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Theorem ablfaclem3 17264
Description: Lemma for ablfac 17265. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
Hypotheses
Ref Expression
ablfac.b  |-  B  =  ( Base `  G
)
ablfac.c  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
ablfac.1  |-  ( ph  ->  G  e.  Abel )
ablfac.2  |-  ( ph  ->  B  e.  Fin )
ablfac.o  |-  O  =  ( od `  G
)
ablfac.a  |-  A  =  { w  e.  Prime  |  w  ||  ( # `  B ) }
ablfac.s  |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  (
p ^ ( p 
pCnt  ( # `  B
) ) ) } )
ablfac.w  |-  W  =  ( g  e.  (SubGrp `  G )  |->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  g ) } )
Assertion
Ref Expression
ablfaclem3  |-  ( ph  ->  ( W `  B
)  =/=  (/) )
Distinct variable groups:    s, p, x, A    g, r, s, S    g, p, w, x, B, r, s    O, p, x    C, g, p, s, w, x    W, p, w, x    ph, p, s, w, x    g, G, p, r, s, w, x
Allowed substitution hints:    ph( g, r)    A( w, g, r)    C( r)    S( x, w, p)    O( w, g, s, r)    W( g, s, r)

Proof of Theorem ablfaclem3
Dummy variables  a 
b  c  f  h  q  t  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzfid 12085 . . . 4  |-  ( ph  ->  ( 1 ... ( # `
 B ) )  e.  Fin )
2 ablfac.a . . . . 5  |-  A  =  { w  e.  Prime  |  w  ||  ( # `  B ) }
3 prmnn 14231 . . . . . . . 8  |-  ( w  e.  Prime  ->  w  e.  NN )
433ad2ant2 1018 . . . . . . 7  |-  ( (
ph  /\  w  e.  Prime  /\  w  ||  ( # `
 B ) )  ->  w  e.  NN )
5 prmz 14232 . . . . . . . . 9  |-  ( w  e.  Prime  ->  w  e.  ZZ )
6 ablfac.1 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  Abel )
7 ablgrp 16929 . . . . . . . . . . 11  |-  ( G  e.  Abel  ->  G  e. 
Grp )
8 ablfac.b . . . . . . . . . . . 12  |-  B  =  ( Base `  G
)
98grpbn0 16205 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  B  =/=  (/) )
106, 7, 93syl 20 . . . . . . . . . 10  |-  ( ph  ->  B  =/=  (/) )
11 ablfac.2 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  Fin )
12 hashnncl 12438 . . . . . . . . . . 11  |-  ( B  e.  Fin  ->  (
( # `  B )  e.  NN  <->  B  =/=  (/) ) )
1311, 12syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ( # `  B
)  e.  NN  <->  B  =/=  (/) ) )
1410, 13mpbird 232 . . . . . . . . 9  |-  ( ph  ->  ( # `  B
)  e.  NN )
15 dvdsle 14042 . . . . . . . . 9  |-  ( ( w  e.  ZZ  /\  ( # `  B )  e.  NN )  -> 
( w  ||  ( # `
 B )  ->  w  <_  ( # `  B
) ) )
165, 14, 15syl2anr 478 . . . . . . . 8  |-  ( (
ph  /\  w  e.  Prime )  ->  ( w  ||  ( # `  B
)  ->  w  <_  (
# `  B )
) )
17163impia 1193 . . . . . . 7  |-  ( (
ph  /\  w  e.  Prime  /\  w  ||  ( # `
 B ) )  ->  w  <_  ( # `
 B ) )
1814nnzd 10989 . . . . . . . . 9  |-  ( ph  ->  ( # `  B
)  e.  ZZ )
19183ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  w  e.  Prime  /\  w  ||  ( # `
 B ) )  ->  ( # `  B
)  e.  ZZ )
20 fznn 11772 . . . . . . . 8  |-  ( (
# `  B )  e.  ZZ  ->  ( w  e.  ( 1 ... ( # `
 B ) )  <-> 
( w  e.  NN  /\  w  <_  ( # `  B
) ) ) )
2119, 20syl 16 . . . . . . 7  |-  ( (
ph  /\  w  e.  Prime  /\  w  ||  ( # `
 B ) )  ->  ( w  e.  ( 1 ... ( # `
 B ) )  <-> 
( w  e.  NN  /\  w  <_  ( # `  B
) ) ) )
224, 17, 21mpbir2and 922 . . . . . 6  |-  ( (
ph  /\  w  e.  Prime  /\  w  ||  ( # `
 B ) )  ->  w  e.  ( 1 ... ( # `  B ) ) )
2322rabssdv 3576 . . . . 5  |-  ( ph  ->  { w  e.  Prime  |  w  ||  ( # `  B ) }  C_  ( 1 ... ( # `
