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Theorem dvhvaddval 40619
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 6892 . . . 4 (ℎ = ðđ → (1st ‘ℎ) = (1st ‘ðđ))
21coeq1d 5858 . . 3 (ℎ = ðđ → ((1st ‘ℎ) ∘ (1st ‘𝑖)) = ((1st ‘ðđ) ∘ (1st ‘𝑖)))
3 fveq2 6892 . . . 4 (ℎ = ðđ → (2nd ‘ℎ) = (2nd ‘ðđ))
43oveq1d 7431 . . 3 (ℎ = ðđ → ((2nd ‘ℎ) âĻĢ (2nd ‘𝑖)) = ((2nd ‘ðđ) âĻĢ (2nd ‘𝑖)))
52, 4opeq12d 4877 . 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 6892 . . . 4 (𝑖 = 𝐚 → (1st ‘𝑖) = (1st ‘𝐚))
76coeq2d 5859 . . 3 (𝑖 = 𝐚 → ((1st ‘ðđ) ∘ (1st ‘𝑖)) = ((1st ‘ðđ) ∘ (1st ‘𝐚)))
8 fveq2 6892 . . . 4 (𝑖 = 𝐚 → (2nd ‘𝑖) = (2nd ‘𝐚))
98oveq2d 7432 . . 3 (𝑖 = 𝐚 → ((2nd ‘ðđ) âĻĢ (2nd ‘𝑖)) = ((2nd ‘ðđ) âĻĢ (2nd ‘𝐚)))
107, 9opeq12d 4877 . 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 40618 . 2 + = (ℎ ∈ (𝑇 × ðļ), 𝑖 ∈ (𝑇 × ðļ) â†Ķ âŸĻ((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) âĻĢ (2nd ‘𝑖))âŸĐ)
13 opex 5460 . 2 âŸĻ((1st ‘ðđ) ∘ (1st ‘𝐚)), ((2nd ‘ðđ) âĻĢ (2nd ‘𝐚))âŸĐ ∈ V
145, 10, 12, 13ovmpo 7578 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 394   = wceq 1533   ∈ wcel 2098  âŸĻcop 4630   × cxp 5670   ∘ ccom 5676  â€˜cfv 6543  (class class class)co 7416   ∈ cmpo 7418  1st c1st 7989  2nd c2nd 7990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421
This theorem is referenced by:  dvhvadd  40621  dvhopaddN  40643
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