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Theorem dvhvaddval 39556
Description: The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.)
Hypothesis
Ref Expression
dvhvaddval.a + = (𝑓 ∈ (𝑇 × ðļ), 𝑔 ∈ (𝑇 × ðļ) â†Ķ âŸĻ((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) âĻĢ (2nd ‘𝑔))âŸĐ)
Assertion
Ref Expression
dvhvaddval ((ðđ ∈ (𝑇 × ðļ) ∧ 𝐚 ∈ (𝑇 × ðļ)) → (ðđ + 𝐚) = âŸĻ((1st ‘ðđ) ∘ (1st ‘𝐚)), ((2nd ‘ðđ) âĻĢ (2nd ‘𝐚))âŸĐ)
Distinct variable groups:   𝑓,𝑔,ðļ   âĻĢ ,𝑓,𝑔   𝑇,𝑓,𝑔
Allowed substitution hints:   + (𝑓,𝑔)   ðđ(𝑓,𝑔)   𝐚(𝑓,𝑔)

Proof of Theorem dvhvaddval
Dummy variables ℎ 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6843 . . . 4 (ℎ = ðđ → (1st ‘ℎ) = (1st ‘ðđ))
21coeq1d 5818 . . 3 (ℎ = ðđ → ((1st ‘ℎ) ∘ (1st ‘𝑖)) = ((1st ‘ðđ) ∘ (1st ‘𝑖)))
3 fveq2 6843 . . . 4 (ℎ = ðđ → (2nd ‘ℎ) = (2nd ‘ðđ))
43oveq1d 7373 . . 3 (ℎ = ðđ → ((2nd ‘ℎ) âĻĢ (2nd ‘𝑖)) = ((2nd ‘ðđ) âĻĢ (2nd ‘𝑖)))
52, 4opeq12d 4839 . 2 (ℎ = ðđ → âŸĻ((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) âĻĢ (2nd ‘𝑖))âŸĐ = âŸĻ((1st ‘ðđ) ∘ (1st ‘𝑖)), ((2nd ‘ðđ) âĻĢ (2nd ‘𝑖))âŸĐ)
6 fveq2 6843 . . . 4 (𝑖 = 𝐚 → (1st ‘𝑖) = (1st ‘𝐚))
76coeq2d 5819 . . 3 (𝑖 = 𝐚 → ((1st ‘ðđ) ∘ (1st ‘𝑖)) = ((1st ‘ðđ) ∘ (1st ‘𝐚)))
8 fveq2 6843 . . . 4 (𝑖 = 𝐚 → (2nd ‘𝑖) = (2nd ‘𝐚))
98oveq2d 7374 . . 3 (𝑖 = 𝐚 → ((2nd ‘ðđ) âĻĢ (2nd ‘𝑖)) = ((2nd ‘ðđ) âĻĢ (2nd ‘𝐚)))
107, 9opeq12d 4839 . 2 (𝑖 = 𝐚 → âŸĻ((1st ‘ðđ) ∘ (1st ‘𝑖)), ((2nd ‘ðđ) âĻĢ (2nd ‘𝑖))âŸĐ = âŸĻ((1st ‘ðđ) ∘ (1st ‘𝐚)), ((2nd ‘ðđ) âĻĢ (2nd ‘𝐚))âŸĐ)
11 dvhvaddval.a . . 3 + = (𝑓 ∈ (𝑇 × ðļ), 𝑔 ∈ (𝑇 × ðļ) â†Ķ âŸĻ((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) âĻĢ (2nd ‘𝑔))âŸĐ)
1211dvhvaddcbv 39555 . 2 + = (ℎ ∈ (𝑇 × ðļ), 𝑖 ∈ (𝑇 × ðļ) â†Ķ âŸĻ((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) âĻĢ (2nd ‘𝑖))âŸĐ)
13 opex 5422 . 2 âŸĻ((1st ‘ðđ) ∘ (1st ‘𝐚)), ((2nd ‘ðđ) âĻĢ (2nd ‘𝐚))âŸĐ ∈ V
145, 10, 12, 13ovmpo 7516 1 ((ðđ ∈ (𝑇 × ðļ) ∧ 𝐚 ∈ (𝑇 × ðļ)) → (ðđ + 𝐚) = âŸĻ((1st ‘ðđ) ∘ (1st ‘𝐚)), ((2nd ‘ðđ) âĻĢ (2nd ‘𝐚))âŸĐ)
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  âŸĻcop 4593   × cxp 5632   ∘ ccom 5638  â€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  1st c1st 7920  2nd c2nd 7921
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 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363
This theorem is referenced by:  dvhvadd  39558  dvhopaddN  39580
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