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Theorem dvhvaddval 36877
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 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩)
Assertion
Ref Expression
dvhvaddval ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸)) → (𝐹 + 𝐺) = ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩)
Distinct variable groups:   𝑓,𝑔,𝐸   ,𝑓,𝑔   𝑇,𝑓,𝑔
Allowed substitution hints:   + (𝑓,𝑔)   𝐹(𝑓,𝑔)   𝐺(𝑓,𝑔)

Proof of Theorem dvhvaddval
Dummy variables 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6348 . . . 4 ( = 𝐹 → (1st) = (1st𝐹))
21coeq1d 5435 . . 3 ( = 𝐹 → ((1st) ∘ (1st𝑖)) = ((1st𝐹) ∘ (1st𝑖)))
3 fveq2 6348 . . . 4 ( = 𝐹 → (2nd) = (2nd𝐹))
43oveq1d 6824 . . 3 ( = 𝐹 → ((2nd) (2nd𝑖)) = ((2nd𝐹) (2nd𝑖)))
52, 4opeq12d 4557 . 2 ( = 𝐹 → ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩ = ⟨((1st𝐹) ∘ (1st𝑖)), ((2nd𝐹) (2nd𝑖))⟩)
6 fveq2 6348 . . . 4 (𝑖 = 𝐺 → (1st𝑖) = (1st𝐺))
76coeq2d 5436 . . 3 (𝑖 = 𝐺 → ((1st𝐹) ∘ (1st𝑖)) = ((1st𝐹) ∘ (1st𝐺)))
8 fveq2 6348 . . . 4 (𝑖 = 𝐺 → (2nd𝑖) = (2nd𝐺))
98oveq2d 6825 . . 3 (𝑖 = 𝐺 → ((2nd𝐹) (2nd𝑖)) = ((2nd𝐹) (2nd𝐺)))
107, 9opeq12d 4557 . 2 (𝑖 = 𝐺 → ⟨((1st𝐹) ∘ (1st𝑖)), ((2nd𝐹) (2nd𝑖))⟩ = ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩)
11 dvhvaddval.a . . 3 + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩)
1211dvhvaddcbv 36876 . 2 + = ( ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
13 opex 5077 . 2 ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩ ∈ V
145, 10, 12, 13ovmpt2 6957 1 ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸)) → (𝐹 + 𝐺) = ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1628  wcel 2135  cop 4323   × cxp 5260  ccom 5266  cfv 6045  (class class class)co 6809  cmpt2 6811  1st c1st 7327  2nd c2nd 7328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pr 5051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-iota 6008  df-fun 6047  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814
This theorem is referenced by:  dvhvadd  36879  dvhopaddN  36901
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