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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 6672 | . . . 4 ⊢ (ℎ = 𝐹 → (1st ‘ℎ) = (1st ‘𝐹)) | |
2 | 1 | coeq1d 5734 | . . 3 ⊢ (ℎ = 𝐹 → ((1st ‘ℎ) ∘ (1st ‘𝑖)) = ((1st ‘𝐹) ∘ (1st ‘𝑖))) |
3 | fveq2 6672 | . . . 4 ⊢ (ℎ = 𝐹 → (2nd ‘ℎ) = (2nd ‘𝐹)) | |
4 | 3 | oveq1d 7173 | . . 3 ⊢ (ℎ = 𝐹 → ((2nd ‘ℎ) ⨣ (2nd ‘𝑖)) = ((2nd ‘𝐹) ⨣ (2nd ‘𝑖))) |
5 | 2, 4 | opeq12d 4813 | . 2 ⊢ (ℎ = 𝐹 → 〈((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))〉 = 〈((1st ‘𝐹) ∘ (1st ‘𝑖)), ((2nd ‘𝐹) ⨣ (2nd ‘𝑖))〉) |
6 | fveq2 6672 | . . . 4 ⊢ (𝑖 = 𝐺 → (1st ‘𝑖) = (1st ‘𝐺)) | |
7 | 6 | coeq2d 5735 | . . 3 ⊢ (𝑖 = 𝐺 → ((1st ‘𝐹) ∘ (1st ‘𝑖)) = ((1st ‘𝐹) ∘ (1st ‘𝐺))) |
8 | fveq2 6672 | . . . 4 ⊢ (𝑖 = 𝐺 → (2nd ‘𝑖) = (2nd ‘𝐺)) | |
9 | 8 | oveq2d 7174 | . . 3 ⊢ (𝑖 = 𝐺 → ((2nd ‘𝐹) ⨣ (2nd ‘𝑖)) = ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))) |
10 | 7, 9 | opeq12d 4813 | . 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 38227 | . 2 ⊢ + = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))〉) |
13 | opex 5358 | . 2 ⊢ 〈((1st ‘𝐹) ∘ (1st ‘𝐺)), ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))〉 ∈ V | |
14 | 5, 10, 12, 13 | ovmpo 7312 | 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 398 = wceq 1537 ∈ wcel 2114 〈cop 4575 × cxp 5555 ∘ ccom 5561 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 1st c1st 7689 2nd c2nd 7690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 |
This theorem is referenced by: dvhvadd 38230 dvhopaddN 38252 |
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