 B ) ) )
242, 23syl5eqss 3543 . . . 4  |-  ( ph  ->  A  C_  ( 1 ... ( # `  B
) ) )
25 ssfi 7759 . . . 4  |-  ( ( ( 1 ... ( # `
 B ) )  e.  Fin  /\  A  C_  ( 1 ... ( # `
 B ) ) )  ->  A  e.  Fin )
261, 24, 25syl2anc 661 . . 3  |-  ( ph  ->  A  e.  Fin )
27 dfin5 3479 . . . . . . . 8  |-  (Word  C  i^i  ( W `  ( S `  q )
) )  =  {
y  e. Word  C  | 
y  e.  ( W `
 ( S `  q ) ) }
28 ablfac.o . . . . . . . . . . . . . 14  |-  O  =  ( od `  G
)
29 ablfac.s . . . . . . . . . . . . . 14  |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  (
p ^ ( p 
pCnt  ( # `  B
) ) ) } )
30 ssrab2 3581 . . . . . . . . . . . . . . . 16  |-  { w  e.  Prime  |  w  ||  ( # `  B ) }  C_  Prime
312, 30eqsstri 3529 . . . . . . . . . . . . . . 15  |-  A  C_  Prime
3231a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  A  C_  Prime )
338, 28, 29, 6, 11, 32ablfac1b 17247 . . . . . . . . . . . . 13  |-  ( ph  ->  G dom DProd  S )
34 fvex 5882 . . . . . . . . . . . . . . . . 17  |-  ( Base `  G )  e.  _V
358, 34eqeltri 2541 . . . . . . . . . . . . . . . 16  |-  B  e. 
_V
3635rabex 4607 . . . . . . . . . . . . . . 15  |-  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  e.  _V
3736, 29dmmpti 5716 . . . . . . . . . . . . . 14  |-  dom  S  =  A
3837a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  S  =  A )
3933, 38dprdf2 17166 . . . . . . . . . . . 12  |-  ( ph  ->  S : A --> (SubGrp `  G ) )
4039ffvelrnda 6032 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  e.  (SubGrp `  G )
)
41 ablfac.c . . . . . . . . . . . 12  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
42 ablfac.w . . . . . . . . . . . 12  |-  W  =  ( g  e.  (SubGrp `  G )  |->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  g ) } )
438, 41, 6, 11, 28, 2, 29, 42ablfaclem1 17262 . . . . . . . . . . 11  |-  ( ( S `  q )  e.  (SubGrp `  G
)  ->  ( W `  ( S `  q
) )  =  {
s  e. Word  C  | 
( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) } )
4440, 43syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  ( W `  ( S `  q ) )  =  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s
)  =  ( S `
 q ) ) } )
45 ssrab2 3581 . . . . . . . . . 10  |-  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) } 
C_ Word  C
4644, 45syl6eqss 3549 . . . . . . . . 9  |-  ( (
ph  /\  q  e.  A )  ->  ( W `  ( S `  q ) )  C_ Word  C )
47 dfss1 3699 . . . . . . . . 9  |-  ( ( W `  ( S `
 q ) ) 
C_ Word  C  <->  (Word  C  i^i  ( W `  ( S `
 q ) ) )  =  ( W `
 ( S `  q ) ) )
4846, 47sylib 196 . . . . . . . 8  |-  ( (
ph  /\  q  e.  A )  ->  (Word  C  i^i  ( W `  ( S `  q ) ) )  =  ( W `  ( S `
 q ) ) )
4927, 48syl5eqr 2512 . . . . . . 7  |-  ( (
ph  /\  q  e.  A )  ->  { y  e. Word  C  |  y  e.  ( W `  ( S `  q ) ) }  =  ( W `  ( S `
 q ) ) )
5049, 44eqtrd 2498 . . . . . 6  |-  ( (
ph  /\  q  e.  A )  ->  { y  e. Word  C  |  y  e.  ( W `  ( S `  q ) ) }  =  {
s  e. Word  C  | 
( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) } )
51 eqid 2457 . . . . . . . . 9  |-  ( Base `  ( Gs  ( S `  q ) ) )  =  ( Base `  ( Gs  ( S `  q ) ) )
52 eqid 2457 . . . . . . . . 9  |-  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  =  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }
536adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  G  e.  Abel )
54 eqid 2457 . . . . . . . . . . 11  |-  ( Gs  ( S `  q ) )  =  ( Gs  ( S `  q ) )
5554subgabl 16970 . . . . . . . . . 10  |-  ( ( G  e.  Abel  /\  ( S `  q )  e.  (SubGrp `  G )
)  ->  ( Gs  ( S `  q )
)  e.  Abel )
5653, 40, 55syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  q  e.  A )  ->  ( Gs  ( S `  q ) )  e.  Abel )
5732sselda 3499 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  q  e.  Prime )
5854subgbas 16331 . . . . . . . . . . . . . 14  |-  ( ( S `  q )  e.  (SubGrp `  G
)  ->  ( S `  q )  =  (
Base `  ( Gs  ( S `  q )
) ) )
5940, 58syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  =  ( Base `  ( Gs  ( S `  q ) ) ) )
6059fveq2d 5876 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( S `  q ) )  =  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) )
618, 28, 29, 6, 11, 32ablfac1a 17246 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( S `  q ) )  =  ( q ^ (
q  pCnt  ( # `  B
) ) ) )
6260, 61eqtr3d 2500 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( Base `  ( Gs  ( S `  q ) ) ) )  =  ( q ^ (
q  pCnt  ( # `  B
) ) ) )
6362oveq2d 6312 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) )  =  ( q  pCnt  (
q ^ ( q 
pCnt  ( # `  B
) ) ) ) )
6414adantr 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 B )  e.  NN )
6557, 64pccld 14385 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( # `  B
) )  e.  NN0 )
6665nn0zd 10988 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( # `  B
) )  e.  ZZ )
67 pcid 14407 . . . . . . . . . . . . . 14  |-  ( ( q  e.  Prime  /\  (
q  pCnt  ( # `  B
) )  e.  ZZ )  ->  ( q  pCnt  ( q ^ ( q 
pCnt  ( # `  B
) ) ) )  =  ( q  pCnt  (
# `  B )
) )
6857, 66, 67syl2anc 661 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( q ^ ( q  pCnt  (
# `  B )
) ) )  =  ( q  pCnt  ( # `
 B ) ) )
6963, 68eqtrd 2498 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) )  =  ( q  pCnt  ( # `
 B ) ) )
7069oveq2d 6312 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  (
q ^ ( q 
pCnt  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) ) )  =  ( q ^
( q  pCnt  ( # `
 B ) ) ) )
7162, 70eqtr4d 2501 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( Base `  ( Gs  ( S `  q ) ) ) )  =  ( q ^ (
q  pCnt  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) ) ) )
7254subggrp 16330 . . . . . . . . . . . 12  |-  ( ( S `  q )  e.  (SubGrp `  G
)  ->  ( Gs  ( S `  q )
)  e.  Grp )
7340, 72syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( Gs  ( S `  q ) )  e.  Grp )
7411adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  B  e.  Fin )
758subgss 16328 . . . . . . . . . . . . . 14  |-  ( ( S `  q )  e.  (SubGrp `  G
)  ->  ( S `  q )  C_  B
)
7640, 75syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  C_  B )
77 ssfi 7759 . . . . . . . . . . . . 13  |-  ( ( B  e.  Fin  /\  ( S `  q ) 
C_  B )  -> 
( S `  q
)  e.  Fin )
7874, 76, 77syl2anc 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  e.  Fin )
7959, 78eqeltrrd 2546 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( Base `  ( Gs  ( S `
 q ) ) )  e.  Fin )
8051pgpfi2 16752 . . . . . . . . . . 11  |-  ( ( ( Gs  ( S `  q ) )  e. 
Grp  /\  ( Base `  ( Gs  ( S `  q ) ) )  e.  Fin )  -> 
( q pGrp  ( Gs  ( S `  q ) )  <->  ( q  e. 
