Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 5594 | . . 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 38237 | . . 3 ⊢ ((𝑅 ∈ 𝐸 ∧ 〈𝐹, 𝑈〉 ∈ (𝑇 × 𝐸)) → (𝑅𝑆〈𝐹, 𝑈〉) = 〈(𝑅‘(1st ‘〈𝐹, 𝑈〉)), (𝑅 ∘ (2nd ‘〈𝐹, 𝑈〉))〉) |
4 | 1, 3 | sylan2 594 | . 2 ⊢ ((𝑅 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑅𝑆〈𝐹, 𝑈〉) = 〈(𝑅‘(1st ‘〈𝐹, 𝑈〉)), (𝑅 ∘ (2nd ‘〈𝐹, 𝑈〉))〉) |
5 | op1stg 7703 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (1st ‘〈𝐹, 𝑈〉) = 𝐹) | |
6 | 5 | fveq2d 6676 | . . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (𝑅‘(1st ‘〈𝐹, 𝑈〉)) = (𝑅‘𝐹)) |
7 | op2ndg 7704 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (2nd ‘〈𝐹, 𝑈〉) = 𝑈) | |
8 | 7 | coeq2d 5735 | . . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (𝑅 ∘ (2nd ‘〈𝐹, 𝑈〉)) = (𝑅 ∘ 𝑈)) |
9 | 6, 8 | opeq12d 4813 | . . 3 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → 〈(𝑅‘(1st ‘〈𝐹, 𝑈〉)), (𝑅 ∘ (2nd ‘〈𝐹, 𝑈〉))〉 = 〈(𝑅‘𝐹), (𝑅 ∘ 𝑈)〉) |
10 | 9 | adantl 484 | . 2 ⊢ ((𝑅 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 〈(𝑅‘(1st ‘〈𝐹, 𝑈〉)), (𝑅 ∘ (2nd ‘〈𝐹, 𝑈〉))〉 = 〈(𝑅‘𝐹), (𝑅 ∘ 𝑈)〉) |
11 | 4, 10 | eqtrd 2858 | 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 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-pow 5268 ax-pr 5332 ax-un 7463 |
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-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 |
This theorem is referenced by: dvhopN 38254 |
Copyright terms: Public domain | W3C validator |