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Theorem lsmvalx 16450
Description: Subspace sum value (for a group or vector space). Extended domain version of lsmval 16459. (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
lsmvalx  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  ( T  .(+)  U )  =  ran  ( x  e.  T ,  y  e.  U  |->  ( x  .+  y ) ) )
Distinct variable groups:    x, y,  .+    x, B, y    x, T, y    x, G, y   
x, U, y
Allowed substitution hints:    .(+) ( x, y)    V( x, y)

Proof of Theorem lsmvalx
Dummy variables  u  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lsmfval.v . . . . 5  |-  B  =  ( Base `  G
)
2 lsmfval.a . . . . 5  |-  .+  =  ( +g  `  G )
3 lsmfval.s . . . . 5  |-  .(+)  =  (
LSSum `  G )
41, 2, 3lsmfval 16449 . . . 4  |-  ( G  e.  V  ->  .(+)  =  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) ) )
54oveqd 6294 . . 3  |-  ( G  e.  V  ->  ( T  .(+)  U )  =  ( T ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) ) U ) )
6 fvex 5869 . . . . . 6  |-  ( Base `  G )  e.  _V
71, 6eqeltri 2546 . . . . 5  |-  B  e. 
_V
87elpw2 4606 . . . 4  |-  ( T  e.  ~P B  <->  T  C_  B
)
97elpw2 4606 . . . 4  |-  ( U  e.  ~P B  <->  U  C_  B
)
10 mpt2exga 6851 . . . . . 6  |-  ( ( T  e.  ~P B  /\  U  e.  ~P B )  ->  (
x  e.  T , 
y  e.  U  |->  ( x  .+  y ) )  e.  _V )
11 rnexg 6708 . . . . . 6  |-  ( ( x  e.  T , 
y  e.  U  |->  ( x  .+  y ) )  e.  _V  ->  ran  ( x  e.  T ,  y  e.  U  |->  ( x  .+  y
) )  e.  _V )
1210, 11syl 16 . . . . 5  |-  ( ( T  e.  ~P B  /\  U  e.  ~P B )  ->  ran  ( x  e.  T ,  y  e.  U  |->  ( x  .+  y
) )  e.  _V )
13 mpt2eq12 6334 . . . . . . 7  |-  ( ( t  =  T  /\  u  =  U )  ->  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y
) )  =  ( x  e.  T , 
y  e.  U  |->  ( x  .+  y ) ) )
1413rneqd 5223 . . . . . 6  |-  ( ( t  =  T  /\  u  =  U )  ->  ran  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) )  =  ran  ( x  e.  T ,  y  e.  U  |->  ( x  .+  y ) ) )
15 eqid 2462 . . . . . 6  |-  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) )  =  ( t  e. 
~P B ,  u  e.  ~P B  |->  ran  (
x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) )
1614, 15ovmpt2ga 6409 . . . . 5  |-  ( ( T  e.  ~P B  /\  U  e.  ~P B  /\  ran  ( x  e.  T ,  y  e.  U  |->  ( x 
.+  y ) )  e.  _V )  -> 
( T ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) ) U )  =  ran  ( x  e.  T ,  y  e.  U  |->  ( x  .+  y
) ) )
1712, 16mpd3an3 1320 . . . 4  |-  ( ( T  e.  ~P B  /\  U  e.  ~P B )  ->  ( T ( t  e. 
~P B ,  u  e.  ~P B  |->  ran  (
x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) ) U )  =  ran  ( x  e.  T ,  y  e.  U  |->  ( x 
.+  y ) ) )
188, 9, 17syl2anbr 480 . . 3  |-  ( ( T  C_  B  /\  U  C_  B )  -> 
( T ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) ) U )  =  ran  ( x  e.  T ,  y  e.  U  |->  ( x  .+  y
) ) )
195, 18sylan9eq 2523 . 2  |-  ( ( G  e.  V  /\  ( T  C_  B  /\  U  C_  B ) )  ->  ( T  .(+)  U )  =  ran  (
x  e.  T , 
y  e.  U  |->  ( x  .+  y ) ) )
20193impb 1187 1  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  ( T  .(+)  U )  =  ran  ( x  e.  T ,  y  e.  U  |->  ( x  .+  y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   _Vcvv 3108    C_ wss 3471   ~Pcpw 4005   ran crn 4995   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   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:  lsmelvalx  16451  lsmssv  16454  lsmval  16459  subglsm  16482
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