Prime  /\  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) )  =  ( q ^ ( q 
pCnt  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) ) ) ) ) )
8173, 79, 80syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  (
q pGrp  ( Gs  ( S `
 q ) )  <-> 
( q  e.  Prime  /\  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) )  =  ( q ^ ( q 
pCnt  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) ) ) ) ) )
8257, 71, 81mpbir2and 922 . . . . . . . . 9  |-  ( (
ph  /\  q  e.  A )  ->  q pGrp  ( Gs  ( S `  q ) ) )
8351, 52, 56, 82, 79pgpfac 17261 . . . . . . . 8  |-  ( (
ph  /\  q  e.  A )  ->  E. s  e. Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp 
i^i  ran pGrp  ) }  (
( Gs  ( S `  q ) ) dom DProd  s  /\  ( ( Gs  ( S `  q ) ) DProd  s )  =  ( Base `  ( Gs  ( S `  q ) ) ) ) )
84 ssrab2 3581 . . . . . . . . . . . . . 14  |-  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  C_  (SubGrp `  ( Gs  ( S `  q ) ) )
85 sswrd 12560 . . . . . . . . . . . . . 14  |-  ( { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp 
i^i  ran pGrp  ) }  C_  (SubGrp `  ( Gs  ( S `
 q ) ) )  -> Word  { r  e.  (SubGrp `  ( Gs  ( S `  q )
) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  C_ Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )
8684, 85ax-mp 5 . . . . . . . . . . . . 13  |- Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  C_ Word  (SubGrp `  ( Gs  ( S `  q ) ) )
8786sseli 3495 . . . . . . . . . . . 12  |-  ( s  e. Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp 
i^i  ran pGrp  ) }  ->  s  e. Word  (SubGrp `  ( Gs  ( S `  q )
) ) )
8840adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q )
) ) )  -> 
( S `  q
)  e.  (SubGrp `  G ) )
8988adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ( S `  q
)  e.  (SubGrp `  G ) )
9054subgdmdprd 17207 . . . . . . . . . . . . . . . . . . 19  |-  ( ( S `  q )  e.  (SubGrp `  G
)  ->  ( ( Gs  ( S `  q ) ) dom DProd  s  <->  ( G dom DProd  s  /\  ran  s  C_ 
~P ( S `  q ) ) ) )
9188, 90syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q )
) ) )  -> 
( ( Gs  ( S `
 q ) ) dom DProd  s  <->  ( G dom DProd  s  /\  ran  s  C_ 
~P ( S `  q ) ) ) )
9291simprbda 623 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  G dom DProd  s )
9391simplbda 624 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ran  s  C_  ~P ( S `  q ) )
9454, 89, 92, 93subgdprd 17208 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ( ( Gs  ( S `
 q ) ) DProd 
s )  =  ( G DProd  s ) )
9559ad2antrr 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ( S `  q
)  =  ( Base `  ( Gs  ( S `  q ) ) ) )
9695eqcomd 2465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ( Base `  ( Gs  ( S `  q ) ) )  =  ( S `  q ) )
9794, 96eqeq12d 2479 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ( ( ( Gs  ( S `  q ) ) DProd  s )  =  ( Base `  ( Gs  ( S `  q ) ) )  <->  ( G DProd  s )  =  ( S `
 q ) ) )
9897biimpd 207 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ( ( ( Gs  ( S `  q ) ) DProd  s )  =  ( Base `  ( Gs  ( S `  q ) ) )  ->  ( G DProd  s )  =  ( S `  q ) ) )
9998, 92jctild 543 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ( ( ( Gs  ( S `  q ) ) DProd  s )  =  ( Base `  ( Gs  ( S `  q ) ) )  ->  ( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) ) )
10099expimpd 603 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q )
) ) )  -> 
( ( ( Gs  ( S `  q ) ) dom DProd  s  /\  ( ( Gs  ( S `
 q ) ) DProd 
s )  =  (
Base `  ( Gs  ( S `  q )
) ) )  -> 
( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) ) )
10187, 100sylan2 474 . . . . . . . . . . 11  |-  ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp 
i^i  ran pGrp  ) } )  ->  ( ( ( Gs  ( S `  q
) ) dom DProd  s  /\  ( ( Gs  ( S `
 q ) ) DProd 
s )  =  (
Base `  ( Gs  ( S `  q )
) ) )  -> 
( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) ) )
102 oveq2 6304 . . . . . . . . . . . . . . . 16  |-  ( r  =  y  ->  (
( Gs  ( S `  q ) )s  r )  =  ( ( Gs  ( S `  q ) )s  y ) )
103102eleq1d 2526 . . . . . . . . . . . . . . 15  |-  ( r  =  y  ->  (
( ( Gs  ( S `
 q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) 
<->  ( ( Gs  ( S `
