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Theorem lsmcomx 15148
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 3181 . . . . . . 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 3181 . . . . . . 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 2283 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
108, 9ablcom 15106 . . . . . . 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 2294 . . . . 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 2584 . . . 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 2701 . . . 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 14951 . . 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 14951 . . . 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 2281 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 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   LSSumclsm 14945   Abelcabel 15090
This theorem is referenced by:  lsmcom  15150
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-lsm 14947  df-cmn 15091  df-abl 15092
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