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Theorem lsmcom2 16464
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 993 . . . . . . . . 9  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  T  C_  ( Z `  U )
)
21sselda 3497 . . . . . . . 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 2460 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
6 lsmsubg.z . . . . . . . 8  |-  Z  =  (Cntz `  G )
75, 6cntzi 16155 . . . . . . 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 2474 . . . . 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 2972 . . . 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 3016 . . . 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 16458 . . . 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 1011 . . 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 16458 . . . . 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 1011 . . 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 2457 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 968    = wceq 1374    e. wcel 1762   E.wrex 2808    C_ wss 3469   ` cfv 5579  (class class class)co 6275   +g cplusg 14544  SubGrpcsubg 15983  Cntzccntz 16141   LSSumclsm 16443
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 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-subg 15986  df-cntz 16143  df-lsm 16445
This theorem is referenced by:  lsmdisj3  16490  lsmdisj3r  16493  lsmdisj3a  16496  lsmdisj3b  16497  pj2f  16505  pj1id  16506
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