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Theorem pgpfaclem3 16583
Description: Lemma for pgpfac 16584. (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 12251 . . 3  |-  (/)  e. Word  C
2 pgpfac.g . . . . . 6  |-  ( ph  ->  G  e.  Abel )
3 ablgrp 16281 . . . . . 6  |-  ( G  e.  Abel  ->  G  e. 
Grp )
4 eqid 2442 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
54dprd0 16527 . . . . . 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 15688 . . . . . . . . 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 2442 . . . . . . . . . . 11  |-  ( Gs  U )  =  ( Gs  U )
1312subgbas 15684 . . . . . . . . . 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 15683 . . . . . . . . . . 11  |-  ( U  e.  (SubGrp `  G
)  ->  ( Gs  U
)  e.  Grp )
178, 16syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( Gs  U )  e.  Grp )
18 grpmnd 15549 . . . . . . . . . 10  |-  ( ( Gs  U )  e.  Grp  ->  ( Gs  U )  e.  Mnd )
19 eqid 2442 . . . . . . . . . . 11  |-  ( Base `  ( Gs  U ) )  =  ( Base `  ( Gs  U ) )
20 eqid 2442 . . . . . . . . . . 11  |-  (gEx `  ( Gs  U ) )  =  (gEx `  ( Gs  U
) )
2119, 20gex1 16089 . . . . . . . . . 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 4311 . . . . . . 7  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  U  ~~  1o )
25 en1eqsn 7541 . . . . . . 7  |-  ( ( ( 0g `  G
)  e.  U  /\  U  ~~  1o )  ->  U  =  { ( 0g `  G ) } )
2611, 24, 25syl2anc 661 . . . . . 6  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  U  =  { ( 0g `  G ) } )
2726eqeq2d 2453 . . . . 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 4295 . . . . 5  |-  ( s  =  (/)  ->  ( G dom DProd  s  <->  G dom DProd  (/) ) )
31 oveq2 6098 . . . . . 6  |-  ( s  =  (/)  ->  ( G DProd 
s )  =  ( G DProd  (/) ) )
3231eqeq1d 2450 . . . . 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 3072 . . 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 16319 . . . . . 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 15681 . . . . . . . . 9  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
418, 40syl 16 . . . . . . . 8  |-  ( ph  ->  U  C_  B )
42 ssfi 7532 . . . . . . . 8  |-  ( ( B  e.  Fin  /\  U  C_  B )  ->  U  e.  Fin )
4338, 41, 42syl2anc 661 . . . . . . 7  |-  ( ph  ->  U  e.  Fin )
4414, 43eqeltrrd 2517 . . . . . 6  |-  ( ph  ->  ( Base `  ( Gs  U ) )  e. 
Fin )
4519, 20gexcl2 16087 . . . . . 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 2442 . . . . . 6  |-  ( od
`  ( Gs  U ) )  =  ( od
`  ( Gs  U ) )
4819, 20, 47gexex 16334 . . . . 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 2442 . . . . 5  |-  (mrCls `  (SubGrp `  ( Gs  U ) ) )  =  (mrCls `  (SubGrp `  ( Gs  U
) ) )
52 eqid 2442 . . . . 5  |-  ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  =  ( (mrCls `  (SubGrp `  ( Gs  U ) ) ) `  {
x } )
53 eqid 2442 . . . . 5  |-  ( 0g
`  ( Gs  U ) )  =  ( 0g
`  ( Gs  U ) )
54 eqid 2442 . . . . 5  |-  ( LSSum `  ( Gs  U ) )  =  ( LSSum `  ( Gs  U
) )
55 pgpfac.p . . . . . . 7  |-  ( ph  ->  P pGrp  G )
56 subgpgp 16095 . . . . . . 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 16580 . . . 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 2519 . . . . 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 2477 . . . . 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 16582 . . . 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 2844 . . 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 2844 . 2  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  U ) )
8335, 82pm2.61dane 2688 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 1369    e. wcel 1756    =/= wne 2605   A.wral 2714   E.wrex 2715   {crab 2718    i^i cin 3326    C_ wss 3327    C. wpss 3328   (/)c0 3636   {csn 3876   class class class wbr 4291   dom cdm 4839   ran crn 4840   ` cfv 5417  (class class class)co 6090   1oc1o 6912    ~~ cen 7306   Fincfn 7309   1c1 9282   NNcn 10321  Word cword 12220   Basecbs 14173   ↾s cress 14174   0gc0g 14377  mrClscmrc 14520   Mndcmnd 15408   Grpcgrp 15409  SubGrpcsubg 15674   odcod 16027  gExcgex 16028   pGrp cpgp 16029   LSSumclsm 16132   Abelcabel 16277  CycGrpccyg 16353   DProd cdprd 16474
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 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-disj 4262  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-rpss 6359  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-tpos 6744  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-omul 6924  df-er 7100  df-ec 7102  df-qs 7106  df-map 7215  df-ixp 7263  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-fsupp 7620  df-sup 7690  df-oi 7723  df-card 8108  df-acn 8111  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-q 10953  df-rp 10991  df-fz 11437  df-fzo 11548  df-fl 11641  df-mod 11708  df-seq 11806  df-exp 11865  df-fac 12051  df-bc 12078  df-hash 12103  df-word 12228  df-concat 12230  df-s1 12231  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-clim 12965  df-sum 13163  df-dvds 13535  df-gcd 13690  df-prm 13763  df-pc 13903  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-0g 14379  df-gsum 14380  df-mre 14523  df-mrc 14524  df-acs 14526  df-mnd 15414  df-mhm 15463  df-submnd 15464  df-grp 15544  df-minusg 15545  df-sbg 15546  df-mulg 15547  df-subg 15677  df-eqg 15679  df-ghm 15744  df-gim 15786  df-ga 15807  df-cntz 15834  df-oppg 15860  df-od 16031  df-gex 16032  df-pgp 16033  df-lsm 16134  df-pj1 16135  df-cmn 16278  df-abl 16279  df-cyg 16354  df-dprd 16476
This theorem is referenced by:  pgpfac  16584
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