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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhopspN | Structured version Visualization version GIF version |
Description: Scalar product of DVecH vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dvhopsp.s | ⊢ 𝑆 = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉) |
Ref | Expression |
---|---|
dvhopspN | ⊢ ((𝑅 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑅𝑆〈𝐹, 𝑈〉) = 〈(𝑅‘𝐹), (𝑅 ∘ 𝑈)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 5726 | . . 3 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → 〈𝐹, 𝑈〉 ∈ (𝑇 × 𝐸)) | |
2 | dvhopsp.s | . . . 4 ⊢ 𝑆 = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉) | |
3 | 2 | dvhvscaval 41082 | . . 3 ⊢ ((𝑅 ∈ 𝐸 ∧ 〈𝐹, 𝑈〉 ∈ (𝑇 × 𝐸)) → (𝑅𝑆〈𝐹, 𝑈〉) = 〈(𝑅‘(1st ‘〈𝐹, 𝑈〉)), (𝑅 ∘ (2nd ‘〈𝐹, 𝑈〉))〉) |
4 | 1, 3 | sylan2 593 | . 2 ⊢ ((𝑅 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑅𝑆〈𝐹, 𝑈〉) = 〈(𝑅‘(1st ‘〈𝐹, 𝑈〉)), (𝑅 ∘ (2nd ‘〈𝐹, 𝑈〉))〉) |
5 | op1stg 8025 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (1st ‘〈𝐹, 𝑈〉) = 𝐹) | |
6 | 5 | fveq2d 6911 | . . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (𝑅‘(1st ‘〈𝐹, 𝑈〉)) = (𝑅‘𝐹)) |
7 | op2ndg 8026 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (2nd ‘〈𝐹, 𝑈〉) = 𝑈) | |
8 | 7 | coeq2d 5876 | . . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (𝑅 ∘ (2nd ‘〈𝐹, 𝑈〉)) = (𝑅 ∘ 𝑈)) |
9 | 6, 8 | opeq12d 4886 | . . 3 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → 〈(𝑅‘(1st ‘〈𝐹, 𝑈〉)), (𝑅 ∘ (2nd ‘〈𝐹, 𝑈〉))〉 = 〈(𝑅‘𝐹), (𝑅 ∘ 𝑈)〉) |
10 | 9 | adantl 481 | . 2 ⊢ ((𝑅 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 〈(𝑅‘(1st ‘〈𝐹, 𝑈〉)), (𝑅 ∘ (2nd ‘〈𝐹, 𝑈〉))〉 = 〈(𝑅‘𝐹), (𝑅 ∘ 𝑈)〉) |
11 | 4, 10 | eqtrd 2775 | 1 ⊢ ((𝑅 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑅𝑆〈𝐹, 𝑈〉) = 〈(𝑅‘𝐹), (𝑅 ∘ 𝑈)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 〈cop 4637 × cxp 5687 ∘ ccom 5693 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8011 2nd c2nd 8012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 |
This theorem is referenced by: dvhopN 41099 |
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