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Theorem pgpfac1lem1 17642
Description: Lemma for pgpfac1 17648. (Contributed by Mario Carneiro, 27-Apr-2016.)
Hypotheses
Ref Expression
pgpfac1.k  |-  K  =  (mrCls `  (SubGrp `  G
) )
pgpfac1.s  |-  S  =  ( K `  { A } )
pgpfac1.b  |-  B  =  ( Base `  G
)
pgpfac1.o  |-  O  =  ( od `  G
)
pgpfac1.e  |-  E  =  (gEx `  G )
pgpfac1.z  |-  .0.  =  ( 0g `  G )
pgpfac1.l  |-  .(+)  =  (
LSSum `  G )
pgpfac1.p  |-  ( ph  ->  P pGrp  G )
pgpfac1.g  |-  ( ph  ->  G  e.  Abel )
pgpfac1.n  |-  ( ph  ->  B  e.  Fin )
pgpfac1.oe  |-  ( ph  ->  ( O `  A
)  =  E )
pgpfac1.u  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
pgpfac1.au  |-  ( ph  ->  A  e.  U )
pgpfac1.w  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
pgpfac1.i  |-  ( ph  ->  ( S  i^i  W
)  =  {  .0.  } )
pgpfac1.ss  |-  ( ph  ->  ( S  .(+)  W ) 
C_  U )
pgpfac1.2  |-  ( ph  ->  A. w  e.  (SubGrp `  G ) ( ( w  C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w )
)
Assertion
Ref Expression
pgpfac1lem1  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  =  U )
Distinct variable groups:    w, A    w, 
.(+)    w, P    w, G    w, U    w, C    w, S    w, W    ph, w    w, K
Allowed substitution hints:    B( w)    E( w)    O( w)    .0. ( w)

Proof of Theorem pgpfac1lem1
StepHypRef Expression
1 pgpfac1.ss . . . 4  |-  ( ph  ->  ( S  .(+)  W ) 
C_  U )
21adantr 466 . . 3  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( S  .(+) 
W )  C_  U
)
3 pgpfac1.g . . . . . 6  |-  ( ph  ->  G  e.  Abel )
4 ablgrp 17370 . . . . . 6  |-  ( G  e.  Abel  ->  G  e. 
Grp )
5 pgpfac1.b . . . . . . 7  |-  B  =  ( Base `  G
)
65subgacs 16803 . . . . . 6  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  B ) )
7 acsmre 15509 . . . . . 6  |-  ( (SubGrp `  G )  e.  (ACS
`  B )  -> 
(SubGrp `  G )  e.  (Moore `  B )
)
83, 4, 6, 74syl 19 . . . . 5  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  B )
)
98adantr 466 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  (SubGrp `  G
)  e.  (Moore `  B ) )
10 eldifi 3593 . . . . . 6  |-  ( C  e.  ( U  \ 
( S  .(+)  W ) )  ->  C  e.  U )
1110adantl 467 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  U )
1211snssd 4148 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  { C }  C_  U )
13 pgpfac1.u . . . . 5  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
1413adantr 466 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  U  e.  (SubGrp `  G ) )
15 pgpfac1.k . . . . 5  |-  K  =  (mrCls `  (SubGrp `  G
) )
1615mrcsscl 15477 . . . 4  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  { C }  C_  U  /\  U  e.  (SubGrp `  G ) )  -> 
( K `  { C } )  C_  U
)
179, 12, 14, 16syl3anc 1264 . . 3  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( K `  { C } ) 
C_  U )
18 pgpfac1.s . . . . . . 7  |-  S  =  ( K `  { A } )
195subgss 16769 . . . . . . . . . 10  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
2013, 19syl 17 . . . . . . . . 9  |-  ( ph  ->  U  C_  B )
21 pgpfac1.au . . . . . . . . 9  |-  ( ph  ->  A  e.  U )
2220, 21sseldd 3471 . . . . . . . 8  |-  ( ph  ->  A  e.  B )
2315mrcsncl 15469 . . . . . . . 8  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  A  e.  B
)  ->  ( K `  { A } )  e.  (SubGrp `  G
) )
248, 22, 23syl2anc 665 . . . . . . 7  |-  ( ph  ->  ( K `  { A } )  e.  (SubGrp `  G ) )
2518, 24syl5eqel 2521 . . . . . 6  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
26 pgpfac1.w . . . . . 6  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
27 pgpfac1.l . . . . . . 7  |-  .(+)  =  (
LSSum `  G )
2827lsmsubg2 17432 . . . . . 6  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )  /\  W  e.  (SubGrp `  G ) )  -> 
( S  .(+)  W )  e.  (SubGrp `  G
) )
293, 25, 26, 28syl3anc 1264 . . . . 5  |-  ( ph  ->  ( S  .(+)  W )  e.  (SubGrp `  G
) )
3029adantr 466 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( S  .(+) 
W )  e.  (SubGrp `  G ) )
3120sselda 3470 . . . . . 6  |-  ( (
ph  /\  C  e.  U )  ->  C  e.  B )
3210, 31sylan2 476 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  B )
