Theorem addrfn 40811

Description: Vector addition produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)

Assertion

Ref	Expression
addrfn	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) Fn ℝ)

Proof of Theorem addrfn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7191	. . 3 ⊢ ((𝐴‘𝑥) + (𝐵‘𝑥)) ∈ V
2		eqid 2823	. . 3 ⊢ (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥))) = (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥)))
3	1, 2	fnmpti 6493	. 2 ⊢ (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥))) Fn ℝ
4		addrval 40805	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) = (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥))))
5	4	fneq1d 6448	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐴+𝑟𝐵) Fn ℝ ↔ (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥))) Fn ℝ))
6	3, 5	mpbiri 260	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) Fn ℝ)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2114   ↦ cmpt 5148   Fn wfn 6352  ‘cfv 6357  (class class class)co 7158  ℝcr 10538   + caddc 10542  +_𝑟cplusr 40796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-cnex 10595  ax-resscn 10596

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-addr 40802

This theorem is referenced by:  addrcom  40814