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Theorem lsmcomx 16988
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 999 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  G  e.  Abel )
2 simpl2 1000 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  T  C_  B )
3 simprl 756 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  y  e.  T )
42, 3sseldd 3500 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  y  e.  B )
5 simpl3 1001 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  U  C_  B )
6 simprr 757 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  z  e.  U )
75, 6sseldd 3500 . . . . . . 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 2457 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
108, 9ablcom 16941 . . . . . . 7  |-  ( ( G  e.  Abel  /\  y  e.  B  /\  z  e.  B )  ->  (
y ( +g  `  G
) z )  =  ( z ( +g  `  G ) y ) )
111, 4, 7, 10syl3anc 1228 . . . . . 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 2471 . . . . 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 2974 . . . 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 3019 . . . 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 16786 . . 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 16786 . . . 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 1202 . . 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 2454 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 973    = wceq 1395    e. wcel 1819   E.wrex 2808    C_ wss 3471   ` cfv 5594  (class class class)co 6296   Basecbs 14643   +g cplusg 14711   LSSumclsm 16780   Abelcabl 16925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-lsm 16782  df-cmn 16926  df-abl 16927
This theorem is referenced by:  lsmcom  16990
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