 q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) ) )
104103cbvrabv 3108 . . . . . . . . . . . . . 14  |-  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  =  { y  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) }
10554subsubg 16350 . . . . . . . . . . . . . . . . . . 19  |-  ( ( S `  q )  e.  (SubGrp `  G
)  ->  ( y  e.  (SubGrp `  ( Gs  ( S `  q )
) )  <->  ( y  e.  (SubGrp `  G )  /\  y  C_  ( S `
 q ) ) ) )
10640, 105syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  q  e.  A )  ->  (
y  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  <-> 
( y  e.  (SubGrp `  G )  /\  y  C_  ( S `  q
) ) ) )
107106simprbda 623 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) ) )  -> 
y  e.  (SubGrp `  G ) )
1081073adant3 1016 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) )  /\  (
( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) )  ->  y  e.  (SubGrp `  G ) )
109403ad2ant1 1017 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) )  /\  (
( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) )  ->  ( S `  q )  e.  (SubGrp `  G ) )
110106simplbda 624 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) ) )  -> 
y  C_  ( S `  q ) )
1111103adant3 1016 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) )  /\  (
( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) )  ->  y  C_  ( S `  q ) )
112 ressabs 14709 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( S `  q
)  e.  (SubGrp `  G )  /\  y  C_  ( S `  q
) )  ->  (
( Gs  ( S `  q ) )s  y )  =  ( Gs  y ) )
113109, 111, 112syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) )  /\  (
( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) )  ->  ( ( Gs  ( S `  q ) )s  y )  =  ( Gs  y ) )
114 simp3 998 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) )  /\  (
( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) )  ->  ( ( Gs  ( S `  q ) )s  y )  e.  (CycGrp 
i^i  ran pGrp  ) )
115113, 114eqeltrrd 2546 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) )  /\  (
( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) )  ->  ( Gs  y
)  e.  (CycGrp  i^i  ran pGrp  ) )
116 oveq2 6304 . . . . . . . . . . . . . . . . . 18  |-  ( r  =  y  ->  ( Gs  r )  =  ( Gs  y ) )
117116eleq1d 2526 . . . . . . . . . . . . . . . . 17  |-  ( r  =  y  ->  (
( Gs  r )  e.  (CycGrp  i^i  ran pGrp  )  <->  ( Gs  y
)  e.  (CycGrp  i^i  ran pGrp  ) ) )
118117, 41elrab2 3259 . . . . . . . . . . . . . . . 16  |-  ( y  e.  C  <->  ( y  e.  (SubGrp `  G )  /\  ( Gs  y )  e.  (CycGrp  i^i  ran pGrp  ) ) )
119108, 115, 118sylanbrc 664 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) )  /\  (
( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) )  ->  y  e.  C )
120119rabssdv 3576 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  q  e.  A )  ->  { y  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) }  C_  C )
121104, 120syl5eqss 3543 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  C_  C )
122 sswrd 12560 . . . . . . . . . . . . 13  |-  ( { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp 
i^i  ran pGrp  ) }  C_  C  -> Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp 
i^i  ran pGrp  ) }  C_ Word  C )
123121, 122syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  -> Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  C_ Word  C )
124123sselda 3499 . . . . . . . . . . 11  |-  ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp 
i^i  ran pGrp  ) } )  ->  s  e. Word  C
)
125101, 124jctild 543 . . . . . . . . . 10  |-  ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp 
i^i  ran pGrp  ) } )  ->  ( ( ( Gs  ( S `  q
) ) dom DProd  s  /\  ( ( Gs  ( S `
 q ) ) DProd 
s )  =  (
Base `  ( Gs  ( S `  q )
) ) )  -> 
( s  e. Word  C  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) ) ) )
126125expimpd 603 . . . . . . . . 9  |-  ( (
ph  /\  q  e.  A )  ->  (
( s  e. Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  /\  ( ( Gs  ( S `  q
) ) dom DProd  s  /\  ( ( Gs  ( S `
 q ) ) DProd 
s )  =  (
Base `  ( Gs  ( S `  q )
) ) ) )  ->  ( s  e. Word  C  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  ( S `
 q ) ) ) ) )
127126reximdv2 2928 . . . . . . . 8  |-  ( (
