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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhvscaval | Structured version Visualization version GIF version |
Description: The scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Nov-2013.) |
Ref | Expression |
---|---|
dvhvscaval.s | ⊢ · = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉) |
Ref | Expression |
---|---|
dvhvscaval | ⊢ ((𝑈 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸)) → (𝑈 · 𝐹) = 〈(𝑈‘(1st ‘𝐹)), (𝑈 ∘ (2nd ‘𝐹))〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6102 | . . 3 ⊢ (𝑡 = 𝑈 → (𝑡‘(1st ‘𝑔)) = (𝑈‘(1st ‘𝑔))) | |
2 | coeq1 5201 | . . 3 ⊢ (𝑡 = 𝑈 → (𝑡 ∘ (2nd ‘𝑔)) = (𝑈 ∘ (2nd ‘𝑔))) | |
3 | 1, 2 | opeq12d 4348 | . 2 ⊢ (𝑡 = 𝑈 → 〈(𝑡‘(1st ‘𝑔)), (𝑡 ∘ (2nd ‘𝑔))〉 = 〈(𝑈‘(1st ‘𝑔)), (𝑈 ∘ (2nd ‘𝑔))〉) |
4 | fveq2 6103 | . . . 4 ⊢ (𝑔 = 𝐹 → (1st ‘𝑔) = (1st ‘𝐹)) | |
5 | 4 | fveq2d 6107 | . . 3 ⊢ (𝑔 = 𝐹 → (𝑈‘(1st ‘𝑔)) = (𝑈‘(1st ‘𝐹))) |
6 | fveq2 6103 | . . . 4 ⊢ (𝑔 = 𝐹 → (2nd ‘𝑔) = (2nd ‘𝐹)) | |
7 | 6 | coeq2d 5206 | . . 3 ⊢ (𝑔 = 𝐹 → (𝑈 ∘ (2nd ‘𝑔)) = (𝑈 ∘ (2nd ‘𝐹))) |
8 | 5, 7 | opeq12d 4348 | . 2 ⊢ (𝑔 = 𝐹 → 〈(𝑈‘(1st ‘𝑔)), (𝑈 ∘ (2nd ‘𝑔))〉 = 〈(𝑈‘(1st ‘𝐹)), (𝑈 ∘ (2nd ‘𝐹))〉) |
9 | dvhvscaval.s | . . 3 ⊢ · = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉) | |
10 | 9 | dvhvscacbv 35405 | . 2 ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈(𝑡‘(1st ‘𝑔)), (𝑡 ∘ (2nd ‘𝑔))〉) |
11 | opex 4859 | . 2 ⊢ 〈(𝑈‘(1st ‘𝐹)), (𝑈 ∘ (2nd ‘𝐹))〉 ∈ V | |
12 | 3, 8, 10, 11 | ovmpt2 6694 | 1 ⊢ ((𝑈 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸)) → (𝑈 · 𝐹) = 〈(𝑈‘(1st ‘𝐹)), (𝑈 ∘ (2nd ‘𝐹))〉) |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
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-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 |
This theorem is referenced by: dvhvsca 35408 dvhopspN 35422 |
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