Theorem addrfv 44465

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	⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ ℝ) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝐴‘𝐶) + (𝐵‘𝐶)))

Proof of Theorem addrfv

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		addrval 44462	. . . 4 ⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) = (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥))))
2	1	fveq1d 6909	. . 3 ⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥)))‘𝐶))
3		fveq2 6907	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴‘𝑥) = (𝐴‘𝐶))
4		fveq2 6907	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵‘𝑥) = (𝐵‘𝐶))
5	3, 4	oveq12d 7449	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴‘𝑥) + (𝐵‘𝑥)) = ((𝐴‘𝐶) + (𝐵‘𝐶)))
6		eqid 2735	. . . 4 ⊢ (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥))) = (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥)))
7		ovex 7464	. . . 4 ⊢ ((𝐴‘𝐶) + (𝐵‘𝐶)) ∈ V
8	5, 6, 7	fvmpt 7016	. . 3 ⊢ (𝐶 ∈ ℝ → ((𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥)))‘𝐶) = ((𝐴‘𝐶) + (𝐵‘𝐶)))
9	2, 8	sylan9eq 2795	. 2 ⊢ (((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷) ∧ 𝐶 ∈ ℝ) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝐴‘𝐶) + (𝐵‘𝐶)))
10	9	3impa 1109	1 ⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ ℝ) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝐴‘𝐶) + (𝐵‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ↦ cmpt 5231  ‘cfv 6563  (class class class)co 7431  ℝcr 11152   + caddc 11156  +_𝑟cplusr 44453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-cnex 11209  ax-resscn 11210

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-addr 44459

This theorem is referenced by:  addrcom  44471