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Theorem pgpfaclem3 17006
Description: Lemma for pgpfac 17007. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
Hypotheses
Ref Expression
pgpfac.b  |-  B  =  ( Base `  G
)
pgpfac.c  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
pgpfac.g  |-  ( ph  ->  G  e.  Abel )
pgpfac.p  |-  ( ph  ->  P pGrp  G )
pgpfac.f  |-  ( ph  ->  B  e.  Fin )
pgpfac.u  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
pgpfac.a  |-  ( ph  ->  A. t  e.  (SubGrp `  G ) ( t 
C.  U  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  t ) ) )
Assertion
Ref Expression
pgpfaclem3  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
Distinct variable groups:    t, s, C    s, r, t, G    ph, t    B, s, t    U, r, s, t
Allowed substitution hints:    ph( s, r)    B( r)    C( r)    P( t, s, r)

Proof of Theorem pgpfaclem3
Dummy variables  w  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wrd0 12546 . . 3  |-  (/)  e. Word  C
2 pgpfac.g . . . . . 6  |-  ( ph  ->  G  e.  Abel )
3 ablgrp 16676 . . . . . 6  |-  ( G  e.  Abel  ->  G  e. 
Grp )
4 eqid 2467 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
54dprd0 16950 . . . . . 6  |-  ( G  e.  Grp  ->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  { ( 0g `  G ) } ) )
62, 3, 53syl 20 . . . . 5  |-  ( ph  ->  ( G dom DProd  (/)  /\  ( G DProd 
(/) )  =  {
( 0g `  G
) } ) )
76adantr 465 . . . 4  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  { ( 0g `  G ) } ) )
8 pgpfac.u . . . . . . . . 9  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
94subg0cl 16081 . . . . . . . . 9  |-  ( U  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  U
)
108, 9syl 16 . . . . . . . 8  |-  ( ph  ->  ( 0g `  G
)  e.  U )
1110adantr 465 . . . . . . 7  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  ( 0g `  G )  e.  U )
12 eqid 2467 . . . . . . . . . . 11  |-  ( Gs  U )  =  ( Gs  U )
1312subgbas 16077 . . . . . . . . . 10  |-  ( U  e.  (SubGrp `  G
)  ->  U  =  ( Base `  ( Gs  U
) ) )
148, 13syl 16 . . . . . . . . 9  |-  ( ph  ->  U  =  ( Base `  ( Gs  U ) ) )
1514adantr 465 . . . . . . . 8  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  U  =  ( Base `  ( Gs  U ) ) )
1612subggrp 16076 . . . . . . . . . . 11  |-  ( U  e.  (SubGrp `  G
)  ->  ( Gs  U
)  e.  Grp )
178, 16syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( Gs  U )  e.  Grp )
18 grpmnd 15934 . . . . . . . . . 10  |-  ( ( Gs  U )  e.  Grp  ->  ( Gs  U )  e.  Mnd )
19 eqid 2467 . . . . . . . . . . 11  |-  ( Base `  ( Gs  U ) )  =  ( Base `  ( Gs  U ) )
20 eqid 2467 . . . . . . . . . . 11  |-  (gEx `  ( Gs  U ) )  =  (gEx `  ( Gs  U
) )
2119, 20gex1 16484 . . . . . . . . . 10  |-  ( ( Gs  U )  e.  Mnd  ->  ( (gEx `  ( Gs  U ) )  =  1  <->  ( Base `  ( Gs  U ) )  ~~  1o ) )
2217, 18, 213syl 20 . . . . . . . . 9  |-  ( ph  ->  ( (gEx `  ( Gs  U ) )  =  1  <->  ( Base `  ( Gs  U ) )  ~~  1o ) )
2322biimpa 484 . . . . . . . 8  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  ( Base `  ( Gs  U ) )  ~~  1o )
2415, 23eqbrtrd 4473 . . . . . . 7  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  U  ~~  1o )
25 en1eqsn 7761 . . . . . . 7  |-  ( ( ( 0g `  G
)  e.  U  /\  U  ~~  1o )  ->  U  =  { ( 0g `  G ) } )
2611, 24, 25syl2anc 661 . . . . . 6  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  U  =  { ( 0g `  G ) } )
2726eqeq2d 2481 . . . . 5  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  (
( G DProd  (/) )  =  U  <->  ( G DProd  (/) )  =  { ( 0g `  G ) } ) )
2827anbi2d 703 . . . 4  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  (
( G dom DProd  (/)  /\  ( G DProd 
(/) )  =  U )  <->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  { ( 0g `  G ) } ) ) )
297, 28mpbird 232 . . 3  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  U ) )
30 breq2 4457 . . . . 5  |-  ( s  =  (/)  ->  ( G dom DProd  s  <->  G dom DProd  (/) ) )
