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Theorem lsmcom2 14966
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 957 . . . . . . . . 9  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  T  C_  ( Z `  U )
)
21sselda 3180 . . . . . . . 8  |-  ( ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  /\  a  e.  T
)  ->  a  e.  ( Z `  U ) )
32adantrr 697 . . . . . . 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 733 . . . . . . 7  |-  ( ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  /\  ( a  e.  T  /\  b  e.  U ) )  -> 
b  e.  U )
5 eqid 2283 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
6 lsmsubg.z . . . . . . . 8  |-  Z  =  (Cntz `  G )
75, 6cntzi 14805 . . . . . . 7  |-  ( ( a  e.  ( Z `
 U )  /\  b  e.  U )  ->  ( a ( +g  `  G ) b )  =  ( b ( +g  `  G ) a ) )
83, 4, 7syl2anc 642 . . . . . 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 2294 . . . . 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 2584 . . . 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 2701 . . . 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 252 . . 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 14960 . . . 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 975 . . 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 14960 . . . . 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 439 . . . 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 975 . . 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 276 . 2  |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `
 U ) )  ->  ( x  e.  ( T  .(+)  U )  <-> 
x  e.  ( U 
.(+)  T ) ) )
2019eqrdv 2281 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   ` cfv 5255  (class class class)co 5858   +g cplusg 13208  SubGrpcsubg 14615  Cntzccntz 14791   LSSumclsm 14945
This theorem is referenced by:  lsmdisj3  14992  lsmdisj3r  14995  lsmdisj3a  14998  lsmdisj3b  14999  pj2f  15007  pj1id  15008
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-reu 2550  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-subg 14618  df-cntz 14793  df-lsm 14947
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