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Theorem lsmcntzr 14989
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 14988 . 2  |-  ( ph  ->  ( ( T  .(+)  U )  C_  ( Z `  S )  <->  ( T  C_  ( Z `  S
)  /\  U  C_  ( Z `  S )
) ) )
7 subgrcl 14626 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
8 grpmnd 14494 . . . . 5  |-  ( G  e.  Grp  ->  G  e.  Mnd )
94, 7, 83syl 18 . . . 4  |-  ( ph  ->  G  e.  Mnd )
10 eqid 2283 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
1110subgss 14622 . . . . 5  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
122, 11syl 15 . . . 4  |-  ( ph  ->  T  C_  ( Base `  G ) )
1310subgss 14622 . . . . 5  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
143, 13syl 15 . . . 4  |-  ( ph  ->  U  C_  ( Base `  G ) )
1510, 1lsmssv 14954 . . . 4  |-  ( ( G  e.  Mnd  /\  T  C_  ( Base `  G
)  /\  U  C_  ( Base `  G ) )  ->  ( T  .(+)  U )  C_  ( Base `  G ) )
169, 12, 14, 15syl3anc 1182 . . 3  |-  ( ph  ->  ( T  .(+)  U ) 
C_  ( Base `  G
) )
1710subgss 14622 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
184, 17syl 15 . . 3  |-  ( ph  ->  S  C_  ( Base `  G ) )
1910, 5cntzrec 14809 . . 3  |-  ( ( ( T  .(+)  U ) 
C_  ( Base `  G
)  /\  S  C_  ( Base `  G ) )  ->  ( ( T 
.(+)  U )  C_  ( Z `  S )  <->  S 
C_  ( Z `  ( T  .(+)  U ) ) ) )
2016, 18, 19syl2anc 642 . 2  |-  ( ph  ->  ( ( T  .(+)  U )  C_  ( Z `  S )  <->  S  C_  ( Z `  ( T  .(+) 
U ) ) ) )
2110, 5cntzrec 14809 . . . 4  |-  ( ( T  C_  ( Base `  G )  /\  S  C_  ( Base `  G
) )  ->  ( T  C_  ( Z `  S )  <->  S  C_  ( Z `  T )
) )
2212, 18, 21syl2anc 642 . . 3  |-  ( ph  ->  ( T  C_  ( Z `  S )  <->  S 
C_  ( Z `  T ) ) )
2310, 5cntzrec 14809 . . . 4  |-  ( ( U  C_  ( Base `  G )  /\  S  C_  ( Base `  G
) )  ->  ( U  C_  ( Z `  S )  <->  S  C_  ( Z `  U )
) )
2414, 18, 23syl2anc 642 . . 3  |-  ( ph  ->  ( U  C_  ( Z `  S )  <->  S 
C_  ( Z `  U ) ) )
2522, 24anbi12d 691 . 2  |-  ( ph  ->  ( ( T  C_  ( Z `  S )  /\  U  C_  ( Z `  S )
)  <->  ( S  C_  ( Z `  T )  /\  S  C_  ( Z `  U )
) ) )
266, 20, 253bitr3d 274 1  |-  ( ph  ->  ( S  C_  ( Z `  ( T  .(+) 
U ) )  <->  ( S  C_  ( Z `  T
)  /\  S  C_  ( Z `  U )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   Mndcmnd 14361   Grpcgrp 14362  SubGrpcsubg 14615  Cntzccntz 14791   LSSumclsm 14945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-subg 14618  df-cntz 14793  df-lsm 14947
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