Theorem addrfv 45177

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 45174	. . . 4 ⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) = (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥))))
2	1	fveq1d 6883	. . 3 ⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥)))‘𝐶))
3		fveq2 6881	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴‘𝑥) = (𝐴‘𝐶))
4		fveq2 6881	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵‘𝑥) = (𝐵‘𝐶))
5	3, 4	oveq12d 7428	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴‘𝑥) + (𝐵‘𝑥)) = ((𝐴‘𝐶) + (𝐵‘𝐶)))
6		eqid 2763	. . . 4 ⊢ (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥))) = (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥)))
7		ovex 7443	. . . 4 ⊢ ((𝐴‘𝐶) + (𝐵‘𝐶)) ∈ V
8	5, 6, 7	fvmpt 6989	. . 3 ⊢ (𝐶 ∈ ℝ → ((𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥)))‘𝐶) = ((𝐴‘𝐶) + (𝐵‘𝐶)))
9	2, 8	sylan9eq 2818	. 2 ⊢ (((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷) ∧ 𝐶 ∈ ℝ) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝐴‘𝐶) + (𝐵‘𝐶)))
10	9	3impa 1127	1 ⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ ℝ) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝐴‘𝐶) + (𝐵‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ↦ cmpt 5192  ‘cfv 6536  (class class class)co 7410  ℝcr 11094   + caddc 11098  +_𝑟cplusr 45165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-cnex 11151  ax-resscn 11152

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-addr 45171

This theorem is referenced by:  addrcom  45183