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Theorem lsmcom2 16173
Description: Subgroup sum commutes. (Contributed by Mario Carneiro, 22-Apr-2016.)
Hypotheses
Ref Expression
lsmsubg.p  |-  .(+)  =  (
LSSum `  G )
lsmsubg.z  |-  Z  =  (Cntz `  G )
Assertion
Ref Expression
lsmcom2  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( T  .(+)  U )  =  ( U 
.(+)  T ) )

Proof of Theorem lsmcom2
Dummy variables  a 
b  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 990 . . . . . . . . 9  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  T  C_  ( Z `  U )
)
21sselda 3375 . . . . . . . 8  |-  ( ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  /\  a  e.  T
)  ->  a  e.  ( Z `  U ) )
32adantrr 716 . . . . . . 7  |-  ( ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  /\  ( a  e.  T  /\  b  e.  U ) )  -> 
a  e.  ( Z `
 U ) )
4 simprr 756 . . . . . . 7  |-  ( ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  /\  ( a  e.  T  /\  b  e.  U ) )  -> 
b  e.  U )
5 eqid 2443 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
6 lsmsubg.z . . . . . . . 8  |-  Z  =  (Cntz `  G )
75, 6cntzi 15866 . . . . . . 7  |-  ( ( a  e.  ( Z `
 U )  /\  b  e.  U )  ->  ( a ( +g  `  G ) b )  =  ( b ( +g  `  G ) a ) )
83, 4, 7syl2anc 661 . . . . . 6  |-  ( ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  /\  ( a  e.  T  /\  b  e.  U ) )  -> 
( a ( +g  `  G ) b )  =  ( b ( +g  `  G ) a ) )
98eqeq2d 2454 . . . . 5  |-  ( ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  /\  ( a  e.  T  /\  b  e.  U ) )  -> 
( x  =  ( a ( +g  `  G
) b )  <->  x  =  ( b ( +g  `  G ) a ) ) )
1092rexbidva 2775 . . . 4  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( E. a  e.  T  E. b  e.  U  x  =  ( a ( +g  `  G ) b )  <->  E. a  e.  T  E. b  e.  U  x  =  ( b
( +g  `  G ) a ) ) )
11 rexcom 2901 . . . 4  |-  ( E. a  e.  T  E. b  e.  U  x  =  ( b ( +g  `  G ) a )  <->  E. b  e.  U  E. a  e.  T  x  =  ( b ( +g  `  G ) a ) )
1210, 11syl6bb 261 . . 3  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( E. a  e.  T  E. b  e.  U  x  =  ( a ( +g  `  G ) b )  <->  E. b  e.  U  E. a  e.  T  x  =  ( b
( +g  `  G ) a ) ) )
13 lsmsubg.p . . . . 5  |-  .(+)  =  (
LSSum `  G )
145, 13lsmelval 16167 . . . 4  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( x  e.  ( T  .(+)  U )  <->  E. a  e.  T  E. b  e.  U  x  =  ( a
( +g  `  G ) b ) ) )
15143adant3 1008 . . 3  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( x  e.  ( T  .(+)  U )  <->  E. a  e.  T  E. b  e.  U  x  =  ( a
( +g  `  G ) b ) ) )
165, 13lsmelval 16167 . . . . 5  |-  ( ( U  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )
)  ->  ( x  e.  ( U  .(+)  T )  <->  E. b  e.  U  E. a  e.  T  x  =  ( b
( +g  `  G ) a ) ) )
1716ancoms 453 . . . 4  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
)  ->  ( x  e.  ( U  .(+)  T )  <->  E. b  e.  U  E. a  e.  T  x  =  ( b
( +g  `  G ) a ) ) )
18173adant3 1008 . . 3  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( x  e.  ( U  .(+)  T )  <->  E. b  e.  U  E. a  e.  T  x  =  ( b
( +g  `  G ) a ) ) )
1912, 15, 183bitr4d 285 . 2  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( x  e.  ( T  .(+)  U )  <-> 
x  e.  ( U 
.(+)  T ) ) )
2019eqrdv 2441 1  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( T  .(+)  U )  =  ( U 
.(+)  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2735    C_ wss 3347   ` cfv 5437  (class class class)co 6110   +g cplusg 14257  SubGrpcsubg 15694  Cntzccntz 15852   LSSumclsm 16152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6596  df-2nd 6597  df-subg 15697  df-cntz 15854  df-lsm 16154
This theorem is referenced by:  lsmdisj3  16199  lsmdisj3r  16202  lsmdisj3a  16205  lsmdisj3b  16206  pj2f  16214  pj1id  16215
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