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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 5722 | . . 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 41101 | . . 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 8026 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (1st ‘〈𝐹, 𝑈〉) = 𝐹) | |
| 6 | 5 | fveq2d 6910 | . . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (𝑅‘(1st ‘〈𝐹, 𝑈〉)) = (𝑅‘𝐹)) |
| 7 | op2ndg 8027 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (2nd ‘〈𝐹, 𝑈〉) = 𝑈) | |
| 8 | 7 | coeq2d 5873 | . . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (𝑅 ∘ (2nd ‘〈𝐹, 𝑈〉)) = (𝑅 ∘ 𝑈)) |
| 9 | 6, 8 | opeq12d 4881 | . . 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 2777 | 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 1540 ∈ wcel 2108 〈cop 4632 × cxp 5683 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8012 2nd c2nd 8013 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 |
| This theorem is referenced by: dvhopN 41118 |
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