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Theorem pgpfac1lem1 16594
Description: Lemma for pgpfac1 16600. (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 465 . . 3  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( S  .(+) 
W )  C_  U
)
3 pgpfac1.g . . . . . 6  |-  ( ph  ->  G  e.  Abel )
4 ablgrp 16301 . . . . . 6  |-  ( G  e.  Abel  ->  G  e. 
Grp )
5 pgpfac1.b . . . . . . 7  |-  B  =  ( Base `  G
)
65subgacs 15735 . . . . . 6  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  B ) )
7 acsmre 14609 . . . . . 6  |-  ( (SubGrp `  G )  e.  (ACS
`  B )  -> 
(SubGrp `  G )  e.  (Moore `  B )
)
83, 4, 6, 74syl 21 . . . . 5  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  B )
)
98adantr 465 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  (SubGrp `  G
)  e.  (Moore `  B ) )
10 eldifi 3497 . . . . . 6  |-  ( C  e.  ( U  \ 
( S  .(+)  W ) )  ->  C  e.  U )
1110adantl 466 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  U )
1211snssd 4037 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  { C }  C_  U )
13 pgpfac1.u . . . . 5  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
1413adantr 465 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  U  e.  (SubGrp `  G ) )
15 pgpfac1.k . . . . 5  |-  K  =  (mrCls `  (SubGrp `  G
) )
1615mrcsscl 14577 . . . 4  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  { C }  C_  U  /\  U  e.  (SubGrp `  G ) )  -> 
( K `  { C } )  C_  U
)
179, 12, 14, 16syl3anc 1218 . . 3  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( K `  { C } ) 
C_  U )
18 pgpfac1.s . . . . . . 7  |-  S  =  ( K `  { A } )
195subgss 15701 . . . . . . . . . 10  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
2013, 19syl 16 . . . . . . . . 9  |-  ( ph  ->  U  C_  B )
21 pgpfac1.au . . . . . . . . 9  |-  ( ph  ->  A  e.  U )
2220, 21sseldd 3376 . . . . . . . 8  |-  ( ph  ->  A  e.  B )
2315mrcsncl 14569 . . . . . . . 8  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  A  e.  B
)  ->  ( K `  { A } )  e.  (SubGrp `  G
) )
248, 22, 23syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( K `  { A } )  e.  (SubGrp `  G ) )
2518, 24syl5eqel 2527 . . . . . 6  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
26 pgpfac1.w . . . . . 6  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
27 pgpfac1.l . . . . . . 7  |-  .(+)  =  (
LSSum `  G )
2827lsmsubg2 16360 . . . . . 6  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )  /\  W  e.  (SubGrp `  G ) )  -> 
( S  .(+)  W )  e.  (SubGrp `  G
) )
293, 25, 26, 28syl3anc 1218 . . . . 5  |-  ( ph  ->  ( S  .(+)  W )  e.  (SubGrp `  G
) )
3029adantr 465 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( S  .(+) 
W )  e.  (SubGrp `  G ) )
3120sselda 3375 . . . . . 6  |-  ( (
ph  /\  C  e.  U )  ->  C  e.  B )
3210, 31sylan2 474 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  B )
3315mrcsncl 14569 . . . . 5  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  C  e.  B
)  ->  ( K `  { C } )  e.  (SubGrp `  G
) )
349, 32, 33syl2anc 661 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( K `  { C } )  e.  (SubGrp `  G
) )
3527lsmlub 16181 . . . 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 1218 . . 3  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( (
( S  .(+)  W ) 
C_  U  /\  ( K `  { C } )  C_  U
)  <->  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  C_  U
) )
372, 17, 36mpbi2and 912 . 2  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C_  U )
3827lsmub1 16174 . . . . . 6  |-  ( ( ( S  .(+)  W )  e.  (SubGrp `  G
)  /\  ( K `  { C } )  e.  (SubGrp `  G
) )  ->  ( S  .(+)  W )  C_  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) )
3930, 34, 38syl2anc 661 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( S  .(+) 
W )  C_  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) )
4027lsmub2 16175 . . . . . . 7  |-  ( ( ( S  .(+)  W )  e.  (SubGrp `  G
)  /\  ( K `  { C } )  e.  (SubGrp `  G
) )  ->  ( K `  { C } )  C_  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) )
4130, 34, 40syl2anc 661 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( K `  { C } ) 
C_  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) ) )
4232snssd 4037 . . . . . . . 8  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  { C }  C_  B )
439, 15, 42mrcssidd 14582 . . . . . . 7  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  { C }  C_  ( K `  { C } ) )
44 snssg 4026 . . . . . . . 8  |-  ( C  e.  B  ->  ( C  e.  ( K `  { C } )  <->  { C }  C_  ( K `  { C } ) ) )
4532, 44syl 16 . . . . . . 7  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( C  e.  ( K `  { C } )  <->  { C }  C_  ( K `  { C } ) ) )
