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Theorem psgnunilem5 15991
Description: Lemma for psgnuni 15996. 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 3636 . . . 4  |-  -.  A  e.  (/)
2 psgnunilem2.id . . . . . . . 8  |-  ( ph  ->  ( G  gsumg  W )  =  (  _I  |`  D )
)
32difeq1d 3468 . . . . . . 7  |-  ( ph  ->  ( ( G  gsumg  W ) 
\  _I  )  =  ( (  _I  |`  D ) 
\  _I  ) )
43dmeqd 5037 . . . . . 6  |-  ( ph  ->  dom  ( ( G 
gsumg  W )  \  _I  )  =  dom  ( (  _I  |`  D )  \  _I  ) )
5 resss 5129 . . . . . . . . 9  |-  (  _I  |`  D )  C_  _I
6 ssdif0 3732 . . . . . . . . 9  |-  ( (  _I  |`  D )  C_  _I  <->  ( (  _I  |`  D )  \  _I  )  =  (/) )
75, 6mpbi 208 . . . . . . . 8  |-  ( (  _I  |`  D )  \  _I  )  =  (/)
87dmeqi 5036 . . . . . . 7  |-  dom  (
(  _I  |`  D ) 
\  _I  )  =  dom  (/)
9 dm0 5048 . . . . . . 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 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 15896 . . . . . . . . 9  |-  ( D  e.  V  ->  G  e.  Grp )
17 grpmnd 15541 . . . . . . . . 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 2438 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
2119, 15, 20symgtrf 15966 . . . . . . . . . . 11  |-  T  C_  ( Base `  G )
22 sswrd 12234 . . . . . . . . . . 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 3352 . . . . . . . . 9  |-  ( ph  ->  W  e. Word  ( Base `  G ) )
26 swrdcl 12307 . . . . . . . . 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 15509 . . . . . . . 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 15877 . . . . . . . 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 12232 . . . . . . . . . 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 6102 . . . . . . . . . 10  |-  ( ph  ->  ( 0..^ ( # `  W ) )  =  ( 0..^ L ) )
3936, 38eleqtrrd 2515 . . . . . . . . 9  |-  ( ph  ->  I  e.  ( 0..^ ( # `  W
) ) )
4035, 39ffvelrnd 5839 . . . . . . . 8  |-  ( ph  ->  ( W `  I
)  e.  T )
4121, 40sseldi 3349 . . . . . . 7  |-  ( ph  ->  ( W `  I
)  e.  ( Base `  G ) )
4215, 20elsymgbas2 15877 . . . . . . . 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 15964 . . . . . . . . . . . 12  |-  ( D  e.  V  ->  { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) }  e.  (SubGrp `  G ) )
47 subgsubm 15694 . . . . . . . . . . . 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 11562 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0..^ L )  C_  (
0 ... L )
5150, 36sseldi 3349 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  I  e.  ( 0 ... L ) )
52 elfzuz3 11442 . . . . . . . . . . . . . . . . . . . 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 11569 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  W )  e.  ( ZZ>= `  I )  ->  ( 0..^ I ) 
C_  ( 0..^ (
# `  W )
) )
5654, 55syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 0..^ I ) 
C_  ( 0..^ (
# `  W )
) )
5756sselda 3351 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  s  e.  ( 0..^ I ) )  ->  s  e.  ( 0..^ ( # `  W
) ) )
5835ffvelrnda 5838 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  s  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  s
)  e.  T )
5921, 58sseldi 3349 . . . . . . . . . . . . . . . 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 5686 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  s  ->  ( W `  k )  =  ( W `  s ) )
6362difeq1d 3468 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  s  ->  (
( W `  k
)  \  _I  )  =  ( ( W `
