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Theorem lsmvalx 16786
Description: Subspace sum value (for a group or vector space). Extended domain version of lsmval 16795. (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 16785 . . . 4  |-  ( G  e.  V  ->  .(+)  =  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) ) )
54oveqd 6313 . . 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 5882 . . . . . 6  |-  ( Base `  G )  e.  _V
71, 6eqeltri 2541 . . . . 5  |-  B  e. 
_V
87elpw2 4620 . . . 4  |-  ( T  e.  ~P B  <->  T  C_  B
)
97elpw2 4620 . . . 4  |-  ( U  e.  ~P B  <->  U  C_  B
)
10 mpt2exga 6875 . . . . . 6  |-  ( ( T  e.  ~P B  /\  U  e.  ~P B )  ->  (
x  e.  T , 
y  e.  U  |->  ( x  .+  y ) )  e.  _V )
11 rnexg 6731 . . . . . 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 6356 . . . . . . 7  |-  ( ( t  =  T  /\  u  =  U )  ->  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y
) )  =  ( x  e.  T , 
y  e.  U  |->  ( x  .+  y ) ) )
1413rneqd 5240 . . . . . 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 2457 . . . . . 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 6431 . . . . 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 1325 . . . 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 2518 . 2  |-  ( ( G  e.  V  /\  ( T  C_  B  /\  U  C_  B ) )  ->  ( T  .(+)  U )  =  ran  (
x  e.  T , 
y  e.  U  |->  ( x  .+  y ) ) )
20193impb 1192 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 973    = wceq 1395    e. wcel 1819   _Vcvv 3109    C_ wss 3471   ~Pcpw 4015   ran crn 5009   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   Basecbs 14644   +g cplusg 14712   LSSumclsm 16781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-lsm 16783
This theorem is referenced by:  lsmelvalx  16787  lsmssv  16790  lsmval  16795  subglsm  16818
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