ph  /\  q  e.  A )  ->  ( E. s  e. Word  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  ( ( Gs  ( S `  q ) ) dom DProd  s  /\  ( ( Gs  ( S `
 q ) ) DProd 
s )  =  (
Base `  ( Gs  ( S `  q )
) ) )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) ) )
12883, 127mpd 15 . . . . . . 7  |-  ( (
ph  /\  q  e.  A )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  ( S `
 q ) ) )
129 rabn0 3814 . . . . . . 7  |-  ( { s  e. Word  C  | 
( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) }  =/=  (/)  <->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) )
130128, 129sylibr 212 . . . . . 6  |-  ( (
ph  /\  q  e.  A )  ->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) }  =/=  (/) )
13150, 130eqnetrd 2750 . . . . 5  |-  ( (
ph  /\  q  e.  A )  ->  { y  e. Word  C  |  y  e.  ( W `  ( S `  q ) ) }  =/=  (/) )
132 rabn0 3814 . . . . 5  |-  ( { y  e. Word  C  | 
y  e.  ( W `
 ( S `  q ) ) }  =/=  (/)  <->  E. y  e. Word  C
y  e.  ( W `
 ( S `  q ) ) )
133131, 132sylib 196 . . . 4  |-  ( (
ph  /\  q  e.  A )  ->  E. y  e. Word  C y  e.  ( W `  ( S `
 q ) ) )
134133ralrimiva 2871 . . 3  |-  ( ph  ->  A. q  e.  A  E. y  e. Word  C y  e.  ( W `  ( S `  q ) ) )
135 eleq1 2529 . . . 4  |-  ( y  =  ( f `  q )  ->  (
y  e.  ( W `
 ( S `  q ) )  <->  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )
136135ac6sfi 7782 . . 3  |-  ( ( A  e.  Fin  /\  A. q  e.  A  E. y  e. Word  C y  e.  ( W `  ( S `  q )
) )  ->  E. f
( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )
13726, 134, 136syl2anc 661 . 2  |-  ( ph  ->  E. f ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) ) )
138 sneq 4042 . . . . . . . . 9  |-  ( q  =  y  ->  { q }  =  { y } )
139 fveq2 5872 . . . . . . . . . 10  |-  ( q  =  y  ->  (
f `  q )  =  ( f `  y ) )
140139dmeqd 5215 . . . . . . . . 9  |-  ( q  =  y  ->  dom  ( f `  q
)  =  dom  (
f `  y )
)
141138, 140xpeq12d 5033 . . . . . . . 8  |-  ( q  =  y  ->  ( { q }  X.  dom  ( f `  q
) )  =  ( { y }  X.  dom  ( f `  y
) ) )
142141cbviunv 4371 . . . . . . 7  |-  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) )  =  U_ y  e.  A  ( { y }  X.  dom  ( f `  y
) )
14326adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  A  e.  Fin )
144 snfi 7615 . . . . . . . . . 10  |-  { y }  e.  Fin
145 simprl 756 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  f : A -->Word  C )
146145ffvelrnda 6032 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) ) )  /\  y  e.  A )  ->  ( f `  y
)  e. Word  C )
147 wrdf 12557 . . . . . . . . . . . 12  |-  ( ( f `  y )  e. Word  C  ->  (
f `  y ) : ( 0..^ (
# `  ( f `  y ) ) ) --> C )
148 fdm 5741 . . . . . . . . . . . 12  |-  ( ( f `  y ) : ( 0..^ (
# `  ( f `  y ) ) ) --> C  ->  dom  ( f `
 y )  =  ( 0..^ ( # `  ( f `  y
) ) ) )
149146, 147, 1483syl 20 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) ) )  /\  y  e.  A )  ->  dom  ( f `  y )  =  ( 0..^ ( # `  (
f `  y )
) ) )
150 fzofi 12086 . . . . . . . . . . 11  |-  ( 0..^ ( # `  (
f `  y )
) )  e.  Fin
151149, 150syl6eqel 2553 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) ) )  /\  y  e.  A )  ->  dom  ( f `  y )  e.  Fin )
152 xpfi 7809 . . . . . . . . . 10  |-  ( ( { y }  e.  Fin  /\  dom  ( f `
 y )  e. 
Fin )  ->  ( { y }  X.  dom  ( f `  y
) )  e.  Fin )
153144, 151, 152sylancr 663 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) ) )  /\  y  e.  A )  ->  ( { y }  X.  dom  ( f `
 y ) )  e.  Fin )
154153ralrimiva 2871 . . . . . . . 8  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  A. y  e.  A  ( {
y }  X.  dom  ( f `  y
) )  e.  Fin )
155 iunfi 7826 . . . . . . . 8  |-  ( ( A  e.  Fin  /\  A. y  e.  A  ( { y }  X.  dom  ( f `  y
) )  e.  Fin )  ->  U_ y  e.  A  ( { y }  X.  dom  ( f `  y
) )  e.  Fin )
156143, 154, 155syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  U_ y  e.  A  ( {
y }  X.  dom  ( f `  y
) )  e.  Fin )
157142, 156syl5eqel 2549 . . . . . 6  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) )  e.  Fin )
158 hashcl 12430 . . . . . 6  |-  ( U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) )  e.  Fin  ->  ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) )  e. 