 s )  \  _I  ) )
6463dmeqd 5037 . . . . . . . . . . . . . . . . . . . 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 294 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  s  ->  ( -.  A  e.  dom  ( ( W `  k )  \  _I  ) 
<->  -.  A  e.  dom  ( ( W `  s )  \  _I  ) ) )
6766cbvralv 2942 . . . . . . . . . . . . . . . . 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 2809 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  s  e.  ( 0..^ I ) )  ->  -.  A  e.  dom  ( ( W `  s )  \  _I  ) )
70 difeq1 3462 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =  ( W `  s )  ->  (
j  \  _I  )  =  ( ( W `
 s )  \  _I  ) )
7170dmeqd 5037 . . . . . . . . . . . . . . . . . 18  |-  ( j  =  ( W `  s )  ->  dom  ( j  \  _I  )  =  dom  ( ( W `  s ) 
\  _I  ) )
7271sseq1d 3378 . . . . . . . . . . . . . . . . 17  |-  ( j  =  ( W `  s )  ->  ( dom  ( j  \  _I  )  C_  ( _V  \  { A } )  <->  dom  ( ( W `  s ) 
\  _I  )  C_  ( _V  \  { A } ) ) )
73 disj2 3721 . . . . . . . . . . . . . . . . . 18  |-  ( ( dom  ( ( W `
 s )  \  _I  )  i^i  { A } )  =  (/)  <->  dom  ( ( W `  s )  \  _I  )  C_  ( _V  \  { A } ) )
74 disjsn 3931 . . . . . . . . . . . . . . . . . 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 3112 . . . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 0..^ I )  |->  ( W `  s ) )  =  ( s  e.  ( 0..^ I )  |->  ( W `  s ) )
8078, 79fmptd 5862 . . . . . . . . . . . . 13  |-  ( ph  ->  ( s  e.  ( 0..^ I )  |->  ( W `  s ) ) : ( 0..^ I ) --> { j  e.  ( Base `  G
)  |  dom  (
j  \  _I  )  C_  ( _V  \  { A } ) } )
8137oveq2d 6102 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 0 ... ( # `
 W ) )  =  ( 0 ... L ) )
8251, 81eleqtrrd 2515 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  I  e.  ( 0 ... ( # `  W
) ) )
83 swrd0val 12309 . . . . . . . . . . . . . . . 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 5739 . . . . . . . . . . . . . . . 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 5151 . . . . . . . . . . . . . . . 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 2474 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( W substr  <. 0 ,  I >. )  =  ( s  e.  ( 0..^ I )  |->  ( W `
 s ) ) )
9089feq1d 5541 . . . . . . . . . . . . 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 12231 . . . . . . . . . . 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 15507 . . . . . . . . . 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 3462 . . . . . . . . . . . . . 14  |-  ( j  =  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  ->  ( j  \  _I  )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  )
)
9897dmeqd 5037 . . . . . . . . . . . . 13  |-  ( j  =  ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  ->  dom  ( j 
\  _I  )  =  dom  ( ( G 
gsumg  ( W substr  <. 0 ,  I >. ) )  \  _I  ) )
9998sseq1d 3378 . . . . . . . . . . . 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 3112 . . . . . . . . . . 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 3721 . . . . . . . . . . 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 3931 . . . . . . . . . . 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 1355 . . . . . 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 15946 . . . . 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 1218 . . . 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 11579 . . . . . . . . . . . . . . 15  |-  ( I  e.  ( 0..^ L )  <->  ( I  e. 
