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Theorem subrval 27672
Description: Value of the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.)
Assertion
Ref Expression
subrval  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A - r B )  =  ( v  e.  RR  |->  ( ( A `  v
)  -  ( B `
 v ) ) ) )
Distinct variable groups:    v, A    v, B
Allowed substitution hints:    C( v)    D( v)

Proof of Theorem subrval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( A  e.  C  ->  A  e.  _V )
2 elex 2796 . 2  |-  ( B  e.  D  ->  B  e.  _V )
3 fveq1 5524 . . . . 5  |-  ( x  =  A  ->  (
x `  v )  =  ( A `  v ) )
4 fveq1 5524 . . . . 5  |-  ( y  =  B  ->  (
y `  v )  =  ( B `  v ) )
53, 4oveqan12d 5877 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x `  v )  -  (
y `  v )
)  =  ( ( A `  v )  -  ( B `  v ) ) )
65mpteq2dv 4107 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( v  e.  RR  |->  ( ( x `  v )  -  (
y `  v )
) )  =  ( v  e.  RR  |->  ( ( A `  v
)  -  ( B `
 v ) ) ) )
7 df-subr 27669 . . 3  |-  - r  =  ( x  e. 
_V ,  y  e. 
_V  |->  ( v  e.  RR  |->  ( ( x `
 v )  -  ( y `  v
) ) ) )
8 reex 8828 . . . 4  |-  RR  e.  _V
98mptex 5746 . . 3  |-  ( v  e.  RR  |->  ( ( A `  v )  -  ( B `  v ) ) )  e.  _V
106, 7, 9ovmpt2a 5978 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A - r B )  =  ( v  e.  RR  |->  ( ( A `  v
)  -  ( B `
 v ) ) ) )
111, 2, 10syl2an 463 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A - r B )  =  ( v  e.  RR  |->  ( ( A `  v
)  -  ( B `
 v ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   RRcr 8736    - cmin 9037   - rcminusr 27663
This theorem is referenced by:  subrfv  27675  subrfn  27678
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-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-pr 4214  ax-cnex 8793  ax-resscn 8794
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-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-subr 27669
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