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Theorem dvhvaddval 40693
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 6896 . . . 4 ( = 𝐹 → (1st) = (1st𝐹))
21coeq1d 5864 . . 3 ( = 𝐹 → ((1st) ∘ (1st𝑖)) = ((1st𝐹) ∘ (1st𝑖)))
3 fveq2 6896 . . . 4 ( = 𝐹 → (2nd) = (2nd𝐹))
43oveq1d 7434 . . 3 ( = 𝐹 → ((2nd) (2nd𝑖)) = ((2nd𝐹) (2nd𝑖)))
52, 4opeq12d 4883 . 2 ( = 𝐹 → ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩ = ⟨((1st𝐹) ∘ (1st𝑖)), ((2nd𝐹) (2nd𝑖))⟩)
6 fveq2 6896 . . . 4 (𝑖 = 𝐺 → (1st𝑖) = (1st𝐺))
76coeq2d 5865 . . 3 (𝑖 = 𝐺 → ((1st𝐹) ∘ (1st𝑖)) = ((1st𝐹) ∘ (1st𝐺)))
8 fveq2 6896 . . . 4 (𝑖 = 𝐺 → (2nd𝑖) = (2nd𝐺))
98oveq2d 7435 . . 3 (𝑖 = 𝐺 → ((2nd𝐹) (2nd𝑖)) = ((2nd𝐹) (2nd𝐺)))
107, 9opeq12d 4883 . 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 40692 . 2 + = ( ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
13 opex 5466 . 2 ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩ ∈ V
145, 10, 12, 13ovmpo 7581 1 ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸)) → (𝐹 + 𝐺) = ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cop 4636   × cxp 5676  ccom 5682  cfv 6549  (class class class)co 7419  cmpo 7421  1st c1st 7992  2nd c2nd 7993
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 5300  ax-nul 5307  ax-pr 5429
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 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424
This theorem is referenced by:  dvhvadd  40695  dvhopaddN  40717
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