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Theorem ablfac2 16612
Description: Choose generators for each cyclic group in ablfac 16611. (Contributed by Mario Carneiro, 28-Apr-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 )
ablfac2.m  |-  .x.  =  (.g
`  G )
ablfac2.s  |-  S  =  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  k
) ) ) )
Assertion
Ref Expression
ablfac2  |-  ( ph  ->  E. w  e. Word  B
( S : dom  w
--> C  /\  G dom DProd  S  /\  ( G DProd  S
)  =  B ) )
Distinct variable groups:    S, r    k, n, r, w, B    .x. , k, w    C, k, n, w    ph, k, n, w    k, G, n, r, w
Allowed substitution hints:    ph( r)    C( r)    S( w, k, n)    .x. ( n, r)

Proof of Theorem ablfac2
Dummy variables  s  x  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ablfac.b . . 3  |-  B  =  ( Base `  G
)
2 ablfac.c . . 3  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
3 ablfac.1 . . 3  |-  ( ph  ->  G  e.  Abel )
4 ablfac.2 . . 3  |-  ( ph  ->  B  e.  Fin )
51, 2, 3, 4ablfac 16611 . 2  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  B ) )
6 wrdf 12261 . . . . . . . . . 10  |-  ( s  e. Word  C  ->  s : ( 0..^ (
# `  s )
) --> C )
76ad2antlr 726 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  s : ( 0..^ (
# `  s )
) --> C )
8 fdm 5584 . . . . . . . . 9  |-  ( s : ( 0..^ (
# `  s )
) --> C  ->  dom  s  =  ( 0..^ ( # `  s
) ) )
97, 8syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  dom  s  =  ( 0..^ ( # `  s
) ) )
10 fzofi 11817 . . . . . . . 8  |-  ( 0..^ ( # `  s
) )  e.  Fin
119, 10syl6eqel 2531 . . . . . . 7  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  dom  s  e.  Fin )
129feq2d 5568 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  (
s : dom  s --> C 
<->  s : ( 0..^ ( # `  s
) ) --> C ) )
137, 12mpbird 232 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  s : dom  s --> C )
1413ffvelrnda 5864 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  e.  C
)
15 oveq2 6120 . . . . . . . . . . . . . 14  |-  ( r  =  ( s `  k )  ->  ( Gs  r )  =  ( Gs  ( s `  k
) ) )
1615eleq1d 2509 . . . . . . . . . . . . 13  |-  ( r  =  ( s `  k )  ->  (
( Gs  r )  e.  (CycGrp  i^i  ran pGrp  )  <->  ( Gs  (
s `  k )
)  e.  (CycGrp  i^i  ran pGrp  ) ) )
1716, 2elrab2 3140 . . . . . . . . . . . 12  |-  ( ( s `  k )  e.  C  <->  ( (
s `  k )  e.  (SubGrp `  G )  /\  ( Gs  ( s `  k ) )  e.  (CycGrp  i^i  ran pGrp  ) ) )
1817simplbi 460 . . . . . . . . . . 11  |-  ( ( s `  k )  e.  C  ->  (
s `  k )  e.  (SubGrp `  G )
)
1914, 18syl 16 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  e.  (SubGrp `  G ) )
201subgss 15703 . . . . . . . . . 10  |-  ( ( s `  k )  e.  (SubGrp `  G
)  ->  ( s `  k )  C_  B
)
2119, 20syl 16 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  C_  B
)
22 inss1 3591 . . . . . . . . . . . . 13  |-  (CycGrp  i^i  ran pGrp  )  C_ CycGrp
2317simprbi 464 . . . . . . . . . . . . . 14  |-  ( ( s `  k )  e.  C  ->  ( Gs  ( s `  k
) )  e.  (CycGrp 
i^i  ran pGrp  ) )
2414, 23syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( Gs  ( s `
 k ) )  e.  (CycGrp  i^i  ran pGrp  ) )
2522, 24sseldi 3375 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( Gs  ( s `
 k ) )  e. CycGrp )
