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Theorem pgpfac1lem1 15325
Description: Lemma for pgpfac1 15331. (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 451 . . 3  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( S  .(+) 
W )  C_  U
)
3 pgpfac1.g . . . . . . 7  |-  ( ph  ->  G  e.  Abel )
4 ablgrp 15110 . . . . . . 7  |-  ( G  e.  Abel  ->  G  e. 
Grp )
53, 4syl 15 . . . . . 6  |-  ( ph  ->  G  e.  Grp )
6 pgpfac1.b . . . . . . 7  |-  B  =  ( Base `  G
)
76subgacs 14668 . . . . . 6  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  B ) )
8 acsmre 13570 . . . . . 6  |-  ( (SubGrp `  G )  e.  (ACS
`  B )  -> 
(SubGrp `  G )  e.  (Moore `  B )
)
95, 7, 83syl 18 . . . . 5  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  B )
)
109adantr 451 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  (SubGrp `  G
)  e.  (Moore `  B ) )
11 eldifi 3311 . . . . . 6  |-  ( C  e.  ( U  \ 
( S  .(+)  W ) )  ->  C  e.  U )
1211adantl 452 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  U )
1312snssd 3776 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  { C }  C_  U )
14 pgpfac1.u . . . . 5  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
1514adantr 451 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  U  e.  (SubGrp `  G ) )
16 pgpfac1.k . . . . 5  |-  K  =  (mrCls `  (SubGrp `  G
) )
1716mrcsscl 13538 . . . 4  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  { C }  C_  U  /\  U  e.  (SubGrp `  G ) )  -> 
( K `  { C } )  C_  U
)
1810, 13, 15, 17syl3anc 1182 . . 3  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( K `  { C } ) 
C_  U )
19 pgpfac1.s . . . . . . 7  |-  S  =  ( K `  { A } )
206subgss 14638 . . . . . . . . . 10  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
2114, 20syl 15 . . . . . . . . 9  |-  ( ph  ->  U  C_  B )
22 pgpfac1.au . . . . . . . . 9  |-  ( ph  ->  A  e.  U )
2321, 22sseldd 3194 . . . . . . . 8  |-  ( ph  ->  A  e.  B )
2416mrcsncl 13530 . . . . . . . 8  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  A  e.  B
)  ->  ( K `  { A } )  e.  (SubGrp `  G
) )
259, 23, 24syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( K `  { A } )  e.  (SubGrp `  G ) )
2619, 25syl5eqel 2380 . . . . . 6  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
27 pgpfac1.w . . . . . 6  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
28 pgpfac1.l . . . . . . 7  |-  .(+)  =  (
LSSum `  G )
2928lsmsubg2 15167 . . . . . 6  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )  /\  W  e.  (SubGrp `  G ) )  -> 
( S  .(+)  W )  e.  (SubGrp `  G
) )
303, 26, 27, 29syl3anc 1182 . . . . 5  |-  ( ph  ->  ( S  .(+)  W )  e.  (SubGrp `  G
) )
3130adantr 451 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( S  .(+) 
W )  e.  (SubGrp `  G ) )
3221sselda 3193 . . . . . 6  |-  ( (
ph  /\  C  e.  U )  ->  C  e.  B )
3311, 32sylan2 460 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  B )
3416mrcsncl 13530 . . . . 5  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  C  e.  B
)  ->  ( K `  { C } )  e.  (SubGrp `  G
) )
3510, 33, 34syl2anc 642 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( K `  { C } )  e.  (SubGrp `  G
) )
3628lsmlub 14990 . . . 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 1182 . . 3  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( (
( S  .(+)  W ) 
C_  U  /\  ( K `  { C } )  C_  U
)  <->  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  C_  U
) )
382, 18, 37mpbi2and 887 . 2  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C_  U )
3928lsmub2 14984 . . . . . . 7  |-  ( ( ( S  .(+)  W )  e.  (SubGrp `  G
)  /\  ( K `  { C } )  e.  (SubGrp `  G
) )  ->  ( K `  { C } )  C_  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) )
4031, 35, 39syl2anc 642 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( K `  { C } ) 
C_  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) ) )
4133snssd 3776 . . . . . . . 8  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  { C }  C_  B )
4216mrcssid 13535 . . . . . . . 8  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  { C }  C_  B )  ->  { C }  C_  ( K `  { C } ) )
4310, 41, 42syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  { C }  C_  ( K `  { C } ) )
44 snssg 3767 . . . . . . . 8  |-  ( C  e.  B  ->  ( C  e.  ( K `  { C } )  <->  { C }  C_  ( K `  { C } ) ) )
4533, 44syl 15 . . . . . . 7  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( C  e.  ( K `  { C } )  <->  { C }  C_  ( K `  { C } ) ) )
4643, 45mpbird 223 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  ( K `  { C } ) )
4740, 46sseldd 3194 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  C  e.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) )
48 eldifn 3312 . . . . . 6  |-  ( C  e.  ( U  \ 
( S  .(+)  W ) )  ->  -.  C  e.  ( S  .(+)  W ) )
4948adantl 452 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  -.  C  e.  ( S  .(+)  W ) )
5028lsmub1 14983 . . . . . . 7  |-  ( ( ( S  .(+)  W )  e.  (SubGrp `  G
)  /\  ( K `  { C } )  e.  (SubGrp `  G
) )  ->  ( S  .(+)  W )  C_  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) )
5131, 35, 50syl2anc 642 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( S  .(+) 
W )  C_  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) )
