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Theorem lsmcom2 15209
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 959 . . . . . . . . 9  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  T  C_  ( Z `  U )
)
21sselda 3284 . . . . . . . 8  |-  ( ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  /\  a  e.  T
)  ->  a  e.  ( Z `  U ) )
32adantrr 698 . . . . . . 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 734 . . . . . . 7  |-  ( ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  /\  ( a  e.  T  /\  b  e.  U ) )  -> 
b  e.  U )
5 eqid 2380 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
6 lsmsubg.z . . . . . . . 8  |-  Z  =  (Cntz `  G )
75, 6cntzi 15048 . . . . . . 7  |-  ( ( a  e.  ( Z `
 U )  /\  b  e.  U )  ->  ( a ( +g  `  G ) b )  =  ( b ( +g  `  G ) a ) )
83, 4, 7syl2anc 643 . . . . . 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 2391 . . . . 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 2683 . . . 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 2805 . . . 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 253 . . 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 15203 . . . 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 977 . . 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 15203 . . . . 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 440 . . . 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 977 . . 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 277 . 2  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( x  e.  ( T  .(+)  U )  <-> 
x  e.  ( U 
.(+)  T ) ) )
2019eqrdv 2378 1  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( T  .(+)  U )  =  ( U 
.(+)  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   E.wrex 2643    C_ wss 3256   ` cfv 5387  (class class class)co 6013   +g cplusg 13449  SubGrpcsubg 14858  Cntzccntz 15034   LSSumclsm 15188
This theorem is referenced by:  lsmdisj3  15235  lsmdisj3r  15238  lsmdisj3a  15241  lsmdisj3b  15242  pj2f  15250  pj1id  15251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-subg 14861  df-cntz 15036  df-lsm 15190
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