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Theorem subrval 26820
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 2830 . 2  |-  ( A  e.  C  ->  A  e.  _V )
2 elex 2830 . 2  |-  ( B  e.  D  ->  B  e.  _V )
3 fveq1 5562 . . . . 5  |-  ( x  =  A  ->  (
x `  v )  =  ( A `  v ) )
4 fveq1 5562 . . . . 5  |-  ( y  =  B  ->  (
y `  v )  =  ( B `  v ) )
53, 4oveqan12d 5919 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x `  v )  -  (
y `  v )
)  =  ( ( A `  v )  -  ( B `  v ) ) )
65mpteq2dv 4144 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( v  e.  RR  |->  ( ( x `  v )  -  (
y `  v )
) )  =  ( v  e.  RR  |->  ( ( A `  v
)  -  ( B `
 v ) ) ) )
7 df-subr 26817 . . 3  |-  - r  =  ( x  e. 
_V ,  y  e. 
_V  |->  ( v  e.  RR  |->  ( ( x `
 v )  -  ( y `  v
) ) ) )
8 reex 8873 . . . 4  |-  RR  e.  _V
98mptex 5787 . . 3  |-  ( v  e.  RR  |->  ( ( A `  v )  -  ( B `  v ) ) )  e.  _V
106, 7, 9ovmpt2a 6020 . 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 1633    e. wcel 1701   _Vcvv 2822    e. cmpt 4114   ` cfv 5292  (class class class)co 5900   RRcr 8781    - cmin 9082   - rcminusr 26811
This theorem is referenced by:  subrfv  26823  subrfn  26826
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-cnex 8838  ax-resscn 8839
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-subr 26817
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