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Theorem lsmcom2 14982
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 3193 . . . . . . . 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 2296 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
6 lsmsubg.z . . . . . . . 8  |-  Z  =  (Cntz `  G )
75, 6cntzi 14821 . . . . . . 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 2307 . . . . 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 2597 . . . 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 2714 . . . 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 14976 . . . 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 14976 . . . . 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 2294 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 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   ` cfv 5271  (class class class)co 5874   +g cplusg 13224  SubGrpcsubg 14631  Cntzccntz 14807   LSSumclsm 14961
This theorem is referenced by:  lsmdisj3  15008  lsmdisj3r  15011  lsmdisj3a  15014  lsmdisj3b  15015  pj2f  15023  pj1id  15024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-subg 14634  df-cntz 14809  df-lsm 14963
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