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Theorem pgpfaclem3 15596
Description: Lemma for pgpfac 15597. (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 11687 . . 3  |-  (/)  e. Word  C
2 pgpfac.g . . . . . 6  |-  ( ph  ->  G  e.  Abel )
3 ablgrp 15372 . . . . . 6  |-  ( G  e.  Abel  ->  G  e. 
Grp )
4 eqid 2404 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
54dprd0 15544 . . . . . 6  |-  ( G  e.  Grp  ->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  { ( 0g `  G ) } ) )
62, 3, 53syl 19 . . . . 5  |-  ( ph  ->  ( G dom DProd  (/)  /\  ( G DProd 
(/) )  =  {
( 0g `  G
) } ) )
76adantr 452 . . . 4  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  { ( 0g `  G ) } ) )
8 pgpfac.u . . . . . . . . 9  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
94subg0cl 14907 . . . . . . . . 9  |-  ( U  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  U
)
108, 9syl 16 . . . . . . . 8  |-  ( ph  ->  ( 0g `  G
)  e.  U )
1110adantr 452 . . . . . . 7  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  ( 0g `  G )  e.  U )
12 eqid 2404 . . . . . . . . . . 11  |-  ( Gs  U )  =  ( Gs  U )
1312subgbas 14903 . . . . . . . . . 10  |-  ( U  e.  (SubGrp `  G
)  ->  U  =  ( Base `  ( Gs  U
) ) )
148, 13syl 16 . . . . . . . . 9  |-  ( ph  ->  U  =  ( Base `  ( Gs  U ) ) )
1514adantr 452 . . . . . . . 8  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  U  =  ( Base `  ( Gs  U ) ) )
1612subggrp 14902 . . . . . . . . . . 11  |-  ( U  e.  (SubGrp `  G
)  ->  ( Gs  U
)  e.  Grp )
178, 16syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( Gs  U )  e.  Grp )
18 grpmnd 14772 . . . . . . . . . 10  |-  ( ( Gs  U )  e.  Grp  ->  ( Gs  U )  e.  Mnd )
19 eqid 2404 . . . . . . . . . . 11  |-  ( Base `  ( Gs  U ) )  =  ( Base `  ( Gs  U ) )
20 eqid 2404 . . . . . . . . . . 11  |-  (gEx `  ( Gs  U ) )  =  (gEx `  ( Gs  U
) )
2119, 20gex1 15180 . . . . . . . . . 10  |-  ( ( Gs  U )  e.  Mnd  ->  ( (gEx `  ( Gs  U ) )  =  1  <->  ( Base `  ( Gs  U ) )  ~~  1o ) )
2217, 18, 213syl 19 . . . . . . . . 9  |-  ( ph  ->  ( (gEx `  ( Gs  U ) )  =  1  <->  ( Base `  ( Gs  U ) )  ~~  1o ) )
2322biimpa 471 . . . . . . . 8  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  ( Base `  ( Gs  U ) )  ~~  1o )
2415, 23eqbrtrd 4192 . . . . . . 7  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  U  ~~  1o )
25 en1eqsn 7297 . . . . . . 7  |-  ( ( ( 0g `  G
)  e.  U  /\  U  ~~  1o )  ->  U  =  { ( 0g `  G ) } )
2611, 24, 25syl2anc 643 . . . . . 6  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  U  =  { ( 0g `  G ) } )
2726eqeq2d 2415 . . . . 5  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  (
( G DProd  (/) )  =  U  <->  ( G DProd  (/) )  =  { ( 0g `  G ) } ) )
2827anbi2d 685 . . . 4  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  (
( G dom DProd  (/)  /\  ( G DProd 
(/) )  =  U )  <->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  { ( 0g `  G ) } ) ) )
297, 28mpbird 224 . . 3  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  U ) )
30 breq2 4176 . . . . 5  |-  ( s  =  (/)  ->  ( G dom DProd  s  <->  G dom DProd  (/) ) )
31 oveq2 6048 . . . . . 6  |-  ( s  =  (/)  ->  ( G DProd 
s )  =  ( G DProd  (/) ) )
3231eqeq1d 2412 . . . . 5  |-  ( s  =  (/)  ->  ( ( G DProd  s )  =  U  <->  ( G DProd  (/) )  =  U ) )
3330, 32anbi12d 692 . . . 4  |-  ( s  =  (/)  ->  ( ( G dom DProd  s  /\  ( G DProd  s )  =  U )  <->  ( G dom DProd  (/)  /\  ( G DProd  (/) )  =  U ) ) )
3433rspcev 3012 . . 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 645 . 2  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =  1 )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  U ) )
3612subgabl 15410 . . . . . 6  |-  ( ( G  e.  Abel  /\  U  e.  (SubGrp `  G )
)  ->  ( Gs  U
)  e.  Abel )
372, 8, 36syl2anc 643 . . . . 5  |-  ( ph  ->  ( Gs  U )  e.  Abel )
38 pgpfac.f . . . . . . . 8  |-  ( ph  ->  B  e.  Fin )
39 pgpfac.b . . . . . . . . . 10  |-  B  =  ( Base `  G
)
4039subgss 14900 . . . . . . . . 9  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
418, 40syl 16 . . . . . . . 8  |-  ( ph  ->  U  C_  B )
42 ssfi 7288 . . . . . . . 8  |-  ( ( B  e.  Fin  /\  U  C_  B )  ->  U  e.  Fin )
4338, 41, 42syl2anc 643 . . . . . . 7  |-  ( ph  ->  U  e.  Fin )
4414, 43eqeltrrd 2479 . . . . . 6  |-  ( ph  ->  ( Base `  ( Gs  U ) )  e. 