3315mrcsncl 15469 . . . . 5  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  C  e.  B
)  ->  ( K `  { C } )  e.  (SubGrp `  G
) )
349, 32, 33syl2anc 665 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( K `  { C } )  e.  (SubGrp `  G
) )
3527lsmlub 17250 . . . 4  |-  ( ( ( S  .(+)  W )  e.  (SubGrp `  G
)  /\  ( K `  { C } )  e.  (SubGrp `  G
)  /\  U  e.  (SubGrp `  G ) )  ->  ( ( ( S  .(+)  W )  C_  U  /\  ( K `
 { C }
)  C_  U )  <->  ( ( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) 
C_  U ) )
3630, 34, 14, 35syl3anc 1264 . . 3  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( (
( S  .(+)  W ) 
C_  U  /\  ( K `  { C } )  C_  U
)  <->  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  C_  U
) )
372, 17, 36mpbi2and 929 . 2  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C_  U )
3827lsmub1 17243 . . . . . 6  |-  ( ( ( S  .(+)  W )  e.  (SubGrp `  G
)  /\  ( K `  { C } )  e.  (SubGrp `  G
) )  ->  ( S  .(+)  W )  C_  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) )
3930, 34, 38syl2anc 665 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( S  .(+) 
W )  C_  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) )
4027lsmub2 17244 . . . . . . 7  |-  ( ( ( S  .(+)  W )  e.  (SubGrp `  G
)  /\  ( K `  { C } )  e.  (SubGrp `  G
) )  ->  ( K `  { C } )  C_  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) )
4130, 34, 40syl2anc 665 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( K `  { C } ) 
C_  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) ) )
4232snssd 4148 . . . . . . . 8  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  { C }  C_  B )
439, 15, 42mrcssidd 15482 . . . . . . 7  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  { C }  C_  ( K `  { C } ) )
44 snssg 4136 . . . . . . . 8  |-  ( C  e.  B  ->  ( C  e.  ( K `  { C } )  <->  { C }  C_  ( K `  { C } ) ) )
4532, 44syl 17 . . . . . . 7  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( C  e.  ( K `  { C } )  <->  { C }  C_  ( K `  { C } ) ) )
4643, 45mpbird 235 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  ( K `  { C } ) )
4741, 46sseldd 3471 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) )
48 eldifn 3594 . . . . . 6  |-  ( C  e.  ( U  \ 
( S  .(+)  W ) )  ->  -.  C  e.  ( S  .(+)  W ) )
4948adantl 467 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  -.  C  e.  ( S  .(+)  W ) )
5039, 47, 49ssnelpssd 3864 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( S  .(+) 
W )  C.  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) )
5127lsmub1 17243 . . . . . . . . 9  |-  ( ( S  e.  (SubGrp `  G )  /\  W  e.  (SubGrp `  G )
)  ->  S  C_  ( S  .(+)  W ) )
5225, 26, 51syl2anc 665 . . . . . . . 8  |-  ( ph  ->  S  C_  ( S  .(+) 
W ) )
5322snssd 4148 . . . . . . . . . . 11  |-  ( ph  ->  { A }  C_  B )
548, 15, 53mrcssidd 15482 . . . . . . . . . 10  |-  ( ph  ->  { A }  C_  ( K `  { A } ) )
5554, 18syl6sseqr 3517 . . . . . . . . 9  |-  ( ph  ->  { A }  C_  S )
56 snssg 4136 . . . . . . . . . 10  |-  ( A  e.  U  ->  ( A  e.  S  <->  { A }  C_  S ) )
5721, 56syl 17 . . . . . . . . 9  |-  ( ph  ->  ( A  e.  S  <->  { A }  C_  S
) )
5855, 57mpbird 235 . . . . . . . 8  |-  ( ph  ->  A  e.  S )
5952, 58sseldd 3471 . . . . . . 7  |-  ( ph  ->  A  e.  ( S 
.(+)  W ) )
6059adantr 466 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  A  e.  ( S  .(+)  W ) )
6139, 60sseldd 3471 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  A  e.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) )
623adantr 466 . . . . . . 7  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  G  e.  Abel )
6327lsmsubg2 17432 . . . . . . 7  |-  ( ( G  e.  Abel  /\  ( S  .(+)  W )  e.  (SubGrp `  G )  /\  ( K `  { C } )  e.  (SubGrp `  G ) )  -> 
( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  e.  (SubGrp `  G ) )
6462, 30, 34, 63syl3anc 1264 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  e.  (SubGrp `  G )
)
65 pgpfac1.2 . . . . . . 7  |-  ( ph  ->  A. w  e.  (SubGrp `  G ) ( ( w  C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w )
)
6665adantr 466 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  A. w  e.  (SubGrp `  G )