NN0 )
159 hashfzo0 12491 . . . . . 6  |-  ( (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) )  e.  NN0  ->  ( # `  (
0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) ) )  =  ( # `  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) ) )
160157, 158, 1593syl 20 . . . . 5  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  ( # `
 ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) )  =  ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )
161 fzofi 12086 . . . . . 6  |-  ( 0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )  e.  Fin
162 hashen 12422 . . . . . 6  |-  ( ( ( 0..^ ( # `  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) ) )  e.  Fin  /\  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) )  e.  Fin )  ->  ( ( # `  ( 0..^ ( # `  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) ) ) )  =  ( # `  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) )  <->  ( 0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) ) 
~~  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) ) )
163161, 157, 162sylancr 663 . . . . 5  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  (
( # `  ( 0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) ) )  =  ( # `  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) )  <->  ( 0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) ) 
~~  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) ) )
164160, 163mpbid 210 . . . 4  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  (
0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) ) 
~~  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) )
165 bren 7544 . . . 4  |-  ( ( 0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) ) 
~~  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) )  <->  E. h  h : ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) -1-1-onto-> U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) )
166164, 165sylib 196 . . 3  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  E. h  h : ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) -1-1-onto-> U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) )
1676adantr 465 . . . . . 6  |-  ( (
ph  /\  ( (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) )  /\  h : ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) -1-1-onto-> U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )  ->  G  e.  Abel )
16811adantr 465 . . . . . 6  |-  ( (
ph  /\  ( (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) )  /\  h : ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) -1-1-onto-> U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )  ->  B  e.  Fin )
169 breq1 4459 . . . . . . . 8  |-  ( w  =  a  ->  (
w  ||  ( # `  B
)  <->  a  ||  ( # `
 B ) ) )
170169cbvrabv 3108 . . . . . . 7  |-  { w  e.  Prime  |  w  ||  ( # `  B ) }  =  { a  e.  Prime  |  a  ||  ( # `  B
) }
1712, 170eqtri 2486 . . . . . 6  |-  A  =  { a  e.  Prime  |  a  ||  ( # `  B ) }
172 fveq2 5872 . . . . . . . . . . 11  |-  ( x  =  c  ->  ( O `  x )  =  ( O `  c ) )
173172breq1d 4466 . . . . . . . . . 10  |-  ( x  =  c  ->  (
( O `  x
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) )  <->  ( O `  c )  ||  (
p ^ ( p 
pCnt  ( # `  B
) ) ) ) )
174173cbvrabv 3108 . . . . . . . . 9  |-  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  =  { c  e.  B  |  ( O `
 c )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }
175 id 22 . . . . . . . . . . . 12  |-  ( p  =  b  ->  p  =  b )
176 oveq1 6303 . . . . . . . . . . . 12  |-  ( p  =  b  ->  (
p  pCnt  ( # `  B
) )  =  ( b  pCnt  ( # `  B
) ) )
177175, 176oveq12d 6314 . . . . . . . . . . 11  |-  ( p  =  b  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  =  ( b ^ (
b  pCnt  ( # `  B
) ) ) )
178177breq2d 4468 . . . . . . . . . 10  |-  ( p  =  b  ->  (
( O `  c
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) )  <->  ( O `  c )  ||  (
b ^ ( b 
pCnt  ( # `  B
) ) ) ) )
179178rabbidv 3101 . . . . . . . . 9  |-  ( p  =  b  ->  { c  e.  B  |  ( O `  c ) 
||  ( p ^
( p  pCnt  ( # `
 B ) ) ) }  =  {
c  e.  B  | 
( O `  c
)  ||  ( b ^ ( b  pCnt  (
# `  B )
) ) } )
180174, 179syl5eq 2510 . . . . . . . 8  |-  ( p  =  b  ->  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  =  { c  e.  B  |  ( O `
 c )  ||  ( b ^ (
b  pCnt  ( # `  B
) ) ) } )
181180cbvmptv 4548 . . . . . . 7  |-  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) } )  =  ( b  e.  A  |->  { c  e.  B  |  ( O `  c ) 
||  ( b ^
( b  pCnt  ( # `
 B ) ) ) } )
18229, 181eqtri 2486 . . . . . 6  |-  S  =  ( b  e.  A  |->  { c  e.  B  |  ( O `  c )  ||  (
b ^ ( b 
pCnt  ( # `  B
) ) ) } )
183 breq2 4460 . . . . . . . . . 10  |-  ( s  =  t  ->  ( G dom DProd  s  <->  G dom DProd  t ) )
184 oveq2 6304 . . . . . . . . . . 11  |-  ( s  =  t  ->  ( G DProd  s )  =  ( G DProd  t ) )
185184eqeq1d 2459 . . . . . . . . . 10  |-  ( s  =  t  ->  (
( G DProd  s )  =  g  <->  ( G DProd  t
)  =  g ) )
186183, 185anbi12d 710 . . . . . . . . 9  |-  ( s  =  t  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  g )  <->  ( G dom DProd  t  /\  ( G DProd 