26 eqid 2443 . . . . . . . . . . . . . 14  |-  ( Base `  ( Gs  ( s `  k ) ) )  =  ( Base `  ( Gs  ( s `  k
) ) )
27 eqid 2443 . . . . . . . . . . . . . 14  |-  (.g `  ( Gs  ( s `  k
) ) )  =  (.g `  ( Gs  ( s `
 k ) ) )
2826, 27iscyg 16377 . . . . . . . . . . . . 13  |-  ( ( Gs  ( s `  k
) )  e. CycGrp  <->  ( ( Gs  ( s `  k
) )  e.  Grp  /\ 
E. x  e.  (
Base `  ( Gs  (
s `  k )
) ) ran  (
n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `
 k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) ) ) )
2928simprbi 464 . . . . . . . . . . . 12  |-  ( ( Gs  ( s `  k
) )  e. CycGrp  ->  E. x  e.  ( Base `  ( Gs  ( s `  k ) ) ) ran  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) ) )
3025, 29syl 16 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  E. x  e.  (
Base `  ( Gs  (
s `  k )
) ) ran  (
n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `
 k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) ) )
31 eqid 2443 . . . . . . . . . . . . . 14  |-  ( Gs  ( s `  k ) )  =  ( Gs  ( s `  k ) )
3231subgbas 15706 . . . . . . . . . . . . 13  |-  ( ( s `  k )  e.  (SubGrp `  G
)  ->  ( s `  k )  =  (
Base `  ( Gs  (
s `  k )
) ) )
3319, 32syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( s `  k )  =  (
Base `  ( Gs  (
s `  k )
) ) )
3433rexeqdv 2945 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( E. x  e.  ( s `  k
) ran  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) )  <->  E. x  e.  ( Base `  ( Gs  ( s `  k
) ) ) ran  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `  k
) ) ) x ) )  =  (
Base `  ( Gs  (
s `  k )
) ) ) )
3530, 34mpbird 232 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  E. x  e.  ( s `  k ) ran  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) ) )
3619ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  /\  x  e.  ( s `  k ) )  /\  n  e.  ZZ )  ->  ( s `  k
)  e.  (SubGrp `  G ) )
37 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  /\  x  e.  ( s `  k ) )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
38 simplr 754 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  /\  x  e.  ( s `  k ) )  /\  n  e.  ZZ )  ->  x  e.  ( s `
 k ) )
39 ablfac2.m . . . . . . . . . . . . . . . 16  |-  .x.  =  (.g
`  G )
4039, 31, 27subgmulg 15716 . . . . . . . . . . . . . . 15  |-  ( ( ( s `  k
)  e.  (SubGrp `  G )  /\  n  e.  ZZ  /\  x  e.  ( s `  k
) )  ->  (
n  .x.  x )  =  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )
4136, 37, 38, 40syl3anc 1218 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  /\  x  e.  ( s `  k ) )  /\  n  e.  ZZ )  ->  ( n  .x.  x
)  =  ( n (.g `  ( Gs  ( s `
 k ) ) ) x ) )
4241mpteq2dva 4399 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  k  e. 
dom  s )  /\  x  e.  ( s `  k ) )  -> 
( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `
 k ) ) ) x ) ) )
4342rneqd 5088 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  k  e. 
dom  s )  /\  x  e.  ( s `  k ) )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ran  (
n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `
 k ) ) ) x ) ) )
4433adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  k  e. 
dom  s )  /\  x  e.  ( s `  k ) )  -> 
( s `  k
)  =  ( Base `  ( Gs  ( s `  k ) ) ) )
4543, 44eqeq12d 2457 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  k  e. 