4643, 45mpbird 232 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  ( K `  { C } ) )
4741, 46sseldd 3376 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) )
48 eldifn 3498 . . . . . 6  |-  ( C  e.  ( U  \ 
( S  .(+)  W ) )  ->  -.  C  e.  ( S  .(+)  W ) )
4948adantl 466 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  -.  C  e.  ( S  .(+)  W ) )
5039, 47, 49ssnelpssd 3762 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( S  .(+) 
W )  C.  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) )
5127lsmub1 16174 . . . . . . . . 9  |-  ( ( S  e.  (SubGrp `  G )  /\  W  e.  (SubGrp `  G )
)  ->  S  C_  ( S  .(+)  W ) )
5225, 26, 51syl2anc 661 . . . . . . . 8  |-  ( ph  ->  S  C_  ( S  .(+) 
W ) )
5322snssd 4037 . . . . . . . . . . 11  |-  ( ph  ->  { A }  C_  B )
548, 15, 53mrcssidd 14582 . . . . . . . . . 10  |-  ( ph  ->  { A }  C_  ( K `  { A } ) )
5554, 18syl6sseqr 3422 . . . . . . . . 9  |-  ( ph  ->  { A }  C_  S )
56 snssg 4026 . . . . . . . . . 10  |-  ( A  e.  U  ->  ( A  e.  S  <->  { A }  C_  S ) )
5721, 56syl 16 . . . . . . . . 9  |-  ( ph  ->  ( A  e.  S  <->  { A }  C_  S
) )
5855, 57mpbird 232 . . . . . . . 8  |-  ( ph  ->  A  e.  S )
5952, 58sseldd 3376 . . . . . . 7  |-  ( ph  ->  A  e.  ( S 
.(+)  W ) )
6059adantr 465 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  A  e.  ( S  .(+)  W ) )
6139, 60sseldd 3376 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  A  e.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) )
623adantr 465 . . . . . . 7  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  G  e.  Abel )
6327lsmsubg2 16360 . . . . . . 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 1218 . . . . . 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 465 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  A. w  e.  (SubGrp `  G )
( ( w  C.  U  /\  A  e.  w
)  ->  -.  ( S  .(+)  W )  C.  w ) )
67 psseq1 3462 . . . . . . . . 9  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  (
w  C.  U  <->  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C.  U ) )
68 eleq2 2504 . . . . . . . . 9  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  ( A  e.  w  <->  A  e.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) ) )
6967, 68anbi12d 710 . . . . . . . 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 3463 . . . . . . . . 9  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  (
( S  .(+)  W ) 
C.  w  <->  ( S  .(+) 
W )  C.  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) ) )
7170notbid 294 . . . . . . . 8  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  ( -.  ( S  .(+)  W ) 
C.  w  <->  -.  ( S  .(+)  W )  C.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) ) )
7269, 71imbi12d 320 . . . . . . 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 3088 . . . . . 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 60 . . . . 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 674 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) 
C.  U  ->  -.  ( S  .(+)  W ) 
C.  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) ) ) )
7650, 75mt2d 117 . . 3  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  -.  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) 
C.  U )
77 npss 3485 . . 3  |-  ( -.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C.  U  <->  ( (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) 
C_  U  ->  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) )  =  U ) )
7876, 77sylib 196 . 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 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2734    \ cdif 3344    i^i cin 3346    C_ wss 3347    C. wpss 3348   {csn 3896   class class class wbr 4311   ` cfv 5437  (class class class)co 6110   Fincfn 7329   Basecbs 14193   0gc0g 14397  Moorecmre 14539  mrClscmrc 14540  ACScacs 14542   Grpcgrp 15429  SubGrpcsubg 15694   odcod 16047  gExcgex 16048   pGrp cpgp 16049   LSSumclsm 16152   Abelcabel 16297
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 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-iin 4193  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-oadd 6943  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-nn 10342  df-2 10399  df-ndx 14196  df-slot 14197  df-base 14198  df-sets 14199  df-ress 14200  df-plusg 14270  df-0g 14399  df-mre 14543  df-mrc 14544  df-acs 14546  df-mnd 15434  df-submnd 15484  df-grp 15564  df-minusg 15565  df-subg 15697  df-cntz 15854  df-lsm 16154  df-cmn 16298  df-abl 16299
This theorem is referenced by:  pgpfac1lem2  16595  pgpfac1lem3  16597
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