  ( W `  s ) ) )
8685reseq1d 5270 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( W  |`  (
0..^ I ) )  =  ( ( s  e.  ( 0..^ (
# `  W )
)  |->  ( W `  s ) )  |`  ( 0..^ I ) ) )
87 resmpt 5321 . . . . . . . . . . . . . . . 16  |-  ( ( 0..^ I )  C_  ( 0..^ ( # `  W
) )  ->  (
( s  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 s ) )  |`  ( 0..^ I ) )  =  ( s  e.  ( 0..^ I )  |->  ( W `  s ) ) )
8854, 55, 873syl 20 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( s  e.  ( 0..^ ( # `  W ) )  |->  ( W `  s ) )  |`  ( 0..^ I ) )  =  ( s  e.  ( 0..^ I )  |->  ( W `  s ) ) )
8984, 86, 883eqtrd 2512 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( W substr  <. 0 ,  I >. )  =  ( s  e.  ( 0..^ I )  |->  ( W `
 s ) ) )
9089feq1d 5715 . . . . . . . . . . . . 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 232 . . . . . . . . . . . 12  |-  ( ph  ->  ( W substr  <. 0 ,  I >. ) : ( 0..^ I ) --> { j  e.  ( Base `  G )  |  dom  ( j  \  _I  )  C_  ( _V  \  { A } ) } )
9291adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( W substr  <. 0 ,  I >. ) : ( 0..^ I ) --> { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) } )
93 iswrdi 12514 . . . . . . . . . . 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 15829 . . . . . . . . . 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 661 . . . . . . . . 9  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  e. 
{ j  e.  (
Base `  G )  |  dom  ( j  \  _I  )  C_  ( _V 
\  { A }
) } )
97 difeq1 3615 . . . . . . . . . . . . . 14  |-  ( j  =  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  ->  ( j  \  _I  )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )
)
9897dmeqd 5203 . . . . . . . . . . . . 13  |-  ( j  =  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  ->  dom  ( j 
\  _I  )  =  dom  ( ( G 
gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  ) )
9998sseq1d 3531 . . . . . . . . . . . 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 3261 . . . . . . . . . . 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 464 . . . . . . . . . 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 3874 . . . . . . . . . . 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 4088 . . . . . . . . . . 11  |-  ( ( dom  ( ( G 
gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  i^i  { A } )  =  (/)  <->  -.  A  e.  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  ) )
104102, 103bitr3i 251 . . . . . . . . . 10  |-  ( dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  C_  ( _V  \  { A } )  <->  -.  A  e.  dom  ( ( G 
gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  ) )
105101, 104sylib 196 . . . . . . . . 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 465 . . . . . . . 8  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  A  e.  dom  ( ( W `
 I )  \  _I  ) )
109106, 108jca 532 . . . . . . 7  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( -.  A  e.  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  /\  A  e.  dom  ( ( W `  I )  \  _I  ) ) )
110109olcd 393 . . . . . 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 1365 . . . . . 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 212 . . . . 5  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( A  e.  dom  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )  \/_  A  e. 
dom  ( ( W `
 I )  \  _I  ) ) )
113 f1omvdco3 16270 . . . . 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 1228 . . . 4  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  A  e.  dom  ( ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) )  \  _I  ) )
11524adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  W  e. Word  T )
116 elfzo0 11827 . . . . . . . . . . . . . . 15  |-  ( I  e.  ( 0..^ L )  <->  ( I  e. 
NN0  /\  L  e.  NN  /\  I  <  L
) )
117116simp2bi 1012 . . . . . . . . . . . . . 14  |-  ( I  e.  ( 0..^ L )  ->  L  e.  NN )
11836, 117syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  L  e.  NN )
11937, 118eqeltrd 2555 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  W
)  e.  NN )
120 wrdfin 12523 . . . . . . . . . . . . 13  |-  ( W  e. Word  T  ->  W  e.  Fin )
121 hashnncl 12400 . . . . . . . . . . . . 13  |-  ( W  e.  Fin  ->  (
( # `  W )  e.  NN  <->  W  =/=  (/) ) )
12224, 120, 1213syl 20 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  W
)  e.  NN  <->  W  =/=  (/) ) )
123119, 122mpbid 210 . . . . . . . . . . 11  |-  ( ph  ->  W  =/=  (/) )
124123adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  W  =/=  (/) )
125 wrdeqcats1 12658 . . . . . . . . . 10  |-  ( ( W  e. Word  T  /\  W  =/=  (/) )  ->  W  =  ( ( W substr  <. 0 ,  ( (
# `  W )  -  1 ) >.
) concat  <" ( W `
 ( ( # `  W )  -  1 ) ) "> ) )
126115, 124, 125syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  W  =  ( ( W substr  <. 0 ,  ( (
