MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ablfaclem3 Unicode version

Theorem ablfaclem3 15324
Description: Lemma for ablfac 15325. (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 11037 . . . 4  |-  ( ph  ->  ( 1 ... ( # `
 B ) )  e.  Fin )
2 ablfac.a . . . . 5  |-  A  =  { w  e.  Prime  |  w  ||  ( # `  B ) }
3 prmnn 12763 . . . . . . . 8  |-  ( w  e.  Prime  ->  w  e.  NN )
433ad2ant2 977 . . . . . . 7  |-  ( (
ph  /\  w  e.  Prime  /\  w  ||  ( # `
 B ) )  ->  w  e.  NN )
5 prmz 12764 . . . . . . . . 9  |-  ( w  e.  Prime  ->  w  e.  ZZ )
6 ablfac.1 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  Abel )
7 ablgrp 15096 . . . . . . . . . . 11  |-  ( G  e.  Abel  ->  G  e. 
Grp )
8 ablfac.b . . . . . . . . . . . 12  |-  B  =  ( Base `  G
)
98grpbn0 14513 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  B  =/=  (/) )
106, 7, 93syl 18 . . . . . . . . . 10  |-  ( ph  ->  B  =/=  (/) )
11 ablfac.2 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  Fin )
12 hashnncl 11356 . . . . . . . . . . 11  |-  ( B  e.  Fin  ->  (
( # `  B )  e.  NN  <->  B  =/=  (/) ) )
1311, 12syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( ( # `  B
)  e.  NN  <->  B  =/=  (/) ) )
1410, 13mpbird 223 . . . . . . . . 9  |-  ( ph  ->  ( # `  B
)  e.  NN )
15 dvdsle 12576 . . . . . . . . 9  |-  ( ( w  e.  ZZ  /\  ( # `  B )  e.  NN )  -> 
( w  ||  ( # `
 B )  ->  w  <_  ( # `  B
) ) )
165, 14, 15syl2anr 464 . . . . . . . 8  |-  ( (
ph  /\  w  e.  Prime )  ->  ( w  ||  ( # `  B
)  ->  w  <_  (
# `  B )
) )
17163impia 1148 . . . . . . 7  |-  ( (
ph  /\  w  e.  Prime  /\  w  ||  ( # `
 B ) )  ->  w  <_  ( # `
 B ) )
1814nnzd 10118 . . . . . . . . 9  |-  ( ph  ->  ( # `  B
)  e.  ZZ )
19183ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  w  e.  Prime  /\  w  ||  ( # `
 B ) )  ->  ( # `  B
)  e.  ZZ )
20 fznn 10854 . . . . . . . 8  |-  ( (
# `  B )  e.  ZZ  ->  ( w  e.  ( 1 ... ( # `
 B ) )  <-> 
( w  e.  NN  /\  w  <_  ( # `  B
) ) ) )
2119, 20syl 15 . . . . . . 7  |-  ( (
ph  /\  w  e.  Prime  /\  w  ||  ( # `
 B ) )  ->  ( w  e.  ( 1 ... ( # `
 B ) )  <-> 
( w  e.  NN  /\  w  <_  ( # `  B
) ) ) )
224, 17, 21mpbir2and 888 . . . . . 6  |-  ( (
ph  /\  w  e.  Prime  /\  w  ||  ( # `
 B ) )  ->  w  e.  ( 1 ... ( # `  B ) ) )
2322rabssdv 3255 . . . . 5  |-  ( ph  ->  { w  e.  Prime  |  w  ||  ( # `  B ) }  C_  ( 1 ... ( # `
 B ) ) )
242, 23syl5eqss 3224 . . . 4  |-  ( ph  ->  A  C_  ( 1 ... ( # `  B
) ) )
25 ssfi 7085 . . . 4  |-  ( ( ( 1 ... ( # `
 B ) )  e.  Fin  /\  A  C_  ( 1 ... ( # `
 B ) ) )  ->  A  e.  Fin )
261, 24, 25syl2anc 642 . . 3  |-  ( ph  ->  A  e.  Fin )
27 dfin5 3162 . . . . . . . 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 3260 . . . . . . . . . . . . . . . 16  |-  { w  e.  Prime  |  w  ||  ( # `  B ) }  C_  Prime
312, 30eqsstri 3210 . . . . . . . . . . . . . . 15  |-  A  C_  Prime
3231a1i 10 . . . . . . . . . . . . . 14  |-  ( ph  ->  A  C_  Prime )
338, 28, 29, 6, 11, 32ablfac1b 15307 . . . . . . . . . . . . 13  |-  ( ph  ->  G dom DProd  S )
34 fvex 5541 . . . . . . . . . . . . . . . . 17  |-  ( Base `  G )  e.  _V
358, 34eqeltri 2355 . . . . . . . . . . . . . . . 16  |-  B  e. 
