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Theorem lsmelvalix 16983
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 2402 . . 3  |-  ( X 
.+  Y )  =  ( X  .+  Y
)
2 rspceov 6316 . . 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 1315 . 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 16982 . . 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 483 . 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 472 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   E.wrex 2754    C_ wss 3413   ` cfv 5568  (class class class)co 6277   Basecbs 14839   +g cplusg 14907   LSSumclsm 16976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-lsm 16978
This theorem is referenced by:  lsmub1x  16988  lsmub2x  16989  lsmelvali  16992  lsmsubm  16995  kercvrlsm  35371
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