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Theorem addrfv 38499
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 38496 . . . 4 ((𝐴𝐸𝐵𝐷) → (𝐴+𝑟𝐵) = (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥))))
21fveq1d 6191 . . 3 ((𝐴𝐸𝐵𝐷) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥)))‘𝐶))
3 fveq2 6189 . . . . 5 (𝑥 = 𝐶 → (𝐴𝑥) = (𝐴𝐶))
4 fveq2 6189 . . . . 5 (𝑥 = 𝐶 → (𝐵𝑥) = (𝐵𝐶))
53, 4oveq12d 6665 . . . 4 (𝑥 = 𝐶 → ((𝐴𝑥) + (𝐵𝑥)) = ((𝐴𝐶) + (𝐵𝐶)))
6 eqid 2621 . . . 4 (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥))) = (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥)))
7 ovex 6675 . . . 4 ((𝐴𝐶) + (𝐵𝐶)) ∈ V
85, 6, 7fvmpt 6280 . . 3 (𝐶 ∈ ℝ → ((𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥)))‘𝐶) = ((𝐴𝐶) + (𝐵𝐶)))
92, 8sylan9eq 2675 . 2 (((𝐴𝐸𝐵𝐷) ∧ 𝐶 ∈ ℝ) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝐴𝐶) + (𝐵𝐶)))
1093impa 1258 1 ((𝐴𝐸𝐵𝐷𝐶 ∈ ℝ) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝐴𝐶) + (𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037   = wceq 1482  wcel 1989  cmpt 4727  cfv 5886  (class class class)co 6647  cr 9932   + caddc 9936  +𝑟cplusr 38487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pr 4904  ax-cnex 9989  ax-resscn 9990
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-addr 38493
This theorem is referenced by:  addrcom  38505
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