_V
3635rabex 4167 . . . . . . . . . . . . . . 15  |-  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  e.  _V
3736, 29dmmpti 5375 . . . . . . . . . . . . . 14  |-  dom  S  =  A
3837a1i 10 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  S  =  A )
3933, 38dprdf2 15244 . . . . . . . . . . . 12  |-  ( ph  ->  S : A --> (SubGrp `  G ) )
40 ffvelrn 5665 . . . . . . . . . . . 12  |-  ( ( S : A --> (SubGrp `  G )  /\  q  e.  A )  ->  ( S `  q )  e.  (SubGrp `  G )
)
4139, 40sylan 457 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  e.  (SubGrp `  G )
)
42 ablfac.c . . . . . . . . . . . 12  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
43 ablfac.w . . . . . . . . . . . 12  |-  W  =  ( g  e.  (SubGrp `  G )  |->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  g ) } )
448, 42, 6, 11, 28, 2, 29, 43ablfaclem1 15322 . . . . . . . . . . 11  |-  ( ( S `  q )  e.  (SubGrp `  G
)  ->  ( W `  ( S `  q
) )  =  {
s  e. Word  C  | 
( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) } )
4541, 44syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  ( W `  ( S `  q ) )  =  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s
)  =  ( S `
 q ) ) } )
46 ssrab2 3260 . . . . . . . . . . 11  |-  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) } 
C_ Word  C
4746a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) } 
C_ Word  C )
4845, 47eqsstrd 3214 . . . . . . . . 9  |-  ( (
ph  /\  q  e.  A )  ->  ( W `  ( S `  q ) )  C_ Word  C )
49 dfss1 3375 . . . . . . . . 9  |-  ( ( W `  ( S `
 q ) ) 
C_ Word  C  <->  (Word  C  i^i  ( W `  ( S `
 q ) ) )  =  ( W `
 ( S `  q ) ) )
5048, 49sylib 188 . . . . . . . 8  |-  ( (
ph  /\  q  e.  A )  ->  (Word  C  i^i  ( W `  ( S `  q ) ) )  =  ( W `  ( S `
 q ) ) )
5127, 50syl5eqr 2331 . . . . . . 7  |-  ( (
ph  /\  q  e.  A )  ->  { y  e. Word  C  |  y  e.  ( W `  ( S `  q ) ) }  =  ( W `  ( S `
 q ) ) )
5251, 45eqtrd 2317 . . . . . 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 ) ) } )
53 eqid 2285 . . . . . . . . 9  |-  ( Base `  ( Gs  ( S `  q ) ) )  =  ( Base `  ( Gs  ( S `  q ) ) )
54 eqid 2285 . . . . . . . . 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  ) }
556adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  G  e.  Abel )
56 eqid 2285 . . . . . . . . . . 11  |-  ( Gs  ( S `  q ) )  =  ( Gs  ( S `  q ) )
5756subgabl 15134 . . . . . . . . . 10  |-  ( ( G  e.  Abel  /\  ( S `  q )  e.  (SubGrp `  G )
)  ->  ( Gs  ( S `  q )
)  e.  Abel )
5855, 41, 57syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  q  e.  A )  ->  ( Gs  ( S `  q ) )  e.  Abel )
5932sselda 3182 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  q  e.  Prime )
6056subgbas 14627 . . . . . . . . . . . . . 14  |-  ( ( S `  q )  e.  (SubGrp `  G
)  ->  ( S `  q )  =  (
Base `  ( Gs  ( S `  q )
) ) )
6141, 60syl 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  =  ( Base `  ( Gs  ( S `  q ) ) ) )
6261fveq2d 5531 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( S `  q ) )  =  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) )
638, 28, 29, 6, 11, 32ablfac1a 15306 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( S `  q ) )  =  ( q ^ (
q  pCnt  ( # `  B
) ) ) )
6462, 63eqtr3d 2319 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( Base `  ( Gs  ( S `  q ) ) ) )  =  ( q ^ (
