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Theorem dvhopspN 39171
Description: Scalar product of DVecH vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
dvhopsp.s 𝑆 = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
Assertion
Ref Expression
dvhopspN ((𝑅𝐸 ∧ (𝐹𝑇𝑈𝐸)) → (𝑅𝑆𝐹, 𝑈⟩) = ⟨(𝑅𝐹), (𝑅𝑈)⟩)
Distinct variable groups:   𝑓,𝑠,𝐸   𝑇,𝑓,𝑠
Allowed substitution hints:   𝑅(𝑓,𝑠)   𝑆(𝑓,𝑠)   𝑈(𝑓,𝑠)   𝐹(𝑓,𝑠)

Proof of Theorem dvhopspN
StepHypRef Expression
1 opelxpi 5637 . . 3 ((𝐹𝑇𝑈𝐸) → ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸))
2 dvhopsp.s . . . 4 𝑆 = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
32dvhvscaval 39155 . . 3 ((𝑅𝐸 ∧ ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸)) → (𝑅𝑆𝐹, 𝑈⟩) = ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩)
41, 3sylan2 594 . 2 ((𝑅𝐸 ∧ (𝐹𝑇𝑈𝐸)) → (𝑅𝑆𝐹, 𝑈⟩) = ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩)
5 op1stg 7875 . . . . 5 ((𝐹𝑇𝑈𝐸) → (1st ‘⟨𝐹, 𝑈⟩) = 𝐹)
65fveq2d 6808 . . . 4 ((𝐹𝑇𝑈𝐸) → (𝑅‘(1st ‘⟨𝐹, 𝑈⟩)) = (𝑅𝐹))
7 op2ndg 7876 . . . . 5 ((𝐹𝑇𝑈𝐸) → (2nd ‘⟨𝐹, 𝑈⟩) = 𝑈)
87coeq2d 5784 . . . 4 ((𝐹𝑇𝑈𝐸) → (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩)) = (𝑅𝑈))
96, 8opeq12d 4817 . . 3 ((𝐹𝑇𝑈𝐸) → ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩ = ⟨(𝑅𝐹), (𝑅𝑈)⟩)
109adantl 483 . 2 ((𝑅𝐸 ∧ (𝐹𝑇𝑈𝐸)) → ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩ = ⟨(𝑅𝐹), (𝑅𝑈)⟩)
114, 10eqtrd 2776 1 ((𝑅𝐸 ∧ (𝐹𝑇𝑈𝐸)) → (𝑅𝑆𝐹, 𝑈⟩) = ⟨(𝑅𝐹), (𝑅𝑈)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1539  wcel 2104  cop 4571   × cxp 5598  ccom 5604  cfv 6458  (class class class)co 7307  cmpo 7309  1st c1st 7861  2nd c2nd 7862
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-sbc 3722  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-iota 6410  df-fun 6460  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-1st 7863  df-2nd 7864
This theorem is referenced by:  dvhopN  39172
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