![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhvaddval | Structured version Visualization version GIF version |
Description: The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) |
Ref | Expression |
---|---|
dvhvaddval.a | ⊢ + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))〉) |
Ref | Expression |
---|---|
dvhvaddval | ⊢ ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸)) → (𝐹 + 𝐺) = 〈((1st ‘𝐹) ∘ (1st ‘𝐺)), ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6411 | . . . 4 ⊢ (ℎ = 𝐹 → (1st ‘ℎ) = (1st ‘𝐹)) | |
2 | 1 | coeq1d 5487 | . . 3 ⊢ (ℎ = 𝐹 → ((1st ‘ℎ) ∘ (1st ‘𝑖)) = ((1st ‘𝐹) ∘ (1st ‘𝑖))) |
3 | fveq2 6411 | . . . 4 ⊢ (ℎ = 𝐹 → (2nd ‘ℎ) = (2nd ‘𝐹)) | |
4 | 3 | oveq1d 6893 | . . 3 ⊢ (ℎ = 𝐹 → ((2nd ‘ℎ) ⨣ (2nd ‘𝑖)) = ((2nd ‘𝐹) ⨣ (2nd ‘𝑖))) |
5 | 2, 4 | opeq12d 4601 | . 2 ⊢ (ℎ = 𝐹 → 〈((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))〉 = 〈((1st ‘𝐹) ∘ (1st ‘𝑖)), ((2nd ‘𝐹) ⨣ (2nd ‘𝑖))〉) |
6 | fveq2 6411 | . . . 4 ⊢ (𝑖 = 𝐺 → (1st ‘𝑖) = (1st ‘𝐺)) | |
7 | 6 | coeq2d 5488 | . . 3 ⊢ (𝑖 = 𝐺 → ((1st ‘𝐹) ∘ (1st ‘𝑖)) = ((1st ‘𝐹) ∘ (1st ‘𝐺))) |
8 | fveq2 6411 | . . . 4 ⊢ (𝑖 = 𝐺 → (2nd ‘𝑖) = (2nd ‘𝐺)) | |
9 | 8 | oveq2d 6894 | . . 3 ⊢ (𝑖 = 𝐺 → ((2nd ‘𝐹) ⨣ (2nd ‘𝑖)) = ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))) |
10 | 7, 9 | opeq12d 4601 | . 2 ⊢ (𝑖 = 𝐺 → 〈((1st ‘𝐹) ∘ (1st ‘𝑖)), ((2nd ‘𝐹) ⨣ (2nd ‘𝑖))〉 = 〈((1st ‘𝐹) ∘ (1st ‘𝐺)), ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))〉) |
11 | dvhvaddval.a | . . 3 ⊢ + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))〉) | |
12 | 11 | dvhvaddcbv 37110 | . 2 ⊢ + = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))〉) |
13 | opex 5123 | . 2 ⊢ 〈((1st ‘𝐹) ∘ (1st ‘𝐺)), ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))〉 ∈ V | |
14 | 5, 10, 12, 13 | ovmpt2 7030 | 1 ⊢ ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸)) → (𝐹 + 𝐺) = 〈((1st ‘𝐹) ∘ (1st ‘𝐺)), ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 〈cop 4374 × cxp 5310 ∘ ccom 5316 ‘cfv 6101 (class class class)co 6878 ↦ cmpt2 6880 1st c1st 7399 2nd c2nd 7400 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-iota 6064 df-fun 6103 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 |
This theorem is referenced by: dvhvadd 37113 dvhopaddN 37135 |
Copyright terms: Public domain | W3C validator |