q  pCnt  ( # `  B
) ) ) )
6564oveq2d 5876 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) )  =  ( q  pCnt  (
q ^ ( q 
pCnt  ( # `  B
) ) ) ) )
6614adantr 451 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 B )  e.  NN )
6759, 66pccld 12905 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( # `  B
) )  e.  NN0 )
6867nn0zd 10117 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( # `  B
) )  e.  ZZ )
69 pcid 12927 . . . . . . . . . . . . . 14  |-  ( ( q  e.  Prime  /\  (
q  pCnt  ( # `  B
) )  e.  ZZ )  ->  ( q  pCnt  ( q ^ ( q 
pCnt  ( # `  B
) ) ) )  =  ( q  pCnt  (
# `  B )
) )
7059, 68, 69syl2anc 642 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( q ^ ( q  pCnt  (
# `  B )
) ) )  =  ( q  pCnt  ( # `
 B ) ) )
7165, 70eqtrd 2317 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) )  =  ( q  pCnt  ( # `
 B ) ) )
7271oveq2d 5876 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  (
q ^ ( q 
pCnt  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) ) )  =  ( q ^
( q  pCnt  ( # `
 B ) ) ) )
7364, 72eqtr4d 2320 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( Base `  ( Gs  ( S `  q ) ) ) )  =  ( q ^ (
q  pCnt  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) ) ) )
7456subggrp 14626 . . . . . . . . . . . 12  |-  ( ( S `  q )  e.  (SubGrp `  G
)  ->  ( Gs  ( S `  q )
)  e.  Grp )
7541, 74syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( Gs  ( S `  q ) )  e.  Grp )
7611adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  B  e.  Fin )
778subgss 14624 . . . . . . . . . . . . . 14  |-  ( ( S `  q )  e.  (SubGrp `  G
)  ->  ( S `  q )  C_  B
)
7841, 77syl 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  C_  B )
79 ssfi 7085 . . . . . . . . . . . . 13  |-  ( ( B  e.  Fin  /\  ( S `  q ) 
C_  B )  -> 
( S `  q
)  e.  Fin )
8076, 78, 79syl2anc 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  e.  Fin )
8161, 80eqeltrrd 2360 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( Base `  ( Gs  ( S `
 q ) ) )  e.  Fin )
8253pgpfi2 14919 . . . . . . . . . . 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 ) ) ) ) ) ) ) ) )
8375, 81, 82syl2anc 642 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  (
q pGrp  ( Gs  ( S `
 q ) )  <-> 
( q  e.  Prime  /\  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) )  =  ( q ^ ( q 
pCnt  ( # `  ( Base `  ( Gs  ( S `
 q ) ) ) ) ) ) ) ) )
8459, 73, 83mpbir2and 888 . . . . . . . . 9  |-  ( (
ph  /\  q  e.  A )  ->  q pGrp  ( Gs  ( S `  q ) ) )
8553, 54, 58, 84, 81pgpfac 15321 . . . . . . . 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 ) ) ) ) )
86 ssrab2 3260 . . . . . . . . . . . . . 14  |-  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  C_  (SubGrp `  ( Gs  ( S `  q ) ) )
87 sswrd 11425 . . . . . . . . . . . . . 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 ) ) ) )
8886, 87ax-mp 8 . . . . . . . . . . . . 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 ) ) )
8988sseli 3178 . . . . . . . . . . . 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 )
) ) )
9041adantr 451 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q )
) ) )  -> 
( S `  q
)  e.  (SubGrp `  G ) )
9190adantr 451 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ( S `  q
