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Theorem dvhvaddval 40474
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 6885 . . . 4 (ℎ = ðđ → (1st ‘ℎ) = (1st ‘ðđ))
21coeq1d 5855 . . 3 (ℎ = ðđ → ((1st ‘ℎ) ∘ (1st ‘𝑖)) = ((1st ‘ðđ) ∘ (1st ‘𝑖)))
3 fveq2 6885 . . . 4 (ℎ = ðđ → (2nd ‘ℎ) = (2nd ‘ðđ))
43oveq1d 7420 . . 3 (ℎ = ðđ → ((2nd ‘ℎ) âĻĢ (2nd ‘𝑖)) = ((2nd ‘ðđ) âĻĢ (2nd ‘𝑖)))
52, 4opeq12d 4876 . 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 6885 . . . 4 (𝑖 = 𝐚 → (1st ‘𝑖) = (1st ‘𝐚))
76coeq2d 5856 . . 3 (𝑖 = 𝐚 → ((1st ‘ðđ) ∘ (1st ‘𝑖)) = ((1st ‘ðđ) ∘ (1st ‘𝐚)))
8 fveq2 6885 . . . 4 (𝑖 = 𝐚 → (2nd ‘𝑖) = (2nd ‘𝐚))
98oveq2d 7421 . . 3 (𝑖 = 𝐚 → ((2nd ‘ðđ) âĻĢ (2nd ‘𝑖)) = ((2nd ‘ðđ) âĻĢ (2nd ‘𝐚)))
107, 9opeq12d 4876 . 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 40473 . 2 + = (ℎ ∈ (𝑇 × ðļ), 𝑖 ∈ (𝑇 × ðļ) â†Ķ âŸĻ((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) âĻĢ (2nd ‘𝑖))âŸĐ)
13 opex 5457 . 2 âŸĻ((1st ‘ðđ) ∘ (1st ‘𝐚)), ((2nd ‘ðđ) âĻĢ (2nd ‘𝐚))âŸĐ ∈ V
145, 10, 12, 13ovmpo 7564 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 395   = wceq 1533   ∈ wcel 2098  âŸĻcop 4629   × cxp 5667   ∘ ccom 5673  â€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  1st c1st 7972  2nd c2nd 7973
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410
This theorem is referenced by:  dvhvadd  40476  dvhopaddN  40498
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