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Theorem lsmelvalix 16452
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 2462 . . 3  |-  ( X 
.+  Y )  =  ( X  .+  Y
)
2 rspceov 6314 . . 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 1308 . 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 16451 . . 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 485 . 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 474 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762   E.wrex 2810    C_ wss 3471   ` cfv 5581  (class class class)co 6277   Basecbs 14481   +g cplusg 14546   LSSumclsm 16445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-lsm 16447
This theorem is referenced by:  lsmub1x  16457  lsmub2x  16458  lsmelvali  16461  lsmsubm  16464  kercvrlsm  30624
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