)  e.  (SubGrp `  G ) )
9256subgdmdprd 15271 . . . . . . . . . . . . . . . . . . 19  |-  ( ( S `  q )  e.  (SubGrp `  G
)  ->  ( ( Gs  ( S `  q ) ) dom DProd  s  <->  ( G dom DProd  s  /\  ran  s  C_ 
~P ( S `  q ) ) ) )
9390, 92syl 15 . . . . . . . . . . . . . . . . . 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 ) ) ) )
9493simprbda 606 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  G dom DProd  s )
9593simplbda 607 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ran  s  C_  ~P ( S `  q ) )
9656, 91, 94, 95subgdprd 15272 . . . . . . . . . . . . . . . 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 ) )
9761ad2antrr 706 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ( S `  q
)  =  ( Base `  ( Gs  ( S `  q ) ) ) )
9897eqcomd 2290 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  q  e.  A )  /\  s  e. Word  (SubGrp `  ( Gs  ( S `  q ) ) ) )  /\  ( Gs  ( S `  q ) ) dom DProd  s )  ->  ( Base `  ( Gs  ( S `  q ) ) )  =  ( S `  q ) )
9996, 98eqeq12d 2299 . . . . . . . . . . . . . . 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 ) ) )
10099biimpd 198 . . . . . . . . . . . . . 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 ) ) )
101100, 94jctild 527 . . . . . . . . . . . . 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 ) ) ) )
102101expimpd 586 . . . . . . . . . . . 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 ) ) ) )
10389, 102sylan2 460 . . . . . . . . . . 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 ) ) ) )
104 oveq2 5868 . . . . . . . . . . . . . . . 16  |-  ( r  =  y  ->  (
( Gs  ( S `  q ) )s  r )  =  ( ( Gs  ( S `  q ) )s  y ) )
105104eleq1d 2351 . . . . . . . . . . . . . . 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  ) ) )
106105cbvrabv 2789 . . . . . . . . . . . . . 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  ) }
10756subsubg 14642 . . . . . . . . . . . . . . . . . . 19  |-  ( ( S `  q )  e.  (SubGrp `  G
)  ->  ( y  e.  (SubGrp `  ( Gs  ( S `  q )
) )  <->  ( y  e.  (SubGrp `  G )  /\  y  C_  ( S `
 q ) ) ) )
10841, 107syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  q  e.  A )  ->  (
y  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  <-> 
( y  e.  (SubGrp `  G )  /\  y  C_  ( S `  q
) ) ) )
109108simprbda 606 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) ) )  -> 
y  e.  (SubGrp `  G ) )
1101093adant3 975 . . . . . . . . . . . . . . . 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 ) )
111413ad2ant1 976 . . . . . . . . . . . . . . . . . 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 ) )
112108simplbda 607 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  q  e.  A )  /\  y  e.  (SubGrp `  ( Gs  ( S `  q )
) ) )  -> 
y  C_  ( S `  q ) )
1131123adant3 975 . . . . . . . . . . . . . . . . . 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 ) )
114 ressabs 13208 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( S `  q
)  e.  (SubGrp `  G )  /\  y  C_  ( S `  q
) )  ->  (
( Gs  ( S `  q ) )s  y )  =  ( Gs  y ) )
115111, 113, 114syl2anc 642 . . . . . . . . . . . . . . . . 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 ) )
116 simp3 957 . . . . . . . . . . . . . . . . 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  ) )
