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

Theorem dvhvaddval 37111
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 6411 . . . 4 ( = 𝐹 → (1st) = (1st𝐹))
21coeq1d 5487 . . 3 ( = 𝐹 → ((1st) ∘ (1st𝑖)) = ((1st𝐹) ∘ (1st𝑖)))
3 fveq2 6411 . . . 4 ( = 𝐹 → (2nd) = (2nd𝐹))
43oveq1d 6893 . . 3 ( = 𝐹 → ((2nd) (2nd𝑖)) = ((2nd𝐹) (2nd𝑖)))
52, 4opeq12d 4601 . 2 ( = 𝐹 → ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩ = ⟨((1st𝐹) ∘ (1st𝑖)), ((2nd𝐹) (2nd𝑖))⟩)
6 fveq2 6411 . . . 4 (𝑖 = 𝐺 → (1st𝑖) = (1st𝐺))
76coeq2d 5488 . . 3 (𝑖 = 𝐺 → ((1st𝐹) ∘ (1st𝑖)) = ((1st𝐹) ∘ (1st𝐺)))
8 fveq2 6411 . . . 4 (𝑖 = 𝐺 → (2nd𝑖) = (2nd𝐺))
98oveq2d 6894 . . 3 (𝑖 = 𝐺 → ((2nd𝐹) (2nd𝑖)) = ((2nd𝐹) (2nd𝐺)))
107, 9opeq12d 4601 . 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 37110 . 2 + = ( ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
13 opex 5123 . 2 ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩ ∈ V
145, 10, 12, 13ovmpt2 7030 1 ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸)) → (𝐹 + 𝐺) = ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  cop 4374   × cxp 5310  ccom 5316  cfv 6101  (class class class)co 6878  cmpt2 6880  1st c1st 7399  2nd c2nd 7400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883
This theorem is referenced by:  dvhvadd  37113  dvhopaddN  37135
  Copyright terms: Public domain W3C validator