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Theorem ablfaclem1 17653
Description: Lemma for ablfac 17656. (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
ablfaclem1  |-  ( U  e.  (SubGrp `  G
)  ->  ( W `  U )  =  {
s  e. Word  C  | 
( G dom DProd  s  /\  ( G DProd  s )  =  U ) } )
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    U, g, s    g, G, p, r, s, w, x
Allowed substitution hints:    ph( g, r)    A( w, g, r)    C( r)    S( x, w, p)    U( x, w, r, p)    O( w, g, s, r)    W( g, s, r)

Proof of Theorem ablfaclem1
StepHypRef Expression
1 eqeq2 2444 . . . 4  |-  ( g  =  U  ->  (
( G DProd  s )  =  g  <->  ( G DProd  s
)  =  U ) )
21anbi2d 708 . . 3  |-  ( g  =  U  ->  (
( G dom DProd  s  /\  ( G DProd  s )  =  g )  <->  ( G dom DProd  s  /\  ( G DProd 
s )  =  U ) ) )
32rabbidv 3079 . 2  |-  ( g  =  U  ->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  g ) }  =  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd 
s )  =  U ) } )
4 ablfac.w . 2  |-  W  =  ( g  e.  (SubGrp `  G )  |->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  g ) } )
5 ablfac.c . . . . 5  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
6 fvex 5891 . . . . 5  |-  (SubGrp `  G )  e.  _V
75, 6rabex2 4578 . . . 4  |-  C  e. 
_V
8 wrdexg 12669 . . . 4  |-  ( C  e.  _V  -> Word  C  e. 
_V )
97, 8ax-mp 5 . . 3  |- Word  C  e. 
_V
109rabex 4576 . 2  |-  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd  s )  =  U ) }  e.  _V
113, 4, 10fvmpt 5964 1  |-  ( U  e.  (SubGrp `  G
)  ->  ( W `  U )  =  {
s  e. Word  C  | 
( G dom DProd  s  /\  ( G DProd  s )  =  U ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   {crab 2786   _Vcvv 3087    i^i cin 3441   class class class wbr 4426    |-> cmpt 4484   dom cdm 4854   ran crn 4855   ` cfv 5601  (class class class)co 6305   Fincfn 7577   ^cexp 12269   #chash 12512  Word cword 12643    || cdvds 14283   Primecprime 14593    pCnt cpc 14749   Basecbs 15084   ↾s cress 15085  SubGrpcsubg 16762   odcod 17116   pGrp cpgp 17118   Abelcabl 17366  CycGrpccyg 17447   DProd cdprd 17560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7482  df-pm 7483  df-neg 9862  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-word 12651
This theorem is referenced by:  ablfaclem2  17654  ablfaclem3  17655  ablfac  17656
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