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Theorem lsmvalx 17291
Description: Subspace sum value (for a group or vector space). Extended domain version of lsmval 17300. (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 17290 . . . 4  |-  ( G  e.  V  ->  .(+)  =  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) ) )
54oveqd 6307 . . 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 5875 . . . . . 6  |-  ( Base `  G )  e.  _V
71, 6eqeltri 2525 . . . . 5  |-  B  e. 
_V
87elpw2 4567 . . . 4  |-  ( T  e.  ~P B  <->  T  C_  B
)
97elpw2 4567 . . . 4  |-  ( U  e.  ~P B  <->  U  C_  B
)
10 mpt2exga 6869 . . . . . 6  |-  ( ( T  e.  ~P B  /\  U  e.  ~P B )  ->  (
x  e.  T , 
y  e.  U  |->  ( x  .+  y ) )  e.  _V )
11 rnexg 6725 . . . . . 6  |-  ( ( x  e.  T , 
y  e.  U  |->  ( x  .+  y ) )  e.  _V  ->  ran  ( x  e.  T ,  y  e.  U  |->  ( x  .+  y
) )  e.  _V )
1210, 11syl 17 . . . . 5  |-  ( ( T  e.  ~P B  /\  U  e.  ~P B )  ->  ran  ( x  e.  T ,  y  e.  U  |->  ( x  .+  y
) )  e.  _V )
13 mpt2eq12 6351 . . . . . . 7  |-  ( ( t  =  T  /\  u  =  U )  ->  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y
) )  =  ( x  e.  T , 
y  e.  U  |->  ( x  .+  y ) ) )
1413rneqd 5062 . . . . . 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 2451 . . . . . 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 6426 . . . . 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 1365 . . . 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 483 . . 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 2505 . 2  |-  ( ( G  e.  V  /\  ( T  C_  B  /\  U  C_  B ) )  ->  ( T  .(+)  U )  =  ran  (
x  e.  T , 
y  e.  U  |->  ( x  .+  y ) ) )
20193impb 1204 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 371    /\ w3a 985    = wceq 1444    e. wcel 1887   _Vcvv 3045    C_ wss 3404   ~Pcpw 3951   ran crn 4835   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292   Basecbs 15121   +g cplusg 15190   LSSumclsm 17286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-lsm 17288
This theorem is referenced by:  lsmelvalx  17292  lsmssv  17295  lsmval  17300  subglsm  17323
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