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Theorem pgpfac1lem1 15624
Description: Lemma for pgpfac1 15630. (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 452 . . 3  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( S  .(+) 
W )  C_  U
)
3 pgpfac1.g . . . . . . 7  |-  ( ph  ->  G  e.  Abel )
4 ablgrp 15409 . . . . . . 7  |-  ( G  e.  Abel  ->  G  e. 
Grp )
53, 4syl 16 . . . . . 6  |-  ( ph  ->  G  e.  Grp )
6 pgpfac1.b . . . . . . 7  |-  B  =  ( Base `  G
)
76subgacs 14967 . . . . . 6  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  B ) )
8 acsmre 13869 . . . . . 6  |-  ( (SubGrp `  G )  e.  (ACS
`  B )  -> 
(SubGrp `  G )  e.  (Moore `  B )
)
95, 7, 83syl 19 . . . . 5  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  B )
)
109adantr 452 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  (SubGrp `  G
)  e.  (Moore `  B ) )
11 eldifi 3461 . . . . . 6  |-  ( C  e.  ( U  \ 
( S  .(+)  W ) )  ->  C  e.  U )
1211adantl 453 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  U )
1312snssd 3935 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  { C }  C_  U )
14 pgpfac1.u . . . . 5  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
1514adantr 452 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  U  e.  (SubGrp `  G ) )
16 pgpfac1.k . . . . 5  |-  K  =  (mrCls `  (SubGrp `  G
) )
1716mrcsscl 13837 . . . 4  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  { C }  C_  U  /\  U  e.  (SubGrp `  G ) )  -> 
( K `  { C } )  C_  U
)
1810, 13, 15, 17syl3anc 1184 . . 3  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( K `  { C } ) 
C_  U )
19 pgpfac1.s . . . . . . 7  |-  S  =  ( K `  { A } )
206subgss 14937 . . . . . . . . . 10  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
2114, 20syl 16 . . . . . . . . 9  |-  ( ph  ->  U  C_  B )
22 pgpfac1.au . . . . . . . . 9  |-  ( ph  ->  A  e.  U )
2321, 22sseldd 3341 . . . . . . . 8  |-  ( ph  ->  A  e.  B )
2416mrcsncl 13829 . . . . . . . 8  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  A  e.  B
)  ->  ( K `  { A } )  e.  (SubGrp `  G
) )
259, 23, 24syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( K `  { A } )  e.  (SubGrp `  G ) )
2619, 25syl5eqel 2519 . . . . . 6  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
27 pgpfac1.w . . . . . 6  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
28 pgpfac1.l . . . . . . 7  |-  .(+)  =  (
LSSum `  G )
2928lsmsubg2 15466 . . . . . 6  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )  /\  W  e.  (SubGrp `  G ) )  -> 
( S  .(+)  W )  e.  (SubGrp `  G
) )
303, 26, 27, 29syl3anc 1184 . . . . 5  |-  ( ph  ->  ( S  .(+)  W )  e.  (SubGrp `  G
) )
3130adantr 452 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( S  .(+) 
W )  e.  (SubGrp `  G ) )
3221sselda 3340 . . . . . 6  |-  ( (
ph  /\  C  e.  U )  ->  C  e.  B )
3311, 32sylan2 461 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  B )
3416mrcsncl 13829 . . . . 5  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  C  e.  B
)  ->  ( K `  { C } )  e.  (SubGrp `  G
) )
3510, 33, 34syl2anc 643 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( K `  { C } )  e.  (SubGrp `  G
) )
3628lsmlub 15289 . . . 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 ) )
3731, 35, 15, 36syl3anc 1184 . . 3  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( (
( S  .(+)  W ) 
C_  U  /\  ( K `  { C } )  C_  U
)  <->  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  C_  U
) )
382, 18, 37mpbi2and 888 . 2  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C_  U )
3928lsmub1 15282 . . . . . 6  |-  ( ( ( S  .(+)  W )  e.  (SubGrp `  G
)  /\  ( K `  { C } )  e.  (SubGrp `  G
) )  ->  ( S  .(+)  W )  C_  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) )
4031, 35, 39syl2anc 643 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( S  .(+) 
W )  C_  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) )
4128lsmub2 15283 . . . . . . 7  |-  ( ( ( S  .(+)  W )  e.  (SubGrp `  G
)  /\  ( K `  { C } )  e.  (SubGrp `  G
) )  ->  ( K `  { C } )  C_  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) )
4231, 35, 41syl2anc 643 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( K `  { C } ) 
C_  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) ) )
4333snssd 3935 . . . . . . . 8  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  { C }  C_  B )
4410, 16, 43mrcssidd 13842 . . . . . . 7  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  { C }  C_  ( K `  { C } ) )
45 snssg 3924 . . . . . . . 8  |-  ( C  e.  B  ->  ( C  e.  ( K `  { C } )  <->  { C }  C_  ( K `  { C } ) ) )
4633, 45syl 16 . . . . . . 7  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( C  e.  ( K `  { C } )  <->  { C }  C_  ( K `  { C } ) ) )
4744, 46mpbird 224 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  ( K `  { C } ) )
