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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 6671 | . . 3 ⊢ (𝑡 = 𝑈 → (𝑡‘(1st ‘𝑔)) = (𝑈‘(1st ‘𝑔))) | |
2 | coeq1 5730 | . . 3 ⊢ (𝑡 = 𝑈 → (𝑡 ∘ (2nd ‘𝑔)) = (𝑈 ∘ (2nd ‘𝑔))) | |
3 | 1, 2 | opeq12d 4813 | . 2 ⊢ (𝑡 = 𝑈 → 〈(𝑡‘(1st ‘𝑔)), (𝑡 ∘ (2nd ‘𝑔))〉 = 〈(𝑈‘(1st ‘𝑔)), (𝑈 ∘ (2nd ‘𝑔))〉) |
4 | 2fveq3 6677 | . . 3 ⊢ (𝑔 = 𝐹 → (𝑈‘(1st ‘𝑔)) = (𝑈‘(1st ‘𝐹))) | |
5 | fveq2 6672 | . . . 4 ⊢ (𝑔 = 𝐹 → (2nd ‘𝑔) = (2nd ‘𝐹)) | |
6 | 5 | coeq2d 5735 | . . 3 ⊢ (𝑔 = 𝐹 → (𝑈 ∘ (2nd ‘𝑔)) = (𝑈 ∘ (2nd ‘𝐹))) |
7 | 4, 6 | opeq12d 4813 | . 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 38236 | . 2 ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈(𝑡‘(1st ‘𝑔)), (𝑡 ∘ (2nd ‘𝑔))〉) |
10 | opex 5358 | . 2 ⊢ 〈(𝑈‘(1st ‘𝐹)), (𝑈 ∘ (2nd ‘𝐹))〉 ∈ V | |
11 | 3, 7, 9, 10 | ovmpo 7312 | 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 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-pr 5332 |
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-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 |
This theorem is referenced by: dvhvsca 38239 dvhopspN 38253 |
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