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Theorem lsmcntzr 15304
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 15303 . 2  |-  ( ph  ->  ( ( T  .(+)  U )  C_  ( Z `  S )  <->  ( T  C_  ( Z `  S
)  /\  U  C_  ( Z `  S )
) ) )
7 subgrcl 14941 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
8 grpmnd 14809 . . . . 5  |-  ( G  e.  Grp  ->  G  e.  Mnd )
94, 7, 83syl 19 . . . 4  |-  ( ph  ->  G  e.  Mnd )
10 eqid 2435 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
1110subgss 14937 . . . . 5  |-  ( T  e.  (SubGrp `  G
)  ->  T  C_  ( Base `  G ) )
122, 11syl 16 . . . 4  |-  ( ph  ->  T  C_  ( Base `  G ) )
1310subgss 14937 . . . . 5  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  ( Base `  G ) )
143, 13syl 16 . . . 4  |-  ( ph  ->  U  C_  ( Base `  G ) )
1510, 1lsmssv 15269 . . . 4  |-  ( ( G  e.  Mnd  /\  T  C_  ( Base `  G
)  /\  U  C_  ( Base `  G ) )  ->  ( T  .(+)  U )  C_  ( Base `  G ) )
169, 12, 14, 15syl3anc 1184 . . 3  |-  ( ph  ->  ( T  .(+)  U ) 
C_  ( Base `  G
) )
1710subgss 14937 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
184, 17syl 16 . . 3  |-  ( ph  ->  S  C_  ( Base `  G ) )
1910, 5cntzrec 15124 . . 3  |-  ( ( ( T  .(+)  U ) 
C_  ( Base `  G
)  /\  S  C_  ( Base `  G ) )  ->  ( ( T 
.(+)  U )  C_  ( Z `  S )  <->  S 
C_  ( Z `  ( T  .(+)  U ) ) ) )
2016, 18, 19syl2anc 643 . 2  |-  ( ph  ->  ( ( T  .(+)  U )  C_  ( Z `  S )  <->  S  C_  ( Z `  ( T  .(+) 
U ) ) ) )
2110, 5cntzrec 15124 . . . 4  |-  ( ( T  C_  ( Base `  G )  /\  S  C_  ( Base `  G
) )  ->  ( T  C_  ( Z `  S )  <->  S  C_  ( Z `  T )
) )
2212, 18, 21syl2anc 643 . . 3  |-  ( ph  ->  ( T  C_  ( Z `  S )  <->  S 
C_  ( Z `  T ) ) )
2310, 5cntzrec 15124 . . . 4  |-  ( ( U  C_  ( Base `  G )  /\  S  C_  ( Base `  G
) )  ->  ( U  C_  ( Z `  S )  <->  S  C_  ( Z `  U )
) )
2414, 18, 23syl2anc 643 . . 3  |-  ( ph  ->  ( U  C_  ( Z `  S )  <->  S 
C_  ( Z `  U ) ) )
2522, 24anbi12d 692 . 2  |-  ( ph  ->  ( ( T  C_  ( Z `  S )  /\  U  C_  ( Z `  S )
)  <->  ( S  C_  ( Z `  T )  /\  S  C_  ( Z `  U )
) ) )
266, 20, 253bitr3d 275 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312   ` cfv 5446  (class class class)co 6073   Basecbs 13461   Mndcmnd 14676   Grpcgrp 14677  SubGrpcsubg 14930  Cntzccntz 15106   LSSumclsm 15260
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-iun 4087  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-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  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-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-subg 14933  df-cntz 15108  df-lsm 15262
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