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Theorem dvhvscaval 36705
 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 6228 . . 3 (𝑡 = 𝑈 → (𝑡‘(1st𝑔)) = (𝑈‘(1st𝑔)))
2 coeq1 5312 . . 3 (𝑡 = 𝑈 → (𝑡 ∘ (2nd𝑔)) = (𝑈 ∘ (2nd𝑔)))
31, 2opeq12d 4441 . 2 (𝑡 = 𝑈 → ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩ = ⟨(𝑈‘(1st𝑔)), (𝑈 ∘ (2nd𝑔))⟩)
4 fveq2 6229 . . . 4 (𝑔 = 𝐹 → (1st𝑔) = (1st𝐹))
54fveq2d 6233 . . 3 (𝑔 = 𝐹 → (𝑈‘(1st𝑔)) = (𝑈‘(1st𝐹)))
6 fveq2 6229 . . . 4 (𝑔 = 𝐹 → (2nd𝑔) = (2nd𝐹))
76coeq2d 5317 . . 3 (𝑔 = 𝐹 → (𝑈 ∘ (2nd𝑔)) = (𝑈 ∘ (2nd𝐹)))
85, 7opeq12d 4441 . 2 (𝑔 = 𝐹 → ⟨(𝑈‘(1st𝑔)), (𝑈 ∘ (2nd𝑔))⟩ = ⟨(𝑈‘(1st𝐹)), (𝑈 ∘ (2nd𝐹))⟩)
9 dvhvscaval.s . . 3 · = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
109dvhvscacbv 36704 . 2 · = (𝑡𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
11 opex 4962 . 2 ⟨(𝑈‘(1st𝐹)), (𝑈 ∘ (2nd𝐹))⟩ ∈ V
123, 8, 10, 11ovmpt2 6838 1 ((𝑈𝐸𝐹 ∈ (𝑇 × 𝐸)) → (𝑈 · 𝐹) = ⟨(𝑈‘(1st𝐹)), (𝑈 ∘ (2nd𝐹))⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ⟨cop 4216   × cxp 5141   ∘ ccom 5147  ‘cfv 5926  (class class class)co 6690   ↦ cmpt2 6692  1st c1st 7208  2nd c2nd 7209 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695 This theorem is referenced by:  dvhvsca  36707  dvhopspN  36721
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