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Theorem dvhvscaval 37119
 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 6411 . . 3 (𝑡 = 𝑈 → (𝑡‘(1st𝑔)) = (𝑈‘(1st𝑔)))
2 coeq1 5484 . . 3 (𝑡 = 𝑈 → (𝑡 ∘ (2nd𝑔)) = (𝑈 ∘ (2nd𝑔)))
31, 2opeq12d 4602 . 2 (𝑡 = 𝑈 → ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩ = ⟨(𝑈‘(1st𝑔)), (𝑈 ∘ (2nd𝑔))⟩)
4 2fveq3 6417 . . 3 (𝑔 = 𝐹 → (𝑈‘(1st𝑔)) = (𝑈‘(1st𝐹)))
5 fveq2 6412 . . . 4 (𝑔 = 𝐹 → (2nd𝑔) = (2nd𝐹))
65coeq2d 5489 . . 3 (𝑔 = 𝐹 → (𝑈 ∘ (2nd𝑔)) = (𝑈 ∘ (2nd𝐹)))
74, 6opeq12d 4602 . 2 (𝑔 = 𝐹 → ⟨(𝑈‘(1st𝑔)), (𝑈 ∘ (2nd𝑔))⟩ = ⟨(𝑈‘(1st𝐹)), (𝑈 ∘ (2nd𝐹))⟩)
8 dvhvscaval.s . . 3 · = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
98dvhvscacbv 37118 . 2 · = (𝑡𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
10 opex 5124 . 2 ⟨(𝑈‘(1st𝐹)), (𝑈 ∘ (2nd𝐹))⟩ ∈ V
113, 7, 9, 10ovmpt2 7031 1 ((𝑈𝐸𝐹 ∈ (𝑇 × 𝐸)) → (𝑈 · 𝐹) = ⟨(𝑈‘(1st𝐹)), (𝑈 ∘ (2nd𝐹))⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ⟨cop 4375   × cxp 5311   ∘ ccom 5317  ‘cfv 6102  (class class class)co 6879   ↦ cmpt2 6881  1st c1st 7400  2nd c2nd 7401 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-iota 6065  df-fun 6104  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884 This theorem is referenced by:  dvhvsca  37121  dvhopspN  37135
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