dom  s )  /\  x  e.  ( s `  k ) )  -> 
( ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `  k
)  <->  ran  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) ) ) )
4645rexbidva 2753 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  ( E. x  e.  ( s `  k
) ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `  k
)  <->  E. x  e.  ( s `  k ) ran  ( n  e.  ZZ  |->  ( n (.g `  ( Gs  ( s `  k ) ) ) x ) )  =  ( Base `  ( Gs  ( s `  k
) ) ) ) )
4735, 46mpbird 232 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  E. x  e.  ( s `  k ) ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `  k
) )
48 ssrexv 3438 . . . . . . . . 9  |-  ( ( s `  k ) 
C_  B  ->  ( E. x  e.  (
s `  k ) ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `
 k )  ->  E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `
 k ) ) )
4921, 47, 48sylc 60 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  k  e.  dom  s )  ->  E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `
 k ) )
5049ralrimiva 2820 . . . . . . 7  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  A. k  e.  dom  s E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n 
.x.  x ) )  =  ( s `  k ) )
51 oveq2 6120 . . . . . . . . . . 11  |-  ( x  =  ( w `  k )  ->  (
n  .x.  x )  =  ( n  .x.  ( w `  k
) ) )
5251mpteq2dv 4400 . . . . . . . . . 10  |-  ( x  =  ( w `  k )  ->  (
n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  k ) ) ) )
5352rneqd 5088 . . . . . . . . 9  |-  ( x  =  ( w `  k )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) ) )
5453eqeq1d 2451 . . . . . . . 8  |-  ( x  =  ( w `  k )  ->  ( ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `
 k )  <->  ran  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  k ) ) )  =  ( s `  k ) ) )
5554ac6sfi 7577 . . . . . . 7  |-  ( ( dom  s  e.  Fin  /\ 
A. k  e.  dom  s E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  ( s `
 k ) )  ->  E. w ( w : dom  s --> B  /\  A. k  e. 
dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )
5611, 50, 55syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  E. w
( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )
57 simprl 755 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  w : dom  s --> B )
589adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  dom  s  =  ( 0..^ ( # `  s ) ) )
5958feq2d 5568 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( w : dom  s --> B  <->  w :
( 0..^ ( # `  s ) ) --> B ) )
6057, 59mpbid 210 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  w :
( 0..^ ( # `  s ) ) --> B )
61 iswrdi 12260 . . . . . . . . . 10  |-  ( w : ( 0..^ (
# `  s )
) --> B  ->  w  e. Word  B )
6260, 61syl 16 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  w  e. Word  B )
63 fdm 5584 . . . . . . . . . . . . . . . 16  |-  ( w : ( 0..^ (
# `  s )
) --> B  ->  dom  w  =  ( 0..^ ( # `  s
) ) )
6460, 63syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  dom  w  =  ( 0..^ ( # `  s ) ) )
6564, 58eqtr4d 2478 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  dom  w  =  dom  s )
6665eleq2d 2510 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( j  e.  dom  w  <->  j  e.  dom  s ) )
6766biimpa 484 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  ( w : dom  s --> B  /\  A. k  e. 
dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  /\  j  e. 
dom  w )  -> 
j  e.  dom  s
)
68 simprr 756 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) )
69 simpl 457 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( k  =  j  /\  n  e.  ZZ )  ->  k  =  j )
7069fveq2d 5716 . . . . . . . . . . . . . . . . . . 19  |-  ( ( k  =  j  /\  n  e.  ZZ )  ->  ( w `  k
)  =  ( w `
 j ) )
7170oveq2d 6128 . . . . . . . . . . . . . . . . . 18  |-  ( ( k  =  j  /\  n  e.  ZZ )  ->  ( n  .x.  (
w `  k )
)  =  ( n 
.x.  ( w `  j ) ) )
7271mpteq2dva 4399 . . . . . . . . . . . . . . . . 17  |-  ( k  =  j  ->  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  j ) ) ) )
7372rneqd 5088 . . . . . . . . . . . . . . . 16  |-  ( k  =  j  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) ) )
74 fveq2 5712 . . . . . . . . . . . . . . . 16  |-  ( k  =  j  ->  (
s `  k )  =  ( s `  j ) )
7573, 74eqeq12d 2457 . . . . . . . . . . . . . . 15  |-  ( k  =  j  ->  ( ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k )  <->  ran  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  j ) ) )  =  ( s `  j ) ) )
7675rspccva 3093 . . . . . . . . . . . . . 14  |-  ( ( A. k  e.  dom  s ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  k
) ) )  =  ( s `  k
)  /\  j  e.  dom  s )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) )  =  ( s `
 j ) )
7768, 76sylan 471 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  ( w : dom  s --> B  /\  A. k  e. 
dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  /\  j  e. 
dom  s )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) )  =  ( s `
 j ) )
7813adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  s : dom  s --> C )
7978ffvelrnda 5864 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  ( w : dom  s --> B  /\  A. k  e. 
dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  /\  j  e. 
dom  s )  -> 
( s `  j
)  e.  C )
8077, 79eqeltrd 2517 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  ( w : dom  s --> B  /\  A. k  e. 
dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  /\  j  e. 
dom  s )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) )  e.  C )
8167, 80syldan 470 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s
)  =  B ) )  /\  ( w : dom  s --> B  /\  A. k  e. 
dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  /\  j  e. 
dom  w )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) )  e.  C )
82 ablfac2.s . . . . . . . . . . . 12  |-  S  =  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  k
) ) ) )
83 fveq2 5712 . . . . . . . . . . . . . . . 16  |-  ( k  =  j  ->  (
w `  k )  =  ( w `  j ) )
8483oveq2d 6128 . . . . . . . . . . . . . . 15  |-  ( k  =  j  ->  (
n  .x.  ( w `  k ) )  =  ( n  .x.  (
w `  j )
) )
8584mpteq2dv 4400 . . . . . . . . . . . . . 14  |-  ( k  =  j  ->  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( n  e.  ZZ  |->  ( n 
.x.  ( w `  j ) ) ) )
8685rneqd 5088 . . . . . . . . . . . . 13  |-  ( k  =  j  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 j ) ) ) )
8786cbvmptv 4404 . . . . . . . . . . . 12  |-  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) ) )  =  ( j  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  j
) ) ) )
8882, 87eqtri 2463 . . . . . . . . . . 11  |-  S  =  ( j  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `  j
) ) ) )
8981, 88fmptd 5888 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  S : dom  w --> C )
90 simprl 755 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  G dom DProd  s )
9190adantr 465 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  G dom DProd  s )
9265raleqdv 2944 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( A. k  e.  dom  w ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k )  <->  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )
9368, 92mpbird 232 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  A. k  e.  dom  w ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) )
94 mpteq12 4392 . . . . . . . . . . . . . 14  |-  ( ( dom  w  =  dom  s  /\  A. k  e. 
dom  w ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) )  ->  ( k  e. 
dom  w  |->  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) ) )  =  ( k  e.  dom  s  |->  ( s `  k
) ) )
9565, 93, 94syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( k  e.  dom  w  |->  ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) ) )  =  ( k  e.  dom  s  |->  ( s `  k
) ) )
9682, 95syl5eq 2487 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  S  =  ( k  e.  dom  s  |->  ( s `  k ) ) )
97 dprdf 16512 . . . . . . . . . . . . . 14  |-  ( G dom DProd  s  ->  s : dom  s --> (SubGrp `  G ) )
9891, 97syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  s : dom  s --> (SubGrp `  G )
)
9998feqmptd 5765 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  s  =  ( k  e.  dom  s  |->  ( s `  k ) ) )
10096, 99eqtr4d 2478 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  S  =  s )
10191, 100breqtrrd 4339 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  G dom DProd  S )
102100oveq2d 6128 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( G DProd  S )  =  ( G DProd 
s ) )
103 simplrr 760 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( G DProd  s )  =  B )
104102, 103eqtrd 2475 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( G DProd  S )  =  B )
10589, 101, 1043jca 1168 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd  S )  =  B ) )
