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 6658 . . . 4 ( = 𝐹 → (1st) = (1st𝐹))
21coeq1d 5701 . . 3 ( = 𝐹 → ((1st) ∘ (1st𝑖)) = ((1st𝐹) ∘ (1st𝑖)))
3 fveq2 6658 . . . 4 ( = 𝐹 → (2nd) = (2nd𝐹))
43oveq1d 7165 . . 3 ( = 𝐹 → ((2nd) (2nd𝑖)) = ((2nd𝐹) (2nd𝑖)))
52, 4opeq12d 4771 . 2 ( = 𝐹 → ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩ = ⟨((1st𝐹) ∘ (1st𝑖)), ((2nd𝐹) (2nd𝑖))⟩)
6 fveq2 6658 . . . 4 (𝑖 = 𝐺 → (1st𝑖) = (1st𝐺))
76coeq2d 5702 . . 3 (𝑖 = 𝐺 → ((1st𝐹) ∘ (1st𝑖)) = ((1st𝐹) ∘ (1st𝐺)))
8 fveq2 6658 . . . 4 (𝑖 = 𝐺 → (2nd𝑖) = (2nd𝐺))
98oveq2d 7166 . . 3 (𝑖 = 𝐺 → ((2nd𝐹) (2nd𝑖)) = ((2nd𝐹) (2nd𝐺)))
107, 9opeq12d 4771 . 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 38687 . 2 + = ( ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
13 opex 5324 . 2 ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩ ∈ V
145, 10, 12, 13ovmpo 7305 1 ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸)) → (𝐹 + 𝐺) = ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ⟨cop 4528   × cxp 5522   ∘ ccom 5528  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152  1st c1st 7691  2nd c2nd 7692 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155 This theorem is referenced by:  dvhvadd  38690  dvhopaddN  38712
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