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Theorem lsmcomx 15164
Description: Subgroup sum commutes (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmcomx.v  |-  B  =  ( Base `  G
)
lsmcomx.s  |-  .(+)  =  (
LSSum `  G )
Assertion
Ref Expression
lsmcomx  |-  ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )

Proof of Theorem lsmcomx
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 958 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  G  e.  Abel )
2 simpl2 959 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  T  C_  B )
3 simprl 732 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  y  e.  T )
42, 3sseldd 3194 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  y  e.  B )
5 simpl3 960 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  U  C_  B )
6 simprr 733 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  z  e.  U )
75, 6sseldd 3194 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  z  e.  B )
8 lsmcomx.v . . . . . . . 8  |-  B  =  ( Base `  G
)
9 eqid 2296 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
108, 9ablcom 15122 . . . . . . 7  |-  ( ( G  e.  Abel  /\  y  e.  B  /\  z  e.  B )  ->  (
y ( +g  `  G
) z )  =  ( z ( +g  `  G ) y ) )
111, 4, 7, 10syl3anc 1182 . . . . . 6  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  (
y ( +g  `  G
) z )  =  ( z ( +g  `  G ) y ) )
1211eqeq2d 2307 . . . . 5  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  (
x  =  ( y ( +g  `  G
) z )  <->  x  =  ( z ( +g  `  G ) y ) ) )
13122rexbidva 2597 . . . 4  |-  ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  ->  ( E. y  e.  T  E. z  e.  U  x  =  ( y
( +g  `  G ) z )  <->  E. y  e.  T  E. z  e.  U  x  =  ( z ( +g  `  G ) y ) ) )
14 rexcom 2714 . . . 4  |-  ( E. y  e.  T  E. z  e.  U  x  =  ( z ( +g  `  G ) y )  <->  E. z  e.  U  E. y  e.  T  x  =  ( z ( +g  `  G ) y ) )
1513, 14syl6bb 252 . . 3  |-  ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  ->  ( E. y  e.  T  E. z  e.  U  x  =  ( y
( +g  `  G ) z )  <->  E. z  e.  U  E. y  e.  T  x  =  ( z ( +g  `  G ) y ) ) )
16 lsmcomx.s . . . 4  |-  .(+)  =  (
LSSum `  G )
178, 9, 16lsmelvalx 14967 . . 3  |-  ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  ->  (
x  e.  ( T 
.(+)  U )  <->  E. y  e.  T  E. z  e.  U  x  =  ( y ( +g  `  G ) z ) ) )
188, 9, 16lsmelvalx 14967 . . . 4  |-  ( ( G  e.  Abel  /\  U  C_  B  /\  T  C_  B )  ->  (
x  e.  ( U 
.(+)  T )  <->  E. z  e.  U  E. y  e.  T  x  =  ( z ( +g  `  G ) y ) ) )
19183com23 1157 . . 3  |-  ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  ->  (
x  e.  ( U 
.(+)  T )  <->  E. z  e.  U  E. y  e.  T  x  =  ( z ( +g  `  G ) y ) ) )
2015, 17, 193bitr4d 276 . 2  |-  ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  ->  (
x  e.  ( T 
.(+)  U )  <->  x  e.  ( U  .(+)  T ) ) )
2120eqrdv 2294 1  |-  ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  ->  ( 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   Basecbs 13164   +g cplusg 13224   LSSumclsm 14961   Abelcabel 15106
This theorem is referenced by:  lsmcom  15166
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-lsm 14963  df-cmn 15107  df-abl 15108
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