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Theorem lsmcntzr 16489
Description: The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
lsmcntz.p  |-  .(+)  =  (
LSSum `  G )
lsmcntz.s  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
lsmcntz.t  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
lsmcntz.u  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
lsmcntz.z  |-  Z  =  (Cntz `  G )
Assertion
Ref Expression
lsmcntzr  |-  ( ph  ->  ( S  C_  ( Z `  ( T  .(+) 
U ) )  <->  ( S  C_  ( Z `  T
)  /\  S  C_  ( Z `  U )
) ) )

Proof of Theorem lsmcntzr
StepHypRef Expression
1 lsmcntz.p . . 3  |-  .(+)  =  (
LSSum `  G )
2 lsmcntz.t . . 3  |-  ( ph  ->  T  e.  (SubGrp `  G ) )
3 lsmcntz.u . . 3  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
4 lsmcntz.s . . 3  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
5 lsmcntz.z . . 3  |-  Z  =  (Cntz `  G )
61, 2, 3, 4, 5lsmcntz 16488 . 2  |-  ( ph  ->  ( ( T  .(+)  U )  C_  ( Z `  S )  <->  ( T  C_  ( Z `  S
)  /\  U  C_  ( Z `  S )
) ) )
7 subgrcl 15996 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
8 grpmnd 15858 . . . . 5  |-  ( G  e.  Grp  ->  G  e.  Mnd )
94, 7, 83syl 20 . . . 4  |-  ( ph  ->  G  e.  Mnd )
10 eqid 2462 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
1110subgss 15992 . . . . 5  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
122, 11syl 16 . . . 4  |-  ( ph  ->  T  C_  ( Base `  G ) )
1310subgss 15992 . . . . 5  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
143, 13syl 16 . . . 4  |-  ( ph  ->  U  C_  ( Base `  G ) )
1510, 1lsmssv 16454 . . . 4  |-  ( ( G  e.  Mnd  /\  T  C_  ( Base `  G
)  /\  U  C_  ( Base `  G ) )  ->  ( T  .(+)  U )  C_  ( Base `  G ) )
169, 12, 14, 15syl3anc 1223 . . 3  |-  ( ph  ->  ( T  .(+)  U ) 
C_  ( Base `  G
) )
1710subgss 15992 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
184, 17syl 16 . . 3  |-  ( ph  ->  S  C_  ( Base `  G ) )
1910, 5cntzrec 16161 . . 3  |-  ( ( ( T  .(+)  U ) 
C_  ( Base `  G
)  /\  S  C_  ( Base `  G ) )  ->  ( ( T 
.(+)  U )  C_  ( Z `  S )  <->  S 
C_  ( Z `  ( T  .(+)  U ) ) ) )
2016, 18, 19syl2anc 661 . 2  |-  ( ph  ->  ( ( T  .(+)  U )  C_  ( Z `  S )  <->  S  C_  ( Z `  ( T  .(+) 
U ) ) ) )
2110, 5cntzrec 16161 . . . 4  |-  ( ( T  C_  ( Base `  G )  /\  S  C_  ( Base `  G
) )  ->  ( T  C_  ( Z `  S )  <->  S  C_  ( Z `  T )
) )
2212, 18, 21syl2anc 661 . . 3  |-  ( ph  ->  ( T  C_  ( Z `  S )  <->  S 
C_  ( Z `  T ) ) )
2310, 5cntzrec 16161 . . . 4  |-  ( ( U  C_  ( Base `  G )  /\  S  C_  ( Base `  G
) )  ->  ( U  C_  ( Z `  S )  <->  S  C_  ( Z `  U )
) )
2414, 18, 23syl2anc 661 . . 3  |-  ( ph  ->  ( U  C_  ( Z `  S )  <->  S 
C_  ( Z `  U ) ) )
2522, 24anbi12d 710 . 2  |-  ( ph  ->  ( ( T  C_  ( Z `  S )  /\  U  C_  ( Z `  S )
)  <->  ( S  C_  ( Z `  T )  /\  S  C_  ( Z `  U )
) ) )
266, 20, 253bitr3d 283 1  |-  ( ph  ->  ( S  C_  ( Z `  ( T  .(+) 
U ) )  <->  ( S  C_  ( Z `  T
)  /\  S  C_  ( Z `  U )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    C_ wss 3471   ` cfv 5581  (class class class)co 6277   Basecbs 14481   Mndcmnd 15717   Grpcgrp 15718  SubGrpcsubg 15985  Cntzccntz 16143   LSSumclsm 16445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-0g 14688  df-mnd 15723  df-submnd 15773  df-grp 15853  df-minusg 15854  df-subg 15988  df-cntz 16145  df-lsm 16447
This theorem is referenced by: (None)
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