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Theorem lsmelvalx 16859
Description: Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 16868. (Contributed by NM, 28-Jan-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
lsmelvalx  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  ( X  e.  ( T  .(+) 
U )  <->  E. y  e.  T  E. z  e.  U  X  =  ( y  .+  z
) ) )
Distinct variable groups:    y, z,  .+    y, B, z    y, T, z    y, X, z   
y, G, z    y, U, z
Allowed substitution hints:    .(+) ( y, z)    V( y, z)

Proof of Theorem lsmelvalx
StepHypRef Expression
1 lsmfval.v . . . 4  |-  B  =  ( Base `  G
)
2 lsmfval.a . . . 4  |-  .+  =  ( +g  `  G )
3 lsmfval.s . . . 4  |-  .(+)  =  (
LSSum `  G )
41, 2, 3lsmvalx 16858 . . 3  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  ( T  .(+)  U )  =  ran  ( y  e.  T ,  z  e.  U  |->  ( y  .+  z ) ) )
54eleq2d 2524 . 2  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  ( X  e.  ( T  .(+) 
U )  <->  X  e.  ran  ( y  e.  T ,  z  e.  U  |->  ( y  .+  z
) ) ) )
6 eqid 2454 . . 3  |-  ( y  e.  T ,  z  e.  U  |->  ( y 
.+  z ) )  =  ( y  e.  T ,  z  e.  U  |->  ( y  .+  z ) )
7 ovex 6298 . . 3  |-  ( y 
.+  z )  e. 
_V
86, 7elrnmpt2 6388 . 2  |-  ( X  e.  ran  ( y  e.  T ,  z  e.  U  |->  ( y 
.+  z ) )  <->  E. y  e.  T  E. z  e.  U  X  =  ( y  .+  z ) )
95, 8syl6bb 261 1  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  ( X  e.  ( T  .(+) 
U )  <->  E. y  e.  T  E. z  e.  U  X  =  ( y  .+  z
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 971    = wceq 1398    e. wcel 1823   E.wrex 2805    C_ wss 3461   ran crn 4989   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   Basecbs 14716   +g cplusg 14784   LSSumclsm 16853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-lsm 16855
This theorem is referenced by:  lsmelvalix  16860  lsmless1x  16863  lsmless2x  16864  lsmelval  16868  lsmsubm  16872  lsmass  16887  lsmcomx  17061  lsmcss  18896
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