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Theorem pgpfac1lem1 16993
Description: Lemma for pgpfac1 16999. (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 16672 . . . . . 6  |-  ( G  e.  Abel  ->  G  e. 
Grp )
5 pgpfac1.b . . . . . . 7  |-  B  =  ( Base `  G
)
65subgacs 16105 . . . . . 6  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  B ) )
7 acsmre 14921 . . . . . 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 3608 . . . . . 6  |-  ( C  e.  ( U  \ 
( S  .(+)  W ) )  ->  C  e.  U )
1110adantl 466 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  U )
1211snssd 4156 . . . 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 14889 . . . 4  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  { C }  C_  U  /\  U  e.  (SubGrp `  G ) )  -> 
( K `  { C } )  C_  U
)
179, 12, 14, 16syl3anc 1227 . . 3  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( K `  { C } ) 
C_  U )
18 pgpfac1.s . . . . . . 7  |-  S  =  ( K `  { A } )
195subgss 16071 . . . . . . . . . 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 3487 . . . . . . . 8  |-  ( ph  ->  A  e.  B )
2315mrcsncl 14881 . . . . . . . 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 2533 . . . . . 6  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
26 pgpfac1.w . . . . . 6  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
27 pgpfac1.l . . . . . . 7  |-  .(+)  =  (
LSSum `  G )
2827lsmsubg2 16734 . . . . . 6  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )  /\  W  e.  (SubGrp `  G ) )  -> 
( S  .(+)  W )  e.  (SubGrp `  G
) )
293, 25, 26, 28syl3anc 1227 . . . . 5  |-  ( ph  ->  ( S  .(+)  W )  e.  (SubGrp `  G
) )
3029adantr 465 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( S  .(+) 
W )  e.  (SubGrp `  G ) )
3120sselda 3486 . . . . . 6  |-  ( (
ph  /\  C  e.  U )  ->  C  e.  B )
3210, 31sylan2 474 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  B )
3315mrcsncl 14881 . . . . 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 16552 . . . 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 1227 . . 3  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( (
( S  .(+)  W ) 
C_  U  /\  ( K `  { C } )  C_  U
)  <->  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  C_  U
) )
372, 17, 36mpbi2and 919 . 2  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C_  U )
3827lsmub1 16545 . . . . . 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 16546 . . . . . . 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 4156 . . . . . . . 8  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  { C }  C_  B )
439, 15, 42mrcssidd 14894 . . . . . . 7  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  { C }  C_  ( K `  { C } ) )
44 snssg 4144 . . . . . . . 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 3487 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) )
48 eldifn 3609 . . . . . 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 3873 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( S  .(+) 
W )  C.  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) )
5127lsmub1 16545 . . . . . . . . 9  |-  ( ( S  e.  (SubGrp `  G )  /\  W  e.  (SubGrp `  G )
)  ->  S  C_  ( S  .(+)  W ) )
5225, 26, 51syl2anc 661 . . . . . . . 8  |-  ( ph  ->  S  C_  ( S  .(+) 
W ) )
5322snssd 4156 . . . . . . . . . . 11  |-  ( ph  ->  { A }  C_  B )
548, 15, 53mrcssidd 14894 . . . . . . . . . 10  |-  ( ph  ->  { A }  C_  ( K `  { A } ) )
5554, 18syl6sseqr 3533 . . . . . . . . 9  |-  ( ph  ->  { A }  C_  S )
56 snssg 4144 . . . . . . . . . 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 3487 . . . . . . 7  |-  ( ph  ->  A  e.  ( S 
.(+)  W ) )
6059adantr 465 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  A  e.  ( S  .(+)  W ) )
6139, 60sseldd 3487 . . . . 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 16734 . . . . . . 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 1227 . . . . . 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 3573 . . . . . . . . 9  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  (
w  C.  U  <->  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C.  U ) )
68 eleq2 2514 . . . . . . . . 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 3574 . . . . . . . . 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 3190 . . . . . 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 3596 . . 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 1381    e. wcel 1802   A.wral 2791    \ cdif 3455    i^i cin 3457    C_ wss 3458    C. wpss 3459   {csn 4010   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   Fincfn 7514   Basecbs 14504   0gc0g 14709  Moorecmre 14851  mrClscmrc 14852  ACScacs 14854   Grpcgrp 15922  SubGrpcsubg 16064   odcod 16418  gExcgex 16419   pGrp cpgp 16420   LSSumclsm 16523   Abelcabl 16668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-0g 14711  df-mre 14855  df-mrc 14856  df-acs 14858  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-submnd 15836  df-grp 15926  df-minusg 15927  df-subg 16067  df-cntz 16224  df-lsm 16525  df-cmn 16669  df-abl 16670
This theorem is referenced by:  pgpfac1lem2  16994  pgpfac1lem3  16996
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