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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhvaddval | Structured version Visualization version GIF version |
Description: The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) |
Ref | Expression |
---|---|
dvhvaddval.a | ⊢ + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))〉) |
Ref | Expression |
---|---|
dvhvaddval | ⊢ ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸)) → (𝐹 + 𝐺) = 〈((1st ‘𝐹) ∘ (1st ‘𝐺)), ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6896 | . . . 4 ⊢ (ℎ = 𝐹 → (1st ‘ℎ) = (1st ‘𝐹)) | |
2 | 1 | coeq1d 5864 | . . 3 ⊢ (ℎ = 𝐹 → ((1st ‘ℎ) ∘ (1st ‘𝑖)) = ((1st ‘𝐹) ∘ (1st ‘𝑖))) |
3 | fveq2 6896 | . . . 4 ⊢ (ℎ = 𝐹 → (2nd ‘ℎ) = (2nd ‘𝐹)) | |
4 | 3 | oveq1d 7434 | . . 3 ⊢ (ℎ = 𝐹 → ((2nd ‘ℎ) ⨣ (2nd ‘𝑖)) = ((2nd ‘𝐹) ⨣ (2nd ‘𝑖))) |
5 | 2, 4 | opeq12d 4883 | . 2 ⊢ (ℎ = 𝐹 → 〈((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))〉 = 〈((1st ‘𝐹) ∘ (1st ‘𝑖)), ((2nd ‘𝐹) ⨣ (2nd ‘𝑖))〉) |
6 | fveq2 6896 | . . . 4 ⊢ (𝑖 = 𝐺 → (1st ‘𝑖) = (1st ‘𝐺)) | |
7 | 6 | coeq2d 5865 | . . 3 ⊢ (𝑖 = 𝐺 → ((1st ‘𝐹) ∘ (1st ‘𝑖)) = ((1st ‘𝐹) ∘ (1st ‘𝐺))) |
8 | fveq2 6896 | . . . 4 ⊢ (𝑖 = 𝐺 → (2nd ‘𝑖) = (2nd ‘𝐺)) | |
9 | 8 | oveq2d 7435 | . . 3 ⊢ (𝑖 = 𝐺 → ((2nd ‘𝐹) ⨣ (2nd ‘𝑖)) = ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))) |
10 | 7, 9 | opeq12d 4883 | . 2 ⊢ (𝑖 = 𝐺 → 〈((1st ‘𝐹) ∘ (1st ‘𝑖)), ((2nd ‘𝐹) ⨣ (2nd ‘𝑖))〉 = 〈((1st ‘𝐹) ∘ (1st ‘𝐺)), ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))〉) |
11 | dvhvaddval.a | . . 3 ⊢ + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))〉) | |
12 | 11 | dvhvaddcbv 40692 | . 2 ⊢ + = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))〉) |
13 | opex 5466 | . 2 ⊢ 〈((1st ‘𝐹) ∘ (1st ‘𝐺)), ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))〉 ∈ V | |
14 | 5, 10, 12, 13 | ovmpo 7581 | 1 ⊢ ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸)) → (𝐹 + 𝐺) = 〈((1st ‘𝐹) ∘ (1st ‘𝐺)), ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 〈cop 4636 × cxp 5676 ∘ ccom 5682 ‘cfv 6549 (class class class)co 7419 ∈ cmpo 7421 1st c1st 7992 2nd c2nd 7993 |
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 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 |
This theorem is referenced by: dvhvadd 40695 dvhopaddN 40717 |
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