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Theorem dvhvscaval 39908
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 6887 . . 3 (𝑡 = 𝑈 → (𝑡‘(1st𝑔)) = (𝑈‘(1st𝑔)))
2 coeq1 5855 . . 3 (𝑡 = 𝑈 → (𝑡 ∘ (2nd𝑔)) = (𝑈 ∘ (2nd𝑔)))
31, 2opeq12d 4880 . 2 (𝑡 = 𝑈 → ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩ = ⟨(𝑈‘(1st𝑔)), (𝑈 ∘ (2nd𝑔))⟩)
4 2fveq3 6893 . . 3 (𝑔 = 𝐹 → (𝑈‘(1st𝑔)) = (𝑈‘(1st𝐹)))
5 fveq2 6888 . . . 4 (𝑔 = 𝐹 → (2nd𝑔) = (2nd𝐹))
65coeq2d 5860 . . 3 (𝑔 = 𝐹 → (𝑈 ∘ (2nd𝑔)) = (𝑈 ∘ (2nd𝐹)))
74, 6opeq12d 4880 . 2 (𝑔 = 𝐹 → ⟨(𝑈‘(1st𝑔)), (𝑈 ∘ (2nd𝑔))⟩ = ⟨(𝑈‘(1st𝐹)), (𝑈 ∘ (2nd𝐹))⟩)
8 dvhvscaval.s . . 3 · = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
98dvhvscacbv 39907 . 2 · = (𝑡𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
10 opex 5463 . 2 ⟨(𝑈‘(1st𝐹)), (𝑈 ∘ (2nd𝐹))⟩ ∈ V
113, 7, 9, 10ovmpo 7563 1 ((𝑈𝐸𝐹 ∈ (𝑇 × 𝐸)) → (𝑈 · 𝐹) = ⟨(𝑈‘(1st𝐹)), (𝑈 ∘ (2nd𝐹))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cop 4633   × cxp 5673  ccom 5679  cfv 6540  (class class class)co 7404  cmpo 7406  1st c1st 7968  2nd c2nd 7969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409
This theorem is referenced by:  dvhvsca  39910  dvhopspN  39924
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