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Theorem addrfn 43940
Description: Vector addition produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
Assertion
Ref Expression
addrfn ((𝐴𝐶𝐵𝐷) → (𝐴+𝑟𝐵) Fn ℝ)

Proof of Theorem addrfn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ovex 7459 . . 3 ((𝐴𝑥) + (𝐵𝑥)) ∈ V
2 eqid 2728 . . 3 (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥))) = (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥)))
31, 2fnmpti 6703 . 2 (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥))) Fn ℝ
4 addrval 43934 . . 3 ((𝐴𝐶𝐵𝐷) → (𝐴+𝑟𝐵) = (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥))))
54fneq1d 6652 . 2 ((𝐴𝐶𝐵𝐷) → ((𝐴+𝑟𝐵) Fn ℝ ↔ (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥))) Fn ℝ))
63, 5mpbiri 257 1 ((𝐴𝐶𝐵𝐷) → (𝐴+𝑟𝐵) Fn ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  cmpt 5235   Fn wfn 6548  cfv 6553  (class class class)co 7426  cr 11145   + caddc 11149  +𝑟cplusr 43925
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-cnex 11202  ax-resscn 11203
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-addr 43931
This theorem is referenced by:  addrcom  43943
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