Fin )
4519, 20gexcl2 15178 . . . . . 6  |-  ( ( ( Gs  U )  e.  Grp  /\  ( Base `  ( Gs  U ) )  e. 
Fin )  ->  (gEx `  ( Gs  U ) )  e.  NN )
4617, 44, 45syl2anc 643 . . . . 5  |-  ( ph  ->  (gEx `  ( Gs  U
) )  e.  NN )
47 eqid 2404 . . . . . 6  |-  ( od
`  ( Gs  U ) )  =  ( od
`  ( Gs  U ) )
4819, 20, 47gexex 15423 . . . . 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 643 . . . 4  |-  ( ph  ->  E. x  e.  (
Base `  ( Gs  U
) ) ( ( od `  ( Gs  U ) ) `  x
)  =  (gEx `  ( Gs  U ) ) )
5049adantr 452 . . 3  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  ->  E. x  e.  ( Base `  ( Gs  U ) ) ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) )
51 eqid 2404 . . . . 5  |-  (mrCls `  (SubGrp `  ( Gs  U ) ) )  =  (mrCls `  (SubGrp `  ( Gs  U
) ) )
52 eqid 2404 . . . . 5  |-  ( (mrCls `  (SubGrp `  ( Gs  U
) ) ) `  { x } )  =  ( (mrCls `  (SubGrp `  ( Gs  U ) ) ) `  {
x } )
53 eqid 2404 . . . . 5  |-  ( 0g
`  ( Gs  U ) )  =  ( 0g
`  ( Gs  U ) )
54 eqid 2404 . . . . 5  |-  ( LSSum `  ( Gs  U ) )  =  ( LSSum `  ( Gs  U
) )
55 pgpfac.p . . . . . . 7  |-  ( ph  ->  P pGrp  G )
56 subgpgp 15186 . . . . . . 7  |-  ( ( P pGrp  G  /\  U  e.  (SubGrp `  G )
)  ->  P pGrp  ( Gs  U ) )
5755, 8, 56syl2anc 643 . . . . . 6  |-  ( ph  ->  P pGrp  ( Gs  U ) )
5857ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  ->  P pGrp  ( Gs  U ) )
5937ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  /\  (
x  e.  ( Base `  ( Gs  U ) )  /\  ( ( od `  ( Gs  U ) ) `  x )  =  (gEx
`  ( Gs  U ) ) ) )  -> 
( Gs  U )  e.  Abel )
6044ad2antrr 707 . . . . 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 734 . . . . 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 733 . . . . 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 15593 . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 711 . . . . 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 736 . . . . 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 737 . . . . . 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 2481 . . . . 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 738 . . . . 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 733 . . . . 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 741 . . . . 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 742 . . . . . 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 2439 . . . . 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 15595 . . . 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 2794 . . 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 2794 . 2  |-  ( (
ph  /\  (gEx `  ( Gs  U ) )  =/=  1 )  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s
)  =  U ) )
8335, 82pm2.61dane 2645 1  |-  ( ph  ->  E. s  e. Word  C
( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670    i^i cin 3279    C_ wss 3280    C. wpss 3281   (/)c0 3588   {csn 3774   class class class wbr 4172   dom cdm 4837   ran crn 4838   ` cfv 5413  (class class class)co 6040   1oc1o 6676    ~~ cen 7065   Fincfn 7068   1c1 8947   NNcn 9956  Word cword 11672   Basecbs 13424   ↾s cress 13425   0gc0g 13678  mrClscmrc 13763   Mndcmnd 14639   Grpcgrp 14640  SubGrpcsubg 14893   odcod 15118  gExcgex 15119   pGrp cpgp 15120   LSSumclsm 15223   Abelcabel 15368  CycGrpccyg 15442   DProd cdprd 15509
This theorem is referenced by:  pgpfac  15597
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-rpss 6481  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-word 11678  df-concat 11679  df-s1 11680  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-eqg 14898  df-ghm 14959  df-gim 15001  df-ga 15022  df-cntz 15071  df-oppg 15097  df-od 15122  df-gex 15123  df-pgp 15124  df-lsm 15225  df-pj1 15226  df-cmn 15369  df-abl 15370  df-cyg 15443  df-dprd 15511
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