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Theorem lsmcomx 16337
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 991 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  G  e.  Abel )
2 simpl2 992 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  T  C_  B )
3 simprl 755 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  y  e.  T )
42, 3sseldd 3356 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  y  e.  B )
5 simpl3 993 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  U  C_  B )
6 simprr 756 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  z  e.  U )
75, 6sseldd 3356 . . . . . . 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 2442 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
108, 9ablcom 16293 . . . . . . 7  |-  ( ( G  e.  Abel  /\  y  e.  B  /\  z  e.  B )  ->  (
y ( +g  `  G
) z )  =  ( z ( +g  `  G ) y ) )
111, 4, 7, 10syl3anc 1218 . . . . . 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 2453 . . . . 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 2755 . . . 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 2881 . . . 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 261 . . 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 16138 . . 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 16138 . . . 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 1193 . . 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 285 . 2  |-  ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  ->  (
x  e.  ( T 
.(+)  U )  <->  x  e.  ( U  .(+)  T ) ) )
2120eqrdv 2440 1  |-  ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2715    C_ wss 3327   ` cfv 5417  (class class class)co 6090   Basecbs 14173   +g cplusg 14237   LSSumclsm 16132   Abelcabel 16277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-lsm 16134  df-cmn 16278  df-abl 16279
This theorem is referenced by:  lsmcom  16339
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