NN0  /\  L  e.  NN  /\  I  <  L
) )
117116simp2bi 1004 . . . . . . . . . . . . . 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 12240 . . . . . . . . . . . . 13  |-  ( W  e. Word  T  ->  W  e.  Fin )
121 hashnncl 12126 . . . . . . . . . . . . 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 12360 . . . . . . . . . 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 6101 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( # `  W
)  -  1 )  =  ( L  - 
1 ) )
128127adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  (
( # `  W )  -  1 )  =  ( L  -  1 ) )
129118nncnd 10330 . . . . . . . . . . . . 13  |-  ( ph  ->  L  e.  CC )
130 ax-1cn 9332 . . . . . . . . . . . . . 14  |-  1  e.  CC
131130a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  1  e.  CC )
132 elfzoelz 11545 . . . . . . . . . . . . . . 15  |-  ( I  e.  ( 0..^ L )  ->  I  e.  ZZ )
13336, 132syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  I  e.  ZZ )
134133zcnd 10740 . . . . . . . . . . . . 13  |-  ( ph  ->  I  e.  CC )
135129, 131, 134subadd2d 9730 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( L  - 
1 )  =  I  <-> 
( I  +  1 )  =  L ) )
136135biimpar 485 . . . . . . . . . . 11  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( L  -  1 )  =  I )
137128, 136eqtrd 2470 . . . . . . . . . 10  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  (
( # `  W )  -  1 )  =  I )
138 opeq2 4055 . . . . . . . . . . . 12  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  <. 0 ,  ( ( # `  W )  -  1 ) >.  =  <. 0 ,  I >. )
139138oveq2d 6102 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  ( W substr  <. 0 ,  ( ( # `  W
)  -  1 )
>. )  =  ( W substr  <. 0 ,  I >. ) )
140 fveq2 5686 . . . . . . . . . . . 12  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  ( W `  ( ( # `
 W )  - 
1 ) )  =  ( W `  I
) )
141140s1eqd 12284 . . . . . . . . . . 11  |-  ( ( ( # `  W
)  -  1 )  =  I  ->  <" ( W `  ( ( # `
 W )  - 
1 ) ) ">  =  <" ( W `  I ) "> )
142139, 141oveq12d 6104 . . . . . . . . . 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 6102 . . . . . . 7  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( G  gsumg  W )  =  ( G  gsumg  ( ( W substr  <. 0 ,  I >. ) concat  <" ( W `  I ) "> ) ) )
14641s1cld 12286 . . . . . . . . 9  |-  ( ph  ->  <" ( W `
 I ) ">  e. Word  ( Base `  G ) )
147 eqid 2438 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
14820, 147gsumccat 15510 . . . . . . . . 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 1218 . . . . . . . 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 15508 . . . . . . . . . . 11  |-  ( ( W `  I )  e.  ( Base `  G
)  ->  ( G  gsumg  <" ( W `  I ) "> )  =  ( W `  I ) )
15241, 151syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( G  gsumg 
<" ( W `  I ) "> )  =  ( W `  I ) )
153152oveq2d 6102 . . . . . . . . 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 15886 . . . . . . . . . 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 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 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 2474 . . . . . 6  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  ( G  gsumg  W )  =  ( ( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) ) )
159158difeq1d 3468 . . . . 5  |-  ( (
ph  /\  ( I  +  1 )  =  L )  ->  (
( G  gsumg  W )  \  _I  )  =  ( (
( G  gsumg  ( W substr  <. 0 ,  I >. ) )  o.  ( W `  I
) )  \  _I  ) )
160159dmeqd 5037 . . . 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 659 . 2  |-  ( ph  ->  -.  ( I  + 
1 )  =  L )
163 fzostep1 11627 . . . 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 1350    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   {crab 2714   _Vcvv 2967    \ cdif 3320    i^i cin 3322    C_ wss 3323   (/)c0 3632   {csn 3872   <.cop 3878   class class class wbr 4287    e. cmpt 4345    _I cid 4626   dom cdm 4835   ran crn 4836    |` cres 4837    o. ccom 4839   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086   Fincfn 7302   CCcc 9272   0cc0 9274   1c1 9275    + caddc 9277    < clt 9410    - cmin 9587   NNcn 10314   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853   ...cfz 11429  ..^cfzo 11540   #chash 12095  Word cword 12213   concat cconcat 12215   <"cs1 12216   substr csubstr 12217   Basecbs 14166   +g cplusg 14230    gsumg cgsu 14371   Mndcmnd 15401   Grpcgrp 15402  SubMndcsubmnd 15455  SubGrpcsubg 15666   SymGrpcsymg 15873  pmTrspcpmtr 15938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1351  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-fzo 11541  df-seq 11799  df-hash 12096  df-word 12221  df-concat 12223  df-s1 12224  df-substr 12225  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-tset 14249  df-0g 14372  df-gsum 14373  df-mnd 15407  df-submnd 15457  df-grp 15536  df-minusg 15537  df-subg 15669  df-symg 15874  df-pmtr 15939
This theorem is referenced by:  psgnunilem2  15992
  Copyright terms: Public domain W3C validator