10662, 105jca 532 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  /\  ( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) ) )  ->  ( w  e. Word  B  /\  ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd  S )  =  B ) ) )
107106ex 434 . . . . . . 7  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  (
( w : dom  s
--> B  /\  A. k  e.  dom  s ran  (
n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) )  ->  ( w  e. Word  B  /\  ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd 
S )  =  B ) ) ) )
108107eximdv 1676 . . . . . 6  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  ( E. w ( w : dom  s --> B  /\  A. k  e.  dom  s ran  ( n  e.  ZZ  |->  ( n  .x.  ( w `
 k ) ) )  =  ( s `
 k ) )  ->  E. w ( w  e. Word  B  /\  ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd  S )  =  B ) ) ) )
10956, 108mpd 15 . . . . 5  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  E. w
( w  e. Word  B  /\  ( S : dom  w
--> C  /\  G dom DProd  S  /\  ( G DProd  S
)  =  B ) ) )
110 df-rex 2742 . . . . 5  |-  ( E. w  e. Word  B ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd  S )  =  B )  <->  E. w
( w  e. Word  B  /\  ( S : dom  w
--> C  /\  G dom DProd  S  /\  ( G DProd  S
)  =  B ) ) )
111109, 110sylibr 212 . . . 4  |-  ( ( ( ph  /\  s  e. Word  C )  /\  ( G dom DProd  s  /\  ( G DProd  s )  =  B ) )  ->  E. w  e. Word  B ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd 
S )  =  B ) )
112111ex 434 . . 3  |-  ( (
ph  /\  s  e. Word  C )  ->  ( ( G dom DProd  s  /\  ( G DProd  s )  =  B )  ->  E. w  e. Word  B ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd 
S )  =  B ) ) )
113112rexlimdva 2862 . 2  |-  ( ph  ->  ( E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  B )  ->  E. w  e. Word  B
( S : dom  w
--> C  /\  G dom DProd  S  /\  ( G DProd  S
)  =  B ) ) )
1145, 113mpd 15 1  |-  ( ph  ->  E. w  e. Word  B
( S : dom  w
--> C  /\  G dom DProd  S  /\  ( G DProd  S
)  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   A.wral 2736   E.wrex 2737   {crab 2740    i^i cin 3348    C_ wss 3349   class class class wbr 4313    e. cmpt 4371   dom cdm 4861   ran crn 4862   -->wf 5435   ` cfv 5439  (class class class)co 6112   Fincfn 7331   0cc0 9303   ZZcz 10667  ..^cfzo 11569   #chash 12124  Word cword 12242   Basecbs 14195   ↾s cress 14196   Grpcgrp 15431  .gcmg 15435  SubGrpcsubg 15696   pGrp cpgp 16051   Abelcabel 16299  CycGrpccyg 16375   DProd cdprd 16497
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 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-disj 4284  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-rpss 6381  df-om 6498  df-1st 6598  df-2nd 6599  df-supp 6712  df-tpos 6766  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-omul 6946  df-er 7122  df-ec 7124  df-qs 7128  df-map 7237  df-pm 7238  df-ixp 7285  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fsupp 7642  df-sup 7712  df-oi 7745  df-card 8130  df-acn 8133  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-n0 10601  df-z 10668  df-uz 10883  df-q 10975  df-rp 11013  df-fz 11459  df-fzo 11570  df-fl 11663  df-mod 11730  df-seq 11828  df-exp 11887  df-fac 12073  df-bc 12100  df-hash 12125  df-word 12250  df-concat 12252  df-s1 12253  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-clim 12987  df-sum 13185  df-dvds 13557  df-gcd 13712  df-prm 13785  df-pc 13925  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-0g 14401  df-gsum 14402  df-mre 14545  df-mrc 14546  df-acs 14548  df-mnd 15436  df-mhm 15485  df-submnd 15486  df-grp 15566  df-minusg 15567  df-sbg 15568  df-mulg 15569  df-subg 15699  df-eqg 15701  df-ghm 15766  df-gim 15808  df-ga 15829  df-cntz 15856  df-oppg 15882  df-od 16053  df-gex 16054  df-pgp 16055  df-lsm 16156  df-pj1 16157  df-cmn 16300  df-abl 16301  df-cyg 16376  df-dprd 16499
This theorem is referenced by:  dchrpt  22628
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