t )  =  g ) ) )
187186cbvrabv 3108 . . . . . . . 8  |-  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  g ) }  =  { t  e. Word  C  |  ( G dom DProd  t  /\  ( G DProd 
t )  =  g ) }
188187mpteq2i 4540 . . . . . . 7  |-  ( g  e.  (SubGrp `  G
)  |->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd 
s )  =  g ) } )  =  ( g  e.  (SubGrp `  G )  |->  { t  e. Word  C  |  ( G dom DProd  t  /\  ( G DProd  t )  =  g ) } )
18942, 188eqtri 2486 . . . . . 6  |-  W  =  ( g  e.  (SubGrp `  G )  |->  { t  e. Word  C  |  ( G dom DProd  t  /\  ( G DProd  t )  =  g ) } )
190 simprll 763 . . . . . 6  |-  ( (
ph  /\  ( (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) )  /\  h : ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) -1-1-onto-> U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )  ->  f : A -->Word  C )
191 simprlr 764 . . . . . . 7  |-  ( (
ph  /\  ( (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) )  /\  h : ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) -1-1-onto-> U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )  ->  A. q  e.  A  ( f `  q
)  e.  ( W `
 ( S `  q ) ) )
192 fveq2 5872 . . . . . . . . . 10  |-  ( q  =  y  ->  ( S `  q )  =  ( S `  y ) )
193192fveq2d 5876 . . . . . . . . 9  |-  ( q  =  y  ->  ( W `  ( S `  q ) )  =  ( W `  ( S `  y )
) )
194139, 193eleq12d 2539 . . . . . . . 8  |-  ( q  =  y  ->  (
( f `  q
)  e.  ( W `
 ( S `  q ) )  <->  ( f `  y )  e.  ( W `  ( S `
 y ) ) ) )
195194cbvralv 3084 . . . . . . 7  |-  ( A. q  e.  A  (
f `  q )  e.  ( W `  ( S `  q )
)  <->  A. y  e.  A  ( f `  y
)  e.  ( W `
 ( S `  y ) ) )
196191, 195sylib 196 . . . . . 6  |-  ( (
ph  /\  ( (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) )  /\  h : ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) -1-1-onto-> U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )  ->  A. y  e.  A  ( f `  y
)  e.  ( W `
 ( S `  y ) ) )
197 simprr 757 . . . . . 6  |-  ( (
ph  /\  ( (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) )  /\  h : ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) -1-1-onto-> U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )  ->  h : ( 0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) ) -1-1-onto-> U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) )
1988, 41, 167, 168, 28, 171, 182, 189, 190, 196, 142, 197ablfaclem2 17263 . . . . 5  |-  ( (
ph  /\  ( (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) )  /\  h : ( 0..^ (
# `  U_ q  e.  A  ( { q }  X.  dom  (
f `  q )
) ) ) -1-1-onto-> U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )  ->  ( W `  B )  =/=  (/) )
199198expr 615 . . . 4  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  (
h : ( 0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) ) -1-1-onto-> U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) )  ->  ( W `  B )  =/=  (/) ) )
200199exlimdv 1725 . . 3  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  ( E. h  h :
( 0..^ ( # `  U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) ) ) ) -1-1-onto-> U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) )  ->  ( W `  B )  =/=  (/) ) )
201166, 200mpd 15 . 2  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  ( W `  B )  =/=  (/) )
202137, 201exlimddv 1727 1  |-  ( ph  ->  ( W `  B
)  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3109    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   {csn 4032   U_ciun 4332   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   dom cdm 5008   ran crn 5009   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296    ~~ cen 7532   Fincfn 7535   0cc0 9509   1c1 9510    <_ cle 9646   NNcn 10556   NN0cn0 10816   ZZcz 10885   ...cfz 11697  ..^cfzo 11820   ^cexp 12168   #chash 12407  Word cword 12537    || cdvds 13997   Primecprime 14228    pCnt cpc 14371   Basecbs 14643   ↾s cress 14644   Grpcgrp 16179  SubGrpcsubg 16321   odcod 16675   pGrp cpgp 16677   Abelcabl 16925  CycGrpccyg 17006   DProd cdprd 17150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-rpss 6579  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-er 7329  df-ec 7331  df-qs 7335  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-fz 11698  df-fzo 11821  df-fl 11931  df-mod 11999  df-seq 12110  df-exp 12169  df-fac 12356  df-bc 12383  df-hash 12408  df-word 12545  df-concat 12547  df-s1 12548  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322  df-sum 13520  df-dvds 13998  df-gcd 14156  df-prm 14229  df-pc 14372  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-0g 14858  df-gsum 14859  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-mhm 16092  df-submnd 16093  df-grp 16183  df-minusg 16184  df-sbg 16185  df-mulg 16186  df-subg 16324  df-eqg 16326  df-ghm 16391  df-gim 16433  df-ga 16454  df-cntz 16481  df-oppg 16507  df-od 16679  df-gex 16680  df-pgp 16681  df-lsm 16782  df-pj1 16783  df-cmn 16926  df-abl 16927  df-cyg 17007  df-dprd 17152
This theorem is referenced by:  ablfac  17265
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