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Theorem addrfv 43785
Description: Vector addition at a value. The operation takes each vector 𝐴 and 𝐵 and forms a new vector whose values are the sum of each of the values of 𝐴 and 𝐵. (Contributed by Andrew Salmon, 27-Jan-2012.)
Assertion
Ref Expression
addrfv ((𝐴𝐸𝐵𝐷𝐶 ∈ ℝ) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝐴𝐶) + (𝐵𝐶)))

Proof of Theorem addrfv
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 addrval 43782 . . . 4 ((𝐴𝐸𝐵𝐷) → (𝐴+𝑟𝐵) = (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥))))
21fveq1d 6886 . . 3 ((𝐴𝐸𝐵𝐷) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥)))‘𝐶))
3 fveq2 6884 . . . . 5 (𝑥 = 𝐶 → (𝐴𝑥) = (𝐴𝐶))
4 fveq2 6884 . . . . 5 (𝑥 = 𝐶 → (𝐵𝑥) = (𝐵𝐶))
53, 4oveq12d 7422 . . . 4 (𝑥 = 𝐶 → ((𝐴𝑥) + (𝐵𝑥)) = ((𝐴𝐶) + (𝐵𝐶)))
6 eqid 2726 . . . 4 (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥))) = (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥)))
7 ovex 7437 . . . 4 ((𝐴𝐶) + (𝐵𝐶)) ∈ V
85, 6, 7fvmpt 6991 . . 3 (𝐶 ∈ ℝ → ((𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥)))‘𝐶) = ((𝐴𝐶) + (𝐵𝐶)))
92, 8sylan9eq 2786 . 2 (((𝐴𝐸𝐵𝐷) ∧ 𝐶 ∈ ℝ) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝐴𝐶) + (𝐵𝐶)))
1093impa 1107 1 ((𝐴𝐸𝐵𝐷𝐶 ∈ ℝ) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝐴𝐶) + (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  cmpt 5224  cfv 6536  (class class class)co 7404  cr 11108   + caddc 11112  +𝑟cplusr 43773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-cnex 11165  ax-resscn 11166
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-addr 43779
This theorem is referenced by:  addrcom  43791
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