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Theorem lsmvalx 15275
Description: Subspace sum value (for a group or vector space). Extended domain version of lsmval 15284. (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 15274 . . . 4  |-  ( G  e.  V  ->  .(+)  =  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) ) )
54oveqd 6100 . . 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 5744 . . . . . 6  |-  ( Base `  G )  e.  _V
71, 6eqeltri 2508 . . . . 5  |-  B  e. 
_V
87elpw2 4366 . . . 4  |-  ( T  e.  ~P B  <->  T  C_  B
)
97elpw2 4366 . . . 4  |-  ( U  e.  ~P B  <->  U  C_  B
)
10 mpt2exga 6426 . . . . . 6  |-  ( ( T  e.  ~P B  /\  U  e.  ~P B )  ->  (
x  e.  T , 
y  e.  U  |->  ( x  .+  y ) )  e.  _V )
11 rnexg 5133 . . . . . 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 6136 . . . . . . 7  |-  ( ( t  =  T  /\  u  =  U )  ->  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y
) )  =  ( x  e.  T , 
y  e.  U  |->  ( x  .+  y ) ) )
1413rneqd 5099 . . . . . 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 2438 . . . . . 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 6205 . . . . 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 1281 . . . 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 468 . . 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 2490 . 2  |-  ( ( G  e.  V  /\  ( T  C_  B  /\  U  C_  B ) )  ->  ( T  .(+)  U )  =  ran  (
x  e.  T , 
y  e.  U  |->  ( x  .+  y ) ) )
20193impb 1150 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 set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801   ran crn 4881   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   Basecbs 13471   +g cplusg 13531   LSSumclsm 15270
This theorem is referenced by:  lsmelvalx  15276  lsmssv  15279  lsmval  15284  subglsm  15307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-lsm 15272
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