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Mathbox for Norm Megill |
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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 6887 | . . 3 ⊢ (𝑡 = 𝑈 → (𝑡‘(1st ‘𝑔)) = (𝑈‘(1st ‘𝑔))) | |
2 | coeq1 5855 | . . 3 ⊢ (𝑡 = 𝑈 → (𝑡 ∘ (2nd ‘𝑔)) = (𝑈 ∘ (2nd ‘𝑔))) | |
3 | 1, 2 | opeq12d 4880 | . 2 ⊢ (𝑡 = 𝑈 → 〈(𝑡‘(1st ‘𝑔)), (𝑡 ∘ (2nd ‘𝑔))〉 = 〈(𝑈‘(1st ‘𝑔)), (𝑈 ∘ (2nd ‘𝑔))〉) |
4 | 2fveq3 6893 | . . 3 ⊢ (𝑔 = 𝐹 → (𝑈‘(1st ‘𝑔)) = (𝑈‘(1st ‘𝐹))) | |
5 | fveq2 6888 | . . . 4 ⊢ (𝑔 = 𝐹 → (2nd ‘𝑔) = (2nd ‘𝐹)) | |
6 | 5 | coeq2d 5860 | . . 3 ⊢ (𝑔 = 𝐹 → (𝑈 ∘ (2nd ‘𝑔)) = (𝑈 ∘ (2nd ‘𝐹))) |
7 | 4, 6 | opeq12d 4880 | . 2 ⊢ (𝑔 = 𝐹 → 〈(𝑈‘(1st ‘𝑔)), (𝑈 ∘ (2nd ‘𝑔))〉 = 〈(𝑈‘(1st ‘𝐹)), (𝑈 ∘ (2nd ‘𝐹))〉) |
8 | dvhvscaval.s | . . 3 ⊢ · = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉) | |
9 | 8 | dvhvscacbv 39907 | . 2 ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈(𝑡‘(1st ‘𝑔)), (𝑡 ∘ (2nd ‘𝑔))〉) |
10 | opex 5463 | . 2 ⊢ 〈(𝑈‘(1st ‘𝐹)), (𝑈 ∘ (2nd ‘𝐹))〉 ∈ V | |
11 | 3, 7, 9, 10 | ovmpo 7563 | 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 397 = wceq 1542 ∈ wcel 2107 〈cop 4633 × cxp 5673 ∘ ccom 5679 ‘cfv 6540 (class class class)co 7404 ∈ cmpo 7406 1st c1st 7968 2nd c2nd 7969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 |
This theorem is referenced by: dvhvsca 39910 dvhopspN 39924 |
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