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

 Description: Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013.)
Hypothesis
Ref Expression
dvhvaddval.a + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩)
Assertion
Ref Expression
dvhvaddcbv + = ( ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
Distinct variable groups:   𝑓,𝑔,,𝑖,𝐸   ,𝑓,𝑔,,𝑖   𝑇,𝑓,𝑔,,𝑖
Allowed substitution hints:   + (𝑓,𝑔,,𝑖)

StepHypRef Expression
1 dvhvaddval.a . 2 + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩)
2 fveq2 6663 . . . . 5 (𝑓 = → (1st𝑓) = (1st))
32coeq1d 5707 . . . 4 (𝑓 = → ((1st𝑓) ∘ (1st𝑔)) = ((1st) ∘ (1st𝑔)))
4 fveq2 6663 . . . . 5 (𝑓 = → (2nd𝑓) = (2nd))
54oveq1d 7171 . . . 4 (𝑓 = → ((2nd𝑓) (2nd𝑔)) = ((2nd) (2nd𝑔)))
63, 5opeq12d 4774 . . 3 (𝑓 = → ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩ = ⟨((1st) ∘ (1st𝑔)), ((2nd) (2nd𝑔))⟩)
7 fveq2 6663 . . . . 5 (𝑔 = 𝑖 → (1st𝑔) = (1st𝑖))
87coeq2d 5708 . . . 4 (𝑔 = 𝑖 → ((1st) ∘ (1st𝑔)) = ((1st) ∘ (1st𝑖)))
9 fveq2 6663 . . . . 5 (𝑔 = 𝑖 → (2nd𝑔) = (2nd𝑖))
109oveq2d 7172 . . . 4 (𝑔 = 𝑖 → ((2nd) (2nd𝑔)) = ((2nd) (2nd𝑖)))
118, 10opeq12d 4774 . . 3 (𝑔 = 𝑖 → ⟨((1st) ∘ (1st𝑔)), ((2nd) (2nd𝑔))⟩ = ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
126, 11cbvmpov 7249 . 2 (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩) = ( ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
131, 12eqtri 2781 1 + = ( ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  ⟨cop 4531   × cxp 5526   ∘ ccom 5532  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158  1st c1st 7697  2nd c2nd 7698 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 5173  ax-nul 5180  ax-pr 5302 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-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-co 5537  df-iota 6299  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161 This theorem is referenced by:  dvhvaddval  38701
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