4842, 47sseldd 3341 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) )
49 eldifn 3462 . . . . . 6  |-  ( C  e.  ( U  \ 
( S  .(+)  W ) )  ->  -.  C  e.  ( S  .(+)  W ) )
5049adantl 453 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  -.  C  e.  ( S  .(+)  W ) )
5140, 48, 50ssnelpssd 3684 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( S  .(+) 
W )  C.  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) )
5228lsmub1 15282 . . . . . . . . 9  |-  ( ( S  e.  (SubGrp `  G )  /\  W  e.  (SubGrp `  G )
)  ->  S  C_  ( S  .(+)  W ) )
5326, 27, 52syl2anc 643 . . . . . . . 8  |-  ( ph  ->  S  C_  ( S  .(+) 
W ) )
5423snssd 3935 . . . . . . . . . . 11  |-  ( ph  ->  { A }  C_  B )
559, 16, 54mrcssidd 13842 . . . . . . . . . 10  |-  ( ph  ->  { A }  C_  ( K `  { A } ) )
5655, 19syl6sseqr 3387 . . . . . . . . 9  |-  ( ph  ->  { A }  C_  S )
57 snssg 3924 . . . . . . . . . 10  |-  ( A  e.  U  ->  ( A  e.  S  <->  { A }  C_  S ) )
5822, 57syl 16 . . . . . . . . 9  |-  ( ph  ->  ( A  e.  S  <->  { A }  C_  S
) )
5956, 58mpbird 224 . . . . . . . 8  |-  ( ph  ->  A  e.  S )
6053, 59sseldd 3341 . . . . . . 7  |-  ( ph  ->  A  e.  ( S 
.(+)  W ) )
6160adantr 452 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  A  e.  ( S  .(+)  W ) )
6240, 61sseldd 3341 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  A  e.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) )
633adantr 452 . . . . . . 7  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  G  e.  Abel )
6428lsmsubg2 15466 . . . . . . 7  |-  ( ( G  e.  Abel  /\  ( S  .(+)  W )  e.  (SubGrp `  G )  /\  ( K `  { C } )  e.  (SubGrp `  G ) )  -> 
( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  e.  (SubGrp `  G ) )
6563, 31, 35, 64syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  e.  (SubGrp `  G )
)
66 pgpfac1.2 . . . . . . 7  |-  ( ph  ->  A. w  e.  (SubGrp `  G ) ( ( w  C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w ) )
6766adantr 452 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  A. w  e.  (SubGrp `  G )
( ( w  C.  U  /\  A  e.  w
)  ->  -.  ( S  .(+)  W )  C.  w ) )
68 psseq1 3426 . . . . . . . . 9  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  (
w  C.  U  <->  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C.  U ) )
69 eleq2 2496 . . . . . . . . 9  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  ( A  e.  w  <->  A  e.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) ) )
7068, 69anbi12d 692 . . . . . . . 8  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  (
( w  C.  U  /\  A  e.  w
)  <->  ( ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C.  U  /\  A  e.  ( ( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) ) ) )
71 psseq2 3427 . . . . . . . . 9  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  (
( S  .(+)  W ) 
C.  w  <->  ( S  .(+) 
W )  C.  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) ) )
7271notbid 286 . . . . . . . 8  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  ( -.  ( S  .(+)  W ) 
C.  w  <->  -.  ( S  .(+)  W )  C.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) ) )
7370, 72imbi12d 312 . . . . . . 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 } ) ) ) ) )
7473rspcv 3040 . . . . . 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 } ) ) ) ) )
7565, 67, 74sylc 58 . . . . 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 } ) ) ) )
7662, 75mpan2d 656 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) 
C.  U  ->  -.  ( S  .(+)  W ) 
C.  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) ) ) )
7751, 76mt2d 111 . . 3  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  -.  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) 
C.  U )
78 npss 3449 . . 3  |-  ( -.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C.  U  <->  ( (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) 
C_  U  ->  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) )  =  U ) )
7977, 78sylib 189 . 2  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) 
C_  U  ->  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) )  =  U ) )
8038, 79mpd 15 1  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  =  U )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    \ cdif 3309    i^i cin 3311    C_ wss 3312    C. wpss 3313   {csn 3806   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Fincfn 7101   Basecbs 13461   0gc0g 13715  Moorecmre 13799  mrClscmrc 13800  ACScacs 13802   Grpcgrp 14677  SubGrpcsubg 14930   odcod 15155  gExcgex 15156   pGrp cpgp 15157   LSSumclsm 15260   Abelcabel 15405
This theorem is referenced by:  pgpfac1lem2  15625  pgpfac1lem3  15627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-subg 14933  df-cntz 15108  df-lsm 15262  df-cmn 15406  df-abl 15407
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