117115, 116eqeltrrd 2360 . . . . . . . . . . . . . . . 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  ) )
118 oveq2 5868 . . . . . . . . . . . . . . . . . 18  |-  ( r  =  y  ->  ( Gs  r )  =  ( Gs  y ) )
119118eleq1d 2351 . . . . . . . . . . . . . . . . 17  |-  ( r  =  y  ->  (
( Gs  r )  e.  (CycGrp  i^i  ran pGrp  )  <->  ( Gs  y
)  e.  (CycGrp  i^i  ran pGrp  ) ) )
120119, 42elrab2 2927 . . . . . . . . . . . . . . . 16  |-  ( y  e.  C  <->  ( y  e.  (SubGrp `  G )  /\  ( Gs  y )  e.  (CycGrp  i^i  ran pGrp  ) ) )
121110, 117, 120sylanbrc 645 . . . . . . . . . . . . . . 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 )
122121rabssdv 3255 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  q  e.  A )  ->  { y  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  y )  e.  (CycGrp  i^i  ran pGrp  ) }  C_  C )
123106, 122syl5eqss 3224 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  { r  e.  (SubGrp `  ( Gs  ( S `  q ) ) )  |  ( ( Gs  ( S `  q ) )s  r )  e.  (CycGrp  i^i  ran pGrp  ) }  C_  C )
124 sswrd 11425 . . . . . . . . . . . . 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 )
125123, 124syl 15 . . . . . . . . . . . 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 )
126125sselda 3182 . . . . . . . . . . 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
)
127103, 126jctild 527 . . . . . . . . . 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 ) ) ) ) )
128127expimpd 586 . . . . . . . . 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 ) ) ) ) )
129128reximdv2 2654 . . . . . . . 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 ) ) ) )
13085, 129mpd 14 . . . . . . 7  |-  ( (
ph  /\  q  e.  A )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  ( S `
 q ) ) )
131 rabn0 3476 . . . . . . 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 ) ) )
132130, 131sylibr 203 . . . . . 6  |-  ( (
ph  /\  q  e.  A )  ->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  ( S `  q ) ) }  =/=  (/) )
13352, 132eqnetrd 2466 . . . . 5  |-  ( (
ph  /\  q  e.  A )  ->  { y  e. Word  C  |  y  e.  ( W `  ( S `  q ) ) }  =/=  (/) )
134 rabn0 3476 . . . . 5  |-  ( { y  e. Word  C  | 
y  e.  ( W `
 ( S `  q ) ) }  =/=  (/)  <->  E. y  e. Word  C
y  e.  ( W `
 ( S `  q ) ) )
135133, 134sylib 188 . . . 4  |-  ( (
ph  /\  q  e.  A )  ->  E. y  e. Word  C y  e.  ( W `  ( S `
 q ) ) )
136135ralrimiva 2628 . . 3  |-  ( ph  ->  A. q  e.  A  E. y  e. Word  C y  e.  ( W `  ( S `  q ) ) )
137 eleq1 2345 . . . 4  |-  ( y  =  ( f `  q )  ->  (
y  e.  ( W `
 ( S `  q ) )  <->  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )
138137ac6sfi 7103 . . 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 ) ) ) )
13926, 136, 138syl2anc 642 . 2  |-  ( ph  ->  E. f ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) ) )
140 sneq 3653 . . . . . . . . . . 11  |-  ( q  =  y  ->  { q }  =  { y } )
141 fveq2 5527 . . . . . . . . . . . 12  |-  ( q  =  y  ->  (
f `  q )  =  ( f `  y ) )
142141dmeqd 4883 . . . . . . . . . . 11  |-  ( q  =  y  ->  dom  ( f `  q
)  =  dom  (
f `  y )
)
143140, 142xpeq12d 4716 . . . . . . . . . 10  |-  ( q  =  y  ->  ( { q }  X.  dom  ( f `  q
) )  =  ( { y }  X.  dom  ( f `  y
) ) )
144143cbviunv 3943 . . . . . . . . 9  |-  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) )  =  U_ y  e.  A  ( { y }  X.  dom  ( f `  y
