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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 5556 | . . 3 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → 〈𝐹, 𝑈〉 ∈ (𝑇 × 𝐸)) | |
2 | opelxpi 5556 | . . 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 38386 | . . 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 598 | . 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 7683 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (1st ‘〈𝐹, 𝑈〉) = 𝐹) | |
7 | 6 | adantr 484 | . . . 4 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (1st ‘〈𝐹, 𝑈〉) = 𝐹) |
8 | op1stg 7683 | . . . . 5 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸) → (1st ‘〈𝐺, 𝑉〉) = 𝐺) | |
9 | 8 | adantl 485 | . . . 4 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (1st ‘〈𝐺, 𝑉〉) = 𝐺) |
10 | 7, 9 | coeq12d 5699 | . . 3 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → ((1st ‘〈𝐹, 𝑈〉) ∘ (1st ‘〈𝐺, 𝑉〉)) = (𝐹 ∘ 𝐺)) |
11 | op2ndg 7684 | . . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (2nd ‘〈𝐹, 𝑈〉) = 𝑈) | |
12 | op2ndg 7684 | . . . 4 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸) → (2nd ‘〈𝐺, 𝑉〉) = 𝑉) | |
13 | 11, 12 | oveqan12d 7154 | . . 3 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → ((2nd ‘〈𝐹, 𝑈〉)𝑃(2nd ‘〈𝐺, 𝑉〉)) = (𝑈𝑃𝑉)) |
14 | 10, 13 | opeq12d 4773 | . 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 2833 | 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 399 = wceq 1538 ∈ wcel 2111 〈cop 4531 × cxp 5517 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 1st c1st 7669 2nd c2nd 7670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 |
This theorem is referenced by: dvhopN 38412 |
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