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Theorem dvhvscaval 38239
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 6672 . . 3 (𝑡 = 𝑈 → (𝑡‘(1st𝑔)) = (𝑈‘(1st𝑔)))
2 coeq1 5731 . . 3 (𝑡 = 𝑈 → (𝑡 ∘ (2nd𝑔)) = (𝑈 ∘ (2nd𝑔)))
31, 2opeq12d 4814 . 2 (𝑡 = 𝑈 → ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩ = ⟨(𝑈‘(1st𝑔)), (𝑈 ∘ (2nd𝑔))⟩)
4 2fveq3 6678 . . 3 (𝑔 = 𝐹 → (𝑈‘(1st𝑔)) = (𝑈‘(1st𝐹)))
5 fveq2 6673 . . . 4 (𝑔 = 𝐹 → (2nd𝑔) = (2nd𝐹))
65coeq2d 5736 . . 3 (𝑔 = 𝐹 → (𝑈 ∘ (2nd𝑔)) = (𝑈 ∘ (2nd𝐹)))
74, 6opeq12d 4814 . 2 (𝑔 = 𝐹 → ⟨(𝑈‘(1st𝑔)), (𝑈 ∘ (2nd𝑔))⟩ = ⟨(𝑈‘(1st𝐹)), (𝑈 ∘ (2nd𝐹))⟩)
8 dvhvscaval.s . . 3 · = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
98dvhvscacbv 38238 . 2 · = (𝑡𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
10 opex 5359 . 2 ⟨(𝑈‘(1st𝐹)), (𝑈 ∘ (2nd𝐹))⟩ ∈ V
113, 7, 9, 10ovmpo 7313 1 ((𝑈𝐸𝐹 ∈ (𝑇 × 𝐸)) → (𝑈 · 𝐹) = ⟨(𝑈‘(1st𝐹)), (𝑈 ∘ (2nd𝐹))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1536  wcel 2113  cop 4576   × cxp 5556  ccom 5562  cfv 6358  (class class class)co 7159  cmpo 7161  1st c1st 7690  2nd c2nd 7691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-iota 6317  df-fun 6360  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164
This theorem is referenced by:  dvhvsca  38241  dvhopspN  38255
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