# `  W )  -  1 ) >.
) concat  <" ( W `
 ( ( # `  W )  -  1 ) ) "> ) )
12737oveq1d 6297 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  W
)  -  1 )  =  ( L  - 
1 ) )
128127adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  (
( # `  W )  -  1 )  =  ( L  -  1 ) )
129118nncnd 10548 . . . . . . . . . . . . 13  |-  ( ph  ->  L  e.  CC )
130 ax-1cn 9546 . . . . . . . . . . . . . 14  |-  1  e.  CC
131130a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  1  e.  CC )
132 elfzoelz 11793 . . . . . . . . . . . . . . 15  |-  ( I  e.  ( 0..^ L )  ->  I  e.  ZZ )
13336, 132syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  I  e.  ZZ )
134133zcnd 10963 . . . . . . . . . . . . 13  |-  ( ph  ->  I  e.  CC )
135129, 131, 134subadd2d 9945 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( L  - 
1 )  =  I  <-> 
( I  +  1 )  =  L ) )
136135biimpar 485 . . . . . . . . . . 11  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( L  -  1 )  =  I )
137128, 136eqtrd 2508 . . . . . . . . . 10  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  (
( # `  W )  -  1 )  =  I )
138 opeq2 4214 . . . . . . . . . . . 12  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  <. 0 ,  ( ( # `  W )  -  1 ) >.  =  <. 0 ,  I >. )
139138oveq2d 6298 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  ( W substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  =  ( W substr  <. 0 ,  I >. ) )
140 fveq2 5864 . . . . . . . . . . . 12  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  ( W `  ( ( # `
 W )  - 
1 ) )  =  ( W `  I
) )
141140s1eqd 12572 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  <" ( W `  ( ( # `
 W )  - 
1 ) ) ">  =  <" ( W `  I ) "> )
142139, 141oveq12d 6300 . . . . . . . . . 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 2508 . . . . . . . 8  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  W  =  ( ( W substr  <. 0 ,  I >. ) concat  <" ( W `  I ) "> ) )
145144oveq2d 6298 . . . . . . 7  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( G  gsumg  W )  =  ( G  gsumg  ( ( W substr  <. 0 ,  I >. ) concat  <" ( W `  I ) "> ) ) )
14641s1cld 12574 . . . . . . . . 9  |-  ( ph  ->  <" ( W `
 I ) ">  e. Word  ( Base `  G ) )
147 eqid 2467 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
14820, 147gsumccat 15832 . . . . . . . . 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 1228 . . . . . . . 8  |-  ( ph  ->  ( G  gsumg  ( ( W substr  <. 0 ,  I >. ) concat  <" ( W `  I ) "> ) )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) ( +g  `  G
) ( G  gsumg  <" ( W `  I ) "> ) ) )
150149adantr 465 . . . . . . 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 15830 . . . . . . . . . . 11  |-  ( ( W `  I )  e.  ( Base `  G
)  ->  ( G  gsumg  <" ( W `  I ) "> )  =  ( W `  I ) )
15241, 151syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( G  gsumg 
<" ( W `  I ) "> )  =  ( W `  I ) )
153152oveq2d 6298 . . . . . . . . 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 16210 . . . . . . . . . 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 661 . . . . . . . . 9  |-  ( ph  ->  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) ( +g  `  G
) ( W `  I ) )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `
 I ) ) )
156153, 155eqtrd 2508 . . . . . . . 8  |-  ( ph  ->  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) ) ( +g  `  G
) ( G  gsumg  <" ( W `  I ) "> ) )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `
 I ) ) )
157156adantr 465 . . . . . . 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 2512 . . . . . 6  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( G  gsumg  W )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) ) )
159158difeq1d 3621 . . . . 5  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  (
( G  gsumg  W )  \  _I  )  =  ( (
( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) )  \  _I  ) )
160159dmeqd 5203 . . . 4  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  dom  ( ( G  gsumg  W ) 
\  _I  )  =  dom  ( ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) )  \  _I  ) )
161114, 160eleqtrrd 2558 . . 3  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  A  e.  dom  ( ( G 
gsumg  W )  \  _I  ) )
16213, 161mtand 659 . 2  |-  ( ph  ->  -.  ( I  + 
1 )  =  L )
163 fzostep1 11886 . . . 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 377 . 2  |-  ( ph  ->  ( -.  ( I  +  1 )  e.  ( 0..^ L )  ->  ( I  + 
1 )  =  L ) )
166162, 165mt3d 125 1  |-  ( ph  ->  ( I  +  1 )  e.  ( 0..^ L ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/_ wxo 1360    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   {crab 2818   _Vcvv 3113    \ cdif 3473    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   class class class wbr 4447    |-> cmpt 4505    _I cid 4790   dom cdm 4999   ran crn 5000    |` cres 5001    o. ccom 5003   -->wf 5582   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   Fincfn 7513   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491    < clt 9624    - cmin 9801   NNcn 10532   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078   ...cfz 11668  ..^cfzo 11788   #chash 12369  Word cword 12496   concat cconcat 12498   <"cs1 12499   substr csubstr 12500   Basecbs 14486   +g cplusg 14551    gsumg cgsu 14692   Mndcmnd 15722   Grpcgrp 15723  SubMndcsubmnd 15776  SubGrpcsubg 15990   SymGrpcsymg 16197  pmTrspcpmtr 16262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-xor 1361  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12072  df-hash 12370  df-word 12504  df-concat 12506  df-s1 12507  df-substr 12508  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-tset 14570  df-0g 14693  df-gsum 14694  df-mnd 15728  df-submnd 15778  df-grp 15858  df-minusg 15859  df-subg 15993  df-symg 16198  df-pmtr 16263
This theorem is referenced by:  psgnunilem2  16316
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