52 ssnelpss 3530 . . . . . 6  |-  ( ( S  .(+)  W )  C_  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  ->  ( ( C  e.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  /\  -.  C  e.  ( S  .(+)  W ) )  ->  ( S  .(+)  W )  C.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) ) )
5351, 52syl 15 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( ( C  e.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  /\  -.  C  e.  ( S  .(+)  W ) )  ->  ( S  .(+)  W )  C.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) ) )
5447, 49, 53mp2and 660 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( S  .(+) 
W )  C.  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) )
5528lsmub1 14983 . . . . . . . . 9  |-  ( ( S  e.  (SubGrp `  G )  /\  W  e.  (SubGrp `  G )
)  ->  S  C_  ( S  .(+)  W ) )
5626, 27, 55syl2anc 642 . . . . . . . 8  |-  ( ph  ->  S  C_  ( S  .(+) 
W ) )
5723snssd 3776 . . . . . . . . . . 11  |-  ( ph  ->  { A }  C_  B )
5816mrcssid 13535 . . . . . . . . . . 11  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  { A }  C_  B )  ->  { A }  C_  ( K `  { A } ) )
599, 57, 58syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  { A }  C_  ( K `  { A } ) )
6059, 19syl6sseqr 3238 . . . . . . . . 9  |-  ( ph  ->  { A }  C_  S )
61 snssg 3767 . . . . . . . . . 10  |-  ( A  e.  U  ->  ( A  e.  S  <->  { A }  C_  S ) )
6222, 61syl 15 . . . . . . . . 9  |-  ( ph  ->  ( A  e.  S  <->  { A }  C_  S
) )
6360, 62mpbird 223 . . . . . . . 8  |-  ( ph  ->  A  e.  S )
6456, 63sseldd 3194 . . . . . . 7  |-  ( ph  ->  A  e.  ( S 
.(+)  W ) )
6564adantr 451 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  A  e.  ( S  .(+)  W ) )
6651, 65sseldd 3194 . . . . 5  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  A  e.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) )
673adantr 451 . . . . . . 7  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  G  e.  Abel )
6828lsmsubg2 15167 . . . . . . 7  |-  ( ( G  e.  Abel  /\  ( S  .(+)  W )  e.  (SubGrp `  G )  /\  ( K `  { C } )  e.  (SubGrp `  G ) )  -> 
( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  e.  (SubGrp `  G ) )
6967, 31, 35, 68syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  e.  (SubGrp `  G )
)
70 pgpfac1.2 . . . . . . 7  |-  ( ph  ->  A. w  e.  (SubGrp `  G ) ( ( w  C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w ) )
7170adantr 451 . . . . . 6  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  A. w  e.  (SubGrp `  G )
( ( w  C.  U  /\  A  e.  w
)  ->  -.  ( S  .(+)  W )  C.  w ) )
72 psseq1 3276 . . . . . . . . 9  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  (
w  C.  U  <->  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C.  U ) )
73 eleq2 2357 . . . . . . . . 9  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  ( A  e.  w  <->  A  e.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) ) )
7472, 73anbi12d 691 . . . . . . . 8  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  (
( w  C.  U  /\  A  e.  w
)  <->  ( ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C.  U  /\  A  e.  ( ( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) ) ) )
75 psseq2 3277 . . . . . . . . 9  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  (
( S  .(+)  W ) 
C.  w  <->  ( S  .(+) 
W )  C.  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) ) )
7675notbid 285 . . . . . . . 8  |-  ( w  =  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) )  ->  ( -.  ( S  .(+)  W ) 
C.  w  <->  -.  ( S  .(+)  W )  C.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) ) ) )
7774, 76imbi12d 311 . . . . . . 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 } ) ) ) ) )
7877rspcv 2893 . . . . . 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 } ) ) ) ) )
7969, 71, 78sylc 56 . . . . 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 } ) ) ) )
8066, 79mpan2d 655 . . . 4  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) 
C.  U  ->  -.  ( S  .(+)  W ) 
C.  ( ( S 
.(+)  W )  .(+)  ( K `
 { C }
) ) ) )
8154, 80mt2d 109 . . 3  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  -.  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) 
C.  U )
82 npss 3299 . . 3  |-  ( -.  ( ( S  .(+)  W )  .(+)  ( K `  { C } ) )  C.  U  <->  ( (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) 
C_  U  ->  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) )  =  U ) )
8381, 82sylib 188 . 2  |-  ( (
ph  /\  C  e.  ( U  \  ( S  .(+)  W ) ) )  ->  ( (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) ) 
C_  U  ->  (
( S  .(+)  W ) 
.(+)  ( K `  { C } ) )  =  U ) )
8438, 83mpd 14 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    \ cdif 3162    i^i cin 3164    C_ wss 3165    C. wpss 3166   {csn 3653   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Fincfn 6879   Basecbs 13164   0gc0g 13416  Moorecmre 13500  mrClscmrc 13501  ACScacs 13503   Grpcgrp 14378  SubGrpcsubg 14631   odcod 14856  gExcgex 14857   pGrp cpgp 14858   LSSumclsm 14961   Abelcabel 15106
This theorem is referenced by:  pgpfac1lem2  15326  pgpfac1lem3  15328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-subg 14634  df-cntz 14809  df-lsm 14963  df-cmn 15107  df-abl 15108
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