31 oveq2 6303 . . . . . 6  |-  ( s  =  (/)  ->  ( G DProd 
s )  =  ( G DProd  (/) ) )
3231eqeq1d 2469 . . . . 5  |-  ( s  =  (/)  ->  ( ( G DProd  s )  =  U  <->  ( G DProd  (/) )  =  U ) )
3330, 32anbi12d 710 . . . 4  |-  ( s  =  (/)  ->  ( ( G dom DProd  s  /\  ( G DProd  s )  =  U )  <->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  U ) ) )
3433rspcev 3219 . . 3  |-  ( (
(/)  e. Word  C  /\  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  U ) )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  U ) )
351, 29, 34sylancr 663 . 2  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  U ) )
3612subgabl 16717 . . . . . 6  |-  ( ( G  e.  Abel  /\  U  e.  (SubGrp `  G )
)  ->  ( Gs  U
)  e.  Abel )
372, 8, 36syl2anc 661 . . . . 5  |-  ( ph  ->  ( Gs  U )  e.  Abel )
38 pgpfac.f . . . . . . . 8  |-  ( ph  ->  B  e.  Fin )
39 pgpfac.b . . . . . . . . . 10  |-  B  =  ( Base `  G
)
4039subgss 16074 . . . . . . . . 9  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
418, 40syl 16 . . . . . . . 8  |-  ( ph  ->  U  C_  B )
42 ssfi 7752 . . . . . . . 8  |-  ( ( B  e.  Fin  /\  U  C_  B )  ->  U  e.  Fin )
4338, 41, 42syl2anc 661 . . . . . . 7  |-  ( ph  ->  U  e.  Fin )
4414, 43eqeltrrd 2556 . . . . . 6  |-  ( ph  ->  ( Base `  ( Gs  U ) )  e. 
Fin )
4519, 20gexcl2 16482 . . . . . 6  |-  ( ( ( Gs  U )  e.  Grp  /\  ( Base `  ( Gs  U ) )  e. 
Fin )  ->  (gEx `  ( Gs  U ) )  e.  NN )
4617, 44, 45syl2anc 661 . . . . 5  |-  ( ph  ->  (gEx `  ( Gs  U
) )  e.  NN )
47 eqid 2467 . . . . . 6  |-  ( od
`  ( Gs  U ) )  =  ( od
`  ( Gs  U ) )
4819, 20, 47gexex 16732 . . . . 5  |-  ( ( ( Gs  U )  e.  Abel  /\  (gEx `  ( Gs  U
) )  e.  NN )  ->  E. x  e.  (
Base `  ( Gs  U
) ) ( ( od `  ( Gs  U ) ) `  x
)  =  (gEx `  ( Gs  U ) ) )
4937, 46, 48syl2anc 661 . . . 4  |-  ( ph  ->  E. x  e.  (
Base `  ( Gs  U
) ) ( ( od `  ( Gs  U ) ) `  x
)  =  (gEx `  ( Gs  U ) ) )
5049adantr 465 . . 3  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  ->  E. x  e.  ( Base `  ( Gs  U ) ) ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) )
51 eqid 2467 . . . . 5  |-  (mrCls `  (SubGrp `  ( Gs  U ) ) )  =  (mrCls `  (SubGrp `  ( Gs  U
) ) )
52 eqid 2467 . . . . 5  |-  ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  =  ( (mrCls `  (SubGrp `  ( Gs  U ) ) ) `  {
x } )
53 eqid 2467 . . . . 5  |-  ( 0g
`  ( Gs  U ) )  =  ( 0g
`  ( Gs  U ) )
54 eqid 2467 . . . . 5  |-  ( LSSum `  ( Gs  U ) )  =  ( LSSum `  ( Gs  U
) )
55 pgpfac.p . . . . . . 7  |-  ( ph  ->  P pGrp  G )
56 subgpgp 16490 . . . . . . 7  |-  ( ( P pGrp  G  /\  U  e.  (SubGrp `  G )
)  ->  P pGrp  ( Gs  U ) )
5755, 8, 56syl2anc 661 . . . . . 6  |-  ( ph  ->  P pGrp  ( Gs  U ) )
5857ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  ->  P pGrp  ( Gs  U ) )
5937ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  -> 
( Gs  U )  e.  Abel )
6044ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  -> 
( Base `  ( Gs  U
) )  e.  Fin )
61 simprr 756 . . . . 5  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  -> 
( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) )
62 simprl 755 . . . . 5  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  ->  x  e.  ( Base `  ( Gs  U ) ) )
6351, 52, 19, 47, 20, 53, 54, 58, 59, 60, 61, 62pgpfac1 17003 . . . 4  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  ->  E. w  e.  (SubGrp `  ( Gs  U ) ) ( ( ( (mrCls `  (SubGrp `  ( Gs  U ) ) ) `  {
x } )  i^i  w )  =  {
( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) )
64 pgpfac.c . . . . 5  |-  C  =  { r  e.  (SubGrp `  G )  |  ( Gs  r )  e.  (CycGrp 
i^i  ran pGrp  ) }
652ad3antrrr 729 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  G  e.  Abel )
6655ad3antrrr 729 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  P pGrp  G )
6738ad3antrrr 729 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  B  e.  Fin )
