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Theorem addrfv 43906
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 43903 . . . 4 ((𝐴𝐸𝐵𝐷) → (𝐴+𝑟𝐵) = (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥))))
21fveq1d 6899 . . 3 ((𝐴𝐸𝐵𝐷) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥)))‘𝐶))
3 fveq2 6897 . . . . 5 (𝑥 = 𝐶 → (𝐴𝑥) = (𝐴𝐶))
4 fveq2 6897 . . . . 5 (𝑥 = 𝐶 → (𝐵𝑥) = (𝐵𝐶))
53, 4oveq12d 7438 . . . 4 (𝑥 = 𝐶 → ((𝐴𝑥) + (𝐵𝑥)) = ((𝐴𝐶) + (𝐵𝐶)))
6 eqid 2728 . . . 4 (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥))) = (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥)))
7 ovex 7453 . . . 4 ((𝐴𝐶) + (𝐵𝐶)) ∈ V
85, 6, 7fvmpt 7005 . . 3 (𝐶 ∈ ℝ → ((𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥)))‘𝐶) = ((𝐴𝐶) + (𝐵𝐶)))
92, 8sylan9eq 2788 . 2 (((𝐴𝐸𝐵𝐷) ∧ 𝐶 ∈ ℝ) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝐴𝐶) + (𝐵𝐶)))
1093impa 1108 1 ((𝐴𝐸𝐵𝐷𝐶 ∈ ℝ) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝐴𝐶) + (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  cmpt 5231  cfv 6548  (class class class)co 7420  cr 11138   + caddc 11142  +𝑟cplusr 43894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-cnex 11195  ax-resscn 11196
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-addr 43900
This theorem is referenced by:  addrcom  43912
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