Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  addrfn Structured version   Visualization version   GIF version

Theorem addrfn 43909
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 7453 . . 3 ((𝐴𝑥) + (𝐵𝑥)) ∈ V
2 eqid 2728 . . 3 (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥))) = (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥)))
31, 2fnmpti 6698 . 2 (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥))) Fn ℝ
4 addrval 43903 . . 3 ((𝐴𝐶𝐵𝐷) → (𝐴+𝑟𝐵) = (𝑥 ∈ ℝ ↦ ((𝐴𝑥) + (𝐵𝑥))))
54fneq1d 6647 . 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 395  wcel 2099  cmpt 5231   Fn wfn 6543  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
  Copyright terms: Public domain W3C validator