( ( w  C.  U  /\  A  e.  w
)  ->  -.  ( S  .(+)  W )  C.  w ) )
67 psseq1 3558 . . . . . . . . 9  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  (
w  C.  U  <->  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C.  U ) )
68 eleq2 2502 . . . . . . . . 9  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  ( A  e.  w  <->  A  e.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) ) )
6967, 68anbi12d 715 . . . . . . . 8  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  (
( w  C.  U  /\  A  e.  w
)  <->  ( ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C.  U  /\  A  e.  ( ( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) ) ) )
70 psseq2 3559 . . . . . . . . 9  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  (
( S  .(+)  W ) 
C.  w  <->  ( S  .(+) 
W )  C.  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) ) )
7170notbid 295 . . . . . . . 8  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  ( -.  ( S  .(+)  W ) 
C.  w  <->  -.  ( S  .(+)  W )  C.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) ) )
7269, 71imbi12d 321 . . . . . . 7  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  (
( ( w  C.  U  /\  A  e.  w
)  ->  -.  ( S  .(+)  W )  C.  w )  <->  ( (
( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C.  U  /\  A  e.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) )  ->  -.  ( S  .(+) 
W )  C.  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) ) ) )
7372rspcv 3184 . . . . . 6  |-  ( ( ( S  .(+)  W ) 
.(+)  ( K `  { C } ) )  e.  (SubGrp `  G
)  ->  ( A. w  e.  (SubGrp `  G
) ( ( w 
C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W ) 
C.  w )  -> 
( ( ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C.  U  /\  A  e.  ( ( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) )  ->  -.  ( S  .(+)  W )  C.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) ) ) )
7464, 66, 73sylc 62 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( (
( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C.  U  /\  A  e.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) )  ->  -.  ( S  .(+) 
W )  C.  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) ) )
7561, 74mpan2d 678 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) 
C.  U  ->  -.  ( S  .(+)  W ) 
C.  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) ) ) )
7650, 75mt2d 120 . . 3  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  -.  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) 
C.  U )
77 npss 3581 . . 3  |-  ( -.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C.  U  <->  ( (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) 
C_  U  ->  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) )  =  U ) )
7876, 77sylib 199 . 2  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) 
C_  U  ->  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) )  =  U ) )
7937, 78mpd 15 1  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  =  U )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782    \ cdif 3439    i^i cin 3441    C_ wss 3442    C. wpss 3443   {csn 4002   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Fincfn 7577   Basecbs 15084   0gc0g 15297  Moorecmre 15439  mrClscmrc 15440  ACScacs 15442   Grpcgrp 16620  SubGrpcsubg 16762   odcod 17116  gExcgex 17117   pGrp cpgp 17118   LSSumclsm 17221   Abelcabl 17366
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-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
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-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  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-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  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-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-0g 15299  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-grp 16624  df-minusg 16625  df-subg 16765  df-cntz 16922  df-lsm 17223  df-cmn 17367  df-abl 17368
This theorem is referenced by:  pgpfac1lem2  17643  pgpfac1lem3  17645
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