688ad3antrrr 729 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  U  e.  (SubGrp `  G )
)
69 pgpfac.a . . . . . 6  |-  ( ph  ->  A. t  e.  (SubGrp `  G ) ( t 
C.  U  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  t ) ) )
7069ad3antrrr 729 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  A. t  e.  (SubGrp `  G )
( t  C.  U  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  t ) ) )
71 simpllr 758 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  (gEx `  ( Gs  U ) )  =/=  1 )
72 simplrl 759 . . . . . 6  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  x  e.  ( Base `  ( Gs  U ) ) )
7368, 13syl 16 . . . . . 6  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  U  =  ( Base `  ( Gs  U ) ) )
7472, 73eleqtrrd 2558 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  x  e.  U )
75 simplrr 760 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  (
( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) )
76 simprl 755 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  w  e.  (SubGrp `  ( Gs  U
) ) )
77 simprrl 763 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  (
( (mrCls `  (SubGrp `  ( Gs  U ) ) ) `
 { x }
)  i^i  w )  =  { ( 0g `  ( Gs  U ) ) } )
78 simprrr 764 . . . . . 6  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  (
( (mrCls `  (SubGrp `  ( Gs  U ) ) ) `
 { x }
) ( LSSum `  ( Gs  U ) ) w )  =  ( Base `  ( Gs  U ) ) )
7978, 73eqtr4d 2511 . . . . 5  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  (
( (mrCls `  (SubGrp `  ( Gs  U ) ) ) `
 { x }
) ( LSSum `  ( Gs  U ) ) w )  =  U )
8039, 64, 65, 66, 67, 68, 70, 12, 51, 47, 20, 53, 54, 71, 74, 75, 76, 77, 79pgpfaclem2 17005 . . . 4  |-  ( ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  ( x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  /\  ( w  e.  (SubGrp `  ( Gs  U ) )  /\  ( ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  i^i  w )  =  { ( 0g `  ( Gs  U ) ) }  /\  ( ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } ) ( LSSum `  ( Gs  U
) ) w )  =  ( Base `  ( Gs  U ) ) ) ) )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  U ) )
8163, 80rexlimddv 2963 . . 3  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
8250, 81rexlimddv 2963 . 2  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  U ) )
8335, 82pm2.61dane 2785 1  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   {crab 2821    i^i cin 3480    C_ wss 3481    C. wpss 3482   (/)c0 3790   {csn 4033   class class class wbr 4453   dom cdm 5005   ran crn 5006   ` cfv 5594  (class class class)co 6295   1oc1o 7135    ~~ cen 7525   Fincfn 7528   1c1 9505   NNcn 10548  Word cword 12515   Basecbs 14507   ↾s cress 14508   0gc0g 14712  mrClscmrc 14855   Mndcmnd 15793   Grpcgrp 15925  SubGrpcsubg 16067   odcod 16422  gExcgex 16423   pGrp cpgp 16424   LSSumclsm 16527   Abelcabl 16672  CycGrpccyg 16753   DProd cdprd 16897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-disj 4424  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-rpss 6575  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-omul 7147  df-er 7323  df-ec 7325  df-qs 7329  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-sup 7913  df-oi 7947  df-card 8332  df-acn 8335  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-word 12523  df-concat 12525  df-s1 12526  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-dvds 13865  df-gcd 14021  df-prm 14094  df-pc 14237  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-0g 14714  df-gsum 14715  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mulg 15932  df-subg 16070  df-eqg 16072  df-ghm 16137  df-gim 16179  df-ga 16200  df-cntz 16227  df-oppg 16253  df-od 16426  df-gex 16427  df-pgp 16428  df-lsm 16529  df-pj1 16530  df-cmn 16673  df-abl 16674  df-cyg 16754  df-dprd 16899
This theorem is referenced by:  pgpfac  17007
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