Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvhvaddval Structured version   Visualization version   GIF version

 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)

Dummy variables 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6103 . . . 4 ( = 𝐹 → (1st) = (1st𝐹))
21coeq1d 5205 . . 3 ( = 𝐹 → ((1st) ∘ (1st𝑖)) = ((1st𝐹) ∘ (1st𝑖)))
3 fveq2 6103 . . . 4 ( = 𝐹 → (2nd) = (2nd𝐹))
43oveq1d 6564 . . 3 ( = 𝐹 → ((2nd) (2nd𝑖)) = ((2nd𝐹) (2nd𝑖)))
52, 4opeq12d 4348 . 2 ( = 𝐹 → ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩ = ⟨((1st𝐹) ∘ (1st𝑖)), ((2nd𝐹) (2nd𝑖))⟩)
6 fveq2 6103 . . . 4 (𝑖 = 𝐺 → (1st𝑖) = (1st𝐺))
76coeq2d 5206 . . 3 (𝑖 = 𝐺 → ((1st𝐹) ∘ (1st𝑖)) = ((1st𝐹) ∘ (1st𝐺)))
8 fveq2 6103 . . . 4 (𝑖 = 𝐺 → (2nd𝑖) = (2nd𝐺))
98oveq2d 6565 . . 3 (𝑖 = 𝐺 → ((2nd𝐹) (2nd𝑖)) = ((2nd𝐹) (2nd𝐺)))
107, 9opeq12d 4348 . 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 35396 . 2 + = ( ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
13 opex 4859 . 2 ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩ ∈ V
145, 10, 12, 13ovmpt2 6694 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 1475   ∈ wcel 1977  ⟨cop 4131   × cxp 5036   ∘ ccom 5042  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  dvhvadd  35399  dvhopaddN  35421
 Copyright terms: Public domain W3C validator