) )
14526adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  A  e.  Fin )
146 snfi 6943 . . . . . . . . . . . 12  |-  { y }  e.  Fin
147 simprl 732 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  f : A -->Word  C )
148 ffvelrn 5665 . . . . . . . . . . . . . . 15  |-  ( ( f : A -->Word  C  /\  y  e.  A )  ->  ( f `  y
)  e. Word  C )
149147, 148sylan 457 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) ) )  /\  y  e.  A )  ->  ( f `  y
)  e. Word  C )
150 wrdf 11421 . . . . . . . . . . . . . 14  |-  ( ( f `  y )  e. Word  C  ->  (
f `  y ) : ( 0..^ (
# `  ( f `  y ) ) ) --> C )
151 fdm 5395 . . . . . . . . . . . . . 14  |-  ( ( f `  y ) : ( 0..^ (
# `  ( f `  y ) ) ) --> C  ->  dom  ( f `
 y )  =  ( 0..^ ( # `  ( f `  y
) ) ) )
152149, 150, 1513syl 18 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) ) )  /\  y  e.  A )  ->  dom  ( f `  y )  =  ( 0..^ ( # `  (
f `  y )
) ) )
153 fzofi 11038 . . . . . . . . . . . . 13  |-  ( 0..^ ( # `  (
f `  y )
) )  e.  Fin
154152, 153syl6eqel 2373 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) ) )  /\  y  e.  A )  ->  dom  ( f `  y )  e.  Fin )
155 xpfi 7130 . . . . . . . . . . . 12  |-  ( ( { y }  e.  Fin  /\  dom  ( f `
 y )  e. 
Fin )  ->  ( { y }  X.  dom  ( f `  y
) )  e.  Fin )
156146, 154, 155sylancr 644 . . . . . . . . . . 11  |-  ( ( ( 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 )
157156ralrimiva 2628 . . . . . . . . . 10  |-  ( (
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 )
158 iunfi 7146 . . . . . . . . . 10  |-  ( ( 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 )
159145, 157, 158syl2anc 642 . . . . . . . . 9  |-  ( (
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 )
160144, 159syl5eqel 2369 . . . . . . . 8  |-  ( (
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 )
161 hashcl 11352 . . . . . . . 8  |-  ( U_ q  e.  A  ( { q }  X.  dom  ( f `  q
) )  e.  Fin  ->  ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) )  e. 
NN0 )
162 hashfzo0 11386 . . . . . . . 8  |-  ( (
# `  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
) ) ) )
163160, 161, 1623syl 18 . . . . . . 7  |-  ( (
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
) ) ) )
164 fzofi 11038 . . . . . . . 8  |-  ( 0..^ ( # `  U_ q  e.  A  ( {
q }  X.  dom  ( f `  q
) ) ) )  e.  Fin
165 hashen 11348 . . . . . . . 8  |-  ( ( ( 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
) ) ) )
166164, 160, 165sylancr 644 . . . . . . 7  |-  ( (
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
) ) ) )
167163, 166mpbid 201 . . . . . 6  |-  ( (
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
) ) )
168 bren 6873 . . . . . 6  |-  ( ( 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
) ) )
169167, 168sylib 188 . . . . 5  |-  ( (
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
) ) )
1706adantr 451 . . . . . . . 8  |-  ( (
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 )
17111adantr 451 . . . . . . . 8  |-  ( (
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 )
172 breq1 4028 . . . . . . . . . 10  |-  ( w  =  a  ->  (
w  ||  ( # `  B
)  <->  a  ||  ( # `
 B ) ) )
173172cbvrabv 2789 . . . . . . . . 9  |-  { w  e.  Prime  |  w  ||  ( # `  B ) }  =  { a  e.  Prime  |  a  ||  ( # `  B
) }
