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Theorem lsmcomx 15463
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 960 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  G  e.  Abel )
2 simpl2 961 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  T  C_  B )
3 simprl 733 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  y  e.  T )
42, 3sseldd 3341 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  y  e.  B )
5 simpl3 962 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  U  C_  B )
6 simprr 734 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  /\  ( y  e.  T  /\  z  e.  U
) )  ->  z  e.  U )
75, 6sseldd 3341 . . . . . . 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 2435 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
108, 9ablcom 15421 . . . . . . 7  |-  ( ( G  e.  Abel  /\  y  e.  B  /\  z  e.  B )  ->  (
y ( +g  `  G
) z )  =  ( z ( +g  `  G ) y ) )
111, 4, 7, 10syl3anc 1184 . . . . . 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 2446 . . . . 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 2738 . . . 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 2861 . . . 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 253 . . 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 15266 . . 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 15266 . . . 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 1159 . . 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 277 . 2  |-  ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  ->  (
x  e.  ( T 
.(+)  U )  <->  x  e.  ( U  .(+)  T ) ) )
2120eqrdv 2433 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2698    C_ wss 3312   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   LSSumclsm 15260   Abelcabel 15405
This theorem is referenced by:  lsmcom  15465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-lsm 15262  df-cmn 15406  df-abl 15407
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