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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 6644 | . . 3 ⊢ (𝑡 = 𝑈 → (𝑡‘(1st ‘𝑔)) = (𝑈‘(1st ‘𝑔))) | |
2 | coeq1 5692 | . . 3 ⊢ (𝑡 = 𝑈 → (𝑡 ∘ (2nd ‘𝑔)) = (𝑈 ∘ (2nd ‘𝑔))) | |
3 | 1, 2 | opeq12d 4773 | . 2 ⊢ (𝑡 = 𝑈 → 〈(𝑡‘(1st ‘𝑔)), (𝑡 ∘ (2nd ‘𝑔))〉 = 〈(𝑈‘(1st ‘𝑔)), (𝑈 ∘ (2nd ‘𝑔))〉) |
4 | 2fveq3 6650 | . . 3 ⊢ (𝑔 = 𝐹 → (𝑈‘(1st ‘𝑔)) = (𝑈‘(1st ‘𝐹))) | |
5 | fveq2 6645 | . . . 4 ⊢ (𝑔 = 𝐹 → (2nd ‘𝑔) = (2nd ‘𝐹)) | |
6 | 5 | coeq2d 5697 | . . 3 ⊢ (𝑔 = 𝐹 → (𝑈 ∘ (2nd ‘𝑔)) = (𝑈 ∘ (2nd ‘𝐹))) |
7 | 4, 6 | opeq12d 4773 | . 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 38394 | . 2 ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈(𝑡‘(1st ‘𝑔)), (𝑡 ∘ (2nd ‘𝑔))〉) |
10 | opex 5321 | . 2 ⊢ 〈(𝑈‘(1st ‘𝐹)), (𝑈 ∘ (2nd ‘𝐹))〉 ∈ V | |
11 | 3, 7, 9, 10 | ovmpo 7289 | 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 399 = wceq 1538 ∈ wcel 2111 〈cop 4531 × cxp 5517 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 1st c1st 7669 2nd c2nd 7670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 |
This theorem is referenced by: dvhvsca 38397 dvhopspN 38411 |
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