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Mirrors > Home > MPE Home > Th. List > Mathboxes > addrval | Structured version Visualization version GIF version |
Description: Value of the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.) |
Ref | Expression |
---|---|
addrval | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) + (𝐵‘𝑣)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3490 | . 2 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V) | |
2 | elex 3490 | . 2 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V) | |
3 | fveq1 6899 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥‘𝑣) = (𝐴‘𝑣)) | |
4 | fveq1 6899 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦‘𝑣) = (𝐵‘𝑣)) | |
5 | 3, 4 | oveqan12d 7443 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥‘𝑣) + (𝑦‘𝑣)) = ((𝐴‘𝑣) + (𝐵‘𝑣))) |
6 | 5 | mpteq2dv 5252 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) + (𝑦‘𝑣))) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) + (𝐵‘𝑣)))) |
7 | df-addr 43903 | . . 3 ⊢ +𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) + (𝑦‘𝑣)))) | |
8 | reex 11235 | . . . 4 ⊢ ℝ ∈ V | |
9 | 8 | mptex 7239 | . . 3 ⊢ (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) + (𝐵‘𝑣))) ∈ V |
10 | 6, 7, 9 | ovmpoa 7580 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴+𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) + (𝐵‘𝑣)))) |
11 | 1, 2, 10 | syl2an 594 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) + (𝐵‘𝑣)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3471 ↦ cmpt 5233 ‘cfv 6551 (class class class)co 7424 ℝcr 11143 + caddc 11147 +𝑟cplusr 43897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pr 5431 ax-cnex 11200 ax-resscn 11201 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-addr 43903 |
This theorem is referenced by: addrfv 43909 addrfn 43912 |
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