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Theorem dvhvscaval 35406
 Description: The scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Nov-2013.)
Hypothesis
Ref Expression
dvhvscaval.s · = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
Assertion
Ref Expression
dvhvscaval ((𝑈𝐸𝐹 ∈ (𝑇 × 𝐸)) → (𝑈 · 𝐹) = ⟨(𝑈‘(1st𝐹)), (𝑈 ∘ (2nd𝐹))⟩)
Distinct variable groups:   𝑓,𝑠,𝐸   𝑇,𝑠,𝑓
Allowed substitution hints:   · (𝑓,𝑠)   𝑈(𝑓,𝑠)   𝐹(𝑓,𝑠)

Proof of Theorem dvhvscaval
Dummy variables 𝑡 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq1 6102 . . 3 (𝑡 = 𝑈 → (𝑡‘(1st𝑔)) = (𝑈‘(1st𝑔)))
2 coeq1 5201 . . 3 (𝑡 = 𝑈 → (𝑡 ∘ (2nd𝑔)) = (𝑈 ∘ (2nd𝑔)))
31, 2opeq12d 4348 . 2 (𝑡 = 𝑈 → ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩ = ⟨(𝑈‘(1st𝑔)), (𝑈 ∘ (2nd𝑔))⟩)
4 fveq2 6103 . . . 4 (𝑔 = 𝐹 → (1st𝑔) = (1st𝐹))
54fveq2d 6107 . . 3 (𝑔 = 𝐹 → (𝑈‘(1st𝑔)) = (𝑈‘(1st𝐹)))
6 fveq2 6103 . . . 4 (𝑔 = 𝐹 → (2nd𝑔) = (2nd𝐹))
76coeq2d 5206 . . 3 (𝑔 = 𝐹 → (𝑈 ∘ (2nd𝑔)) = (𝑈 ∘ (2nd𝐹)))
85, 7opeq12d 4348 . 2 (𝑔 = 𝐹 → ⟨(𝑈‘(1st𝑔)), (𝑈 ∘ (2nd𝑔))⟩ = ⟨(𝑈‘(1st𝐹)), (𝑈 ∘ (2nd𝐹))⟩)
9 dvhvscaval.s . . 3 · = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
109dvhvscacbv 35405 . 2 · = (𝑡𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
11 opex 4859 . 2 ⟨(𝑈‘(1st𝐹)), (𝑈 ∘ (2nd𝐹))⟩ ∈ V
123, 8, 10, 11ovmpt2 6694 1 ((𝑈𝐸𝐹 ∈ (𝑇 × 𝐸)) → (𝑈 · 𝐹) = ⟨(𝑈‘(1st𝐹)), (𝑈 ∘ (2nd𝐹))⟩)
 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:  dvhvsca  35408  dvhopspN  35422
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