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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhopaddN | Structured version Visualization version GIF version |
Description: Sum of DVecH vectors expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dvhopadd.a | ⊢ 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓)𝑃(2nd ‘𝑔))〉) |
Ref | Expression |
---|---|
dvhopaddN | ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (〈𝐹, 𝑈〉𝐴〈𝐺, 𝑉〉) = 〈(𝐹 ∘ 𝐺), (𝑈𝑃𝑉)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 5072 | . . 3 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → 〈𝐹, 𝑈〉 ∈ (𝑇 × 𝐸)) | |
2 | opelxpi 5072 | . . 3 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸) → 〈𝐺, 𝑉〉 ∈ (𝑇 × 𝐸)) | |
3 | dvhopadd.a | . . . 4 ⊢ 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓)𝑃(2nd ‘𝑔))〉) | |
4 | 3 | dvhvaddval 35397 | . . 3 ⊢ ((〈𝐹, 𝑈〉 ∈ (𝑇 × 𝐸) ∧ 〈𝐺, 𝑉〉 ∈ (𝑇 × 𝐸)) → (〈𝐹, 𝑈〉𝐴〈𝐺, 𝑉〉) = 〈((1st ‘〈𝐹, 𝑈〉) ∘ (1st ‘〈𝐺, 𝑉〉)), ((2nd ‘〈𝐹, 𝑈〉)𝑃(2nd ‘〈𝐺, 𝑉〉))〉) |
5 | 1, 2, 4 | syl2an 493 | . 2 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (〈𝐹, 𝑈〉𝐴〈𝐺, 𝑉〉) = 〈((1st ‘〈𝐹, 𝑈〉) ∘ (1st ‘〈𝐺, 𝑉〉)), ((2nd ‘〈𝐹, 𝑈〉)𝑃(2nd ‘〈𝐺, 𝑉〉))〉) |
6 | op1stg 7071 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (1st ‘〈𝐹, 𝑈〉) = 𝐹) | |
7 | 6 | adantr 480 | . . . 4 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (1st ‘〈𝐹, 𝑈〉) = 𝐹) |
8 | op1stg 7071 | . . . . 5 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸) → (1st ‘〈𝐺, 𝑉〉) = 𝐺) | |
9 | 8 | adantl 481 | . . . 4 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (1st ‘〈𝐺, 𝑉〉) = 𝐺) |
10 | 7, 9 | coeq12d 5208 | . . 3 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → ((1st ‘〈𝐹, 𝑈〉) ∘ (1st ‘〈𝐺, 𝑉〉)) = (𝐹 ∘ 𝐺)) |
11 | op2ndg 7072 | . . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (2nd ‘〈𝐹, 𝑈〉) = 𝑈) | |
12 | op2ndg 7072 | . . . 4 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸) → (2nd ‘〈𝐺, 𝑉〉) = 𝑉) | |
13 | 11, 12 | oveqan12d 6568 | . . 3 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → ((2nd ‘〈𝐹, 𝑈〉)𝑃(2nd ‘〈𝐺, 𝑉〉)) = (𝑈𝑃𝑉)) |
14 | 10, 13 | opeq12d 4348 | . 2 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → 〈((1st ‘〈𝐹, 𝑈〉) ∘ (1st ‘〈𝐺, 𝑉〉)), ((2nd ‘〈𝐹, 𝑈〉)𝑃(2nd ‘〈𝐺, 𝑉〉))〉 = 〈(𝐹 ∘ 𝐺), (𝑈𝑃𝑉)〉) |
15 | 5, 14 | eqtrd 2644 | 1 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (〈𝐹, 𝑈〉𝐴〈𝐺, 𝑉〉) = 〈(𝐹 ∘ 𝐺), (𝑈𝑃𝑉)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 × cxp 5036 ∘ ccom 5042 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1st c1st 7057 2nd c2nd 7058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 |
This theorem is referenced by: dvhopN 35423 |
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