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Mirrors > Home > MPE Home > Th. List > Mathboxes > addrfv | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
addrfv | ⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ ℝ) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝐴‘𝐶) + (𝐵‘𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addrval 40791 | . . . 4 ⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) = (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥)))) | |
2 | 1 | fveq1d 6667 | . . 3 ⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥)))‘𝐶)) |
3 | fveq2 6665 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴‘𝑥) = (𝐴‘𝐶)) | |
4 | fveq2 6665 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐵‘𝑥) = (𝐵‘𝐶)) | |
5 | 3, 4 | oveq12d 7168 | . . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴‘𝑥) + (𝐵‘𝑥)) = ((𝐴‘𝐶) + (𝐵‘𝐶))) |
6 | eqid 2821 | . . . 4 ⊢ (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥))) = (𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥))) | |
7 | ovex 7183 | . . . 4 ⊢ ((𝐴‘𝐶) + (𝐵‘𝐶)) ∈ V | |
8 | 5, 6, 7 | fvmpt 6763 | . . 3 ⊢ (𝐶 ∈ ℝ → ((𝑥 ∈ ℝ ↦ ((𝐴‘𝑥) + (𝐵‘𝑥)))‘𝐶) = ((𝐴‘𝐶) + (𝐵‘𝐶))) |
9 | 2, 8 | sylan9eq 2876 | . 2 ⊢ (((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷) ∧ 𝐶 ∈ ℝ) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝐴‘𝐶) + (𝐵‘𝐶))) |
10 | 9 | 3impa 1106 | 1 ⊢ ((𝐴 ∈ 𝐸 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ ℝ) → ((𝐴+𝑟𝐵)‘𝐶) = ((𝐴‘𝐶) + (𝐵‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ↦ cmpt 5139 ‘cfv 6350 (class class class)co 7150 ℝcr 10530 + caddc 10534 +𝑟cplusr 40782 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pr 5322 ax-cnex 10587 ax-resscn 10588 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-addr 40788 |
This theorem is referenced by: addrcom 40800 |
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