1742, 173eqtri 2305 . . . . . . . 8  |-  A  =  { a  e.  Prime  |  a  ||  ( # `  B ) }
175 fveq2 5527 . . . . . . . . . . . . 13  |-  ( x  =  c  ->  ( O `  x )  =  ( O `  c ) )
176175breq1d 4035 . . . . . . . . . . . 12  |-  ( x  =  c  ->  (
( O `  x
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) )  <->  ( O `  c )  ||  (
p ^ ( p 
pCnt  ( # `  B
) ) ) ) )
177176cbvrabv 2789 . . . . . . . . . . 11  |-  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  =  { c  e.  B  |  ( O `
 c )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }
178 id 19 . . . . . . . . . . . . . 14  |-  ( p  =  b  ->  p  =  b )
179 oveq1 5867 . . . . . . . . . . . . . 14  |-  ( p  =  b  ->  (
p  pCnt  ( # `  B
) )  =  ( b  pCnt  ( # `  B
) ) )
180178, 179oveq12d 5878 . . . . . . . . . . . . 13  |-  ( p  =  b  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  =  ( b ^ (
b  pCnt  ( # `  B
) ) ) )
181180breq2d 4037 . . . . . . . . . . . 12  |-  ( p  =  b  ->  (
( O `  c
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) )  <->  ( O `  c )  ||  (
b ^ ( b 
pCnt  ( # `  B
) ) ) ) )
182181rabbidv 2782 . . . . . . . . . . 11  |-  ( p  =  b  ->  { c  e.  B  |  ( O `  c ) 
||  ( p ^
( p  pCnt  ( # `
 B ) ) ) }  =  {
c  e.  B  | 
( O `  c
)  ||  ( b ^ ( b  pCnt  (
# `  B )
) ) } )
183177, 182syl5eq 2329 . . . . . . . . . 10  |-  ( p  =  b  ->  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  =  { c  e.  B  |  ( O `
 c )  ||  ( b ^ (
b  pCnt  ( # `  B
) ) ) } )
184183cbvmptv 4113 . . . . . . . . 9  |-  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) } )  =  ( b  e.  A  |->  { c  e.  B  |  ( O `  c ) 
||  ( b ^
( b  pCnt  ( # `
 B ) ) ) } )
18529, 184eqtri 2305 . . . . . . . 8  |-  S  =  ( b  e.  A  |->  { c  e.  B  |  ( O `  c )  ||  (
b ^ ( b 
pCnt  ( # `  B
) ) ) } )
186 breq2 4029 . . . . . . . . . . . 12  |-  ( s  =  t  ->  ( G dom DProd  s  <->  G dom DProd  t ) )
187 oveq2 5868 . . . . . . . . . . . . 13  |-  ( s  =  t  ->  ( G DProd  s )  =  ( G DProd  t ) )
188187eqeq1d 2293 . . . . . . . . . . . 12  |-  ( s  =  t  ->  (
( G DProd  s )  =  g  <->  ( G DProd  t
)  =  g ) )
189186, 188anbi12d 691 . . . . . . . . . . 11  |-  ( s  =  t  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  g )  <->  ( G dom DProd  t  /\  ( G DProd 
t )  =  g ) ) )
190189cbvrabv 2789 . . . . . . . . . 10  |-  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  g ) }  =  { t  e. Word  C  |  ( G dom DProd  t  /\  ( G DProd 
t )  =  g ) }
191190mpteq2i 4105 . . . . . . . . 9  |-  ( 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 ) } )
19243, 191eqtri 2305 . . . . . . . 8  |-  W  =  ( g  e.  (SubGrp `  G )  |->  { t  e. Word  C  |  ( G dom DProd  t  /\  ( G DProd  t )  =  g ) } )
193 simprll 738 . . . . . . . 8  |-  ( (
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 )
194 simprlr 739 . . . . . . . . 9  |-  ( (
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 ) ) )
195 fveq2 5527 . . . . . . . . . . . 12  |-  ( q  =  y  ->  ( S `  q )  =  ( S `  y ) )
196195fveq2d 5531 . . . . . . . . . . 11  |-  ( q  =  y  ->  ( W `  ( S `  q ) )  =  ( W `  ( S `  y )
) )
197141, 196eleq12d 2353 . . . . . . . . . 10  |-  ( q  =  y  ->  (
( f `  q
)  e.  ( W `
 ( S `  q ) )  <->  ( f `  y )  e.  ( W `  ( S `
 y ) ) ) )
198197cbvralv 2766 . . . . . . . . 9  |-  ( A. q  e.  A  (
