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Theorem lsmelvalix 15267
Description: Subspace sum membership (for a group or vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmfval.v  |-  B  =  ( Base `  G
)
lsmfval.a  |-  .+  =  ( +g  `  G )
lsmfval.s  |-  .(+)  =  (
LSSum `  G )
Assertion
Ref Expression
lsmelvalix  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  ( X  e.  T  /\  Y  e.  U
) )  ->  ( X  .+  Y )  e.  ( T  .(+)  U ) )

Proof of Theorem lsmelvalix
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . 3  |-  ( X 
.+  Y )  =  ( X  .+  Y
)
2 rspceov 6108 . . 3  |-  ( ( X  e.  T  /\  Y  e.  U  /\  ( X  .+  Y )  =  ( X  .+  Y ) )  ->  E. x  e.  T  E. y  e.  U  ( X  .+  Y )  =  ( x  .+  y ) )
31, 2mp3an3 1268 . 2  |-  ( ( X  e.  T  /\  Y  e.  U )  ->  E. x  e.  T  E. y  e.  U  ( X  .+  Y )  =  ( x  .+  y ) )
4 lsmfval.v . . . 4  |-  B  =  ( Base `  G
)
5 lsmfval.a . . . 4  |-  .+  =  ( +g  `  G )
6 lsmfval.s . . . 4  |-  .(+)  =  (
LSSum `  G )
74, 5, 6lsmelvalx 15266 . . 3  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  (
( X  .+  Y
)  e.  ( T 
.(+)  U )  <->  E. x  e.  T  E. y  e.  U  ( X  .+  Y )  =  ( x  .+  y ) ) )
87biimpar 472 . 2  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  E. x  e.  T  E. y  e.  U  ( X  .+  Y )  =  ( x  .+  y
) )  ->  ( X  .+  Y )  e.  ( T  .(+)  U ) )
93, 8sylan2 461 1  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  ( X  e.  T  /\  Y  e.  U
) )  ->  ( X  .+  Y )  e.  ( T  .(+)  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ 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
This theorem is referenced by:  lsmub1x  15272  lsmub2x  15273  lsmelvali  15276  lsmsubm  15279  kercvrlsm  27139
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
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