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Theorem lsmvalx 14950
Description: Subspace sum value (for a group or vector space). Extended domain version of lsmval 14959. (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 14949 . . . 4  |-  ( G  e.  V  ->  .(+)  =  ( t  e.  ~P B ,  u  e.  ~P B  |->  ran  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y ) ) ) )
54oveqd 5875 . . 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 5539 . . . . . 6  |-  ( Base `  G )  e.  _V
71, 6eqeltri 2353 . . . . 5  |-  B  e. 
_V
87elpw2 4175 . . . 4  |-  ( T  e.  ~P B  <->  T  C_  B
)
97elpw2 4175 . . . 4  |-  ( U  e.  ~P B  <->  U  C_  B
)
10 mpt2exga 6197 . . . . . 6  |-  ( ( T  e.  ~P B  /\  U  e.  ~P B )  ->  (
x  e.  T , 
y  e.  U  |->  ( x  .+  y ) )  e.  _V )
11 rnexg 4940 . . . . . 6  |-  ( ( x  e.  T , 
y  e.  U  |->  ( x  .+  y ) )  e.  _V  ->  ran  ( x  e.  T ,  y  e.  U  |->  ( x  .+  y
) )  e.  _V )
1210, 11syl 15 . . . . 5  |-  ( ( T  e.  ~P B  /\  U  e.  ~P B )  ->  ran  ( x  e.  T ,  y  e.  U  |->  ( x  .+  y
) )  e.  _V )
13 mpt2eq12 5908 . . . . . . 7  |-  ( ( t  =  T  /\  u  =  U )  ->  ( x  e.  t ,  y  e.  u  |->  ( x  .+  y
) )  =  ( x  e.  T , 
y  e.  U  |->  ( x  .+  y ) ) )
1413rneqd 4906 . . . . . 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 2283 . . . . . 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 5977 . . . . 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 1278 . . . 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 466 . . 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 2335 . 2  |-  ( ( G  e.  V  /\  ( T  C_  B  /\  U  C_  B ) )  ->  ( T  .(+)  U )  =  ran  (
x  e.  T , 
y  e.  U  |->  ( x  .+  y ) ) )
20193impb 1147 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   ran crn 4690   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148   +g cplusg 13208   LSSumclsm 14945
This theorem is referenced by:  lsmelvalx  14951  lsmssv  14954  lsmval  14959  subglsm  14982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-lsm 14947
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