f `  q )  e.  ( W `  ( S `  q )
)  <->  A. y  e.  A  ( f `  y
)  e.  ( W `
 ( S `  y ) ) )
199194, 198sylib 188 . . . . . . . 8  |-  ( (
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 ) ) )
200 simprr 733 . . . . . . . 8  |-  ( (
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
) ) )
2018, 42, 170, 171, 28, 174, 185, 192, 193, 199, 144, 200ablfaclem2 15323 . . . . . . 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
) ) ) )  ->  ( W `  B )  =/=  (/) )
202201expr 598 . . . . . 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
) )  ->  ( W `  B )  =/=  (/) ) )
203202exlimdv 1666 . . . . 5  |-  ( (
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 )  =/=  (/) ) )
204169, 203mpd 14 . . . 4  |-  ( (
ph  /\  ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) ) )  ->  ( W `  B )  =/=  (/) )
205204ex 423 . . 3  |-  ( ph  ->  ( ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `
 q ) ) )  ->  ( W `  B )  =/=  (/) ) )
206205exlimdv 1666 . 2  |-  ( ph  ->  ( E. f ( f : A -->Word  C  /\  A. q  e.  A  ( f `  q )  e.  ( W `  ( S `  q ) ) )  ->  ( W `  B )  =/=  (/) ) )
207139, 206mpd 14 1  |-  ( ph  ->  ( W `  B
)  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1530    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   E.wrex 2546   {crab 2549   _Vcvv 2790    i^i cin 3153    C_ wss 3154   (/)c0 3457   ~Pcpw 3627   {csn 3642   U_ciun 3907   class class class wbr 4025    e. cmpt 4079    X. cxp 4689   dom cdm 4691   ran crn 4692   -->wf 5253   -1-1-onto->wf1o 5256   ` cfv 5257  (class class class)co 5860    ~~ cen 6862   Fincfn 6865   0cc0 8739   1c1 8740    <_ cle 8870   NNcn 9748   NN0cn0 9967   ZZcz 10026   ...cfz 10784  ..^cfzo 10872   ^cexp 11106   #chash 11339  Word cword 11405    || cdivides 12533   Primecprime 12760    pCnt cpc 12891   Basecbs 13150   ↾s cress 13151   Grpcgrp 14364  SubGrpcsubg 14617   odcod 14842   pGrp cpgp 14844   Abelcabel 15092  CycGrpccyg 15166   DProd cdprd 15233
This theorem is referenced by:  ablfac  15325
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-disj 3996  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-tpos 6236  df-rpss 6279  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-omul 6486  df-er 6662  df-ec 6664  df-qs 6668  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-sup 7196  df-oi 7227  df-card 7574  df-acn 7577  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-n0 9968  df-z 10027  df-uz 10233  df-q 10319  df-rp 10357  df-fz 10785  df-fzo 10873  df-fl 10927  df-mod 10976  df-seq 11049  df-exp 11107  df-fac 11291  df-bc 11318  df-hash 11340  df-word 11411  df-concat 11412  df-s1 11413  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-clim 11964  df-sum 12161  df-dvds 12534  df-gcd 12688  df-prm 12761  df-pc 12892  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-0g 13406  df-gsum 13407  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-mhm 14417  df-submnd 14418  df-grp 14491  df-minusg 14492  df-sbg 14493  df-mulg 14494  df-subg 14620  df-eqg 14622  df-ghm 14683  df-gim 14725  df-ga 14746  df-cntz 14795  df-oppg 14821  df-od 14846  df-gex 14847  df-pgp 14848  df-lsm 14949  df-pj1 14950  df-cmn 15093  df-abl 15094  df-cyg 15167  df-dprd 15235
  Copyright terms: Public domain W3C validator