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Theorem addrfn 42303
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 7340 . . 3 ((𝐴𝑥) + (𝐵𝑥)) ∈ V
2 eqid 2736 . . 3 (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥))) = (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥)))
31, 2fnmpti 6606 . 2 (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥))) Fn ℝ
4 addrval 42297 . . 3 ((𝐴𝐶𝐵𝐷) → (𝐴+𝑟𝐵) = (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥))))
54fneq1d 6557 . 2 ((𝐴𝐶𝐵𝐷) → ((𝐴+𝑟𝐵) Fn ℝ ↔ (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥))) Fn ℝ))
63, 5mpbiri 258 1 ((𝐴𝐶𝐵𝐷) → (𝐴+𝑟𝐵) Fn ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2104  cmpt 5164   Fn wfn 6453  cfv 6458  (class class class)co 7307  cr 10920   + caddc 10924  +𝑟cplusr 42288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-cnex 10977  ax-resscn 10978
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3305  df-rab 3306  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-addr 42294
This theorem is referenced by:  addrcom  42306
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