Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvhopspN Structured version   Visualization version   GIF version

Theorem dvhopspN 38311
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 5573 . . 3 ((𝐹𝑇𝑈𝐸) → ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸))
2 dvhopsp.s . . . 4 𝑆 = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
32dvhvscaval 38295 . . 3 ((𝑅𝐸 ∧ ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸)) → (𝑅𝑆𝐹, 𝑈⟩) = ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩)
41, 3sylan2 595 . 2 ((𝑅𝐸 ∧ (𝐹𝑇𝑈𝐸)) → (𝑅𝑆𝐹, 𝑈⟩) = ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩)
5 op1stg 7684 . . . . 5 ((𝐹𝑇𝑈𝐸) → (1st ‘⟨𝐹, 𝑈⟩) = 𝐹)
65fveq2d 6655 . . . 4 ((𝐹𝑇𝑈𝐸) → (𝑅‘(1st ‘⟨𝐹, 𝑈⟩)) = (𝑅𝐹))
7 op2ndg 7685 . . . . 5 ((𝐹𝑇𝑈𝐸) → (2nd ‘⟨𝐹, 𝑈⟩) = 𝑈)
87coeq2d 5714 . . . 4 ((𝐹𝑇𝑈𝐸) → (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩)) = (𝑅𝑈))
96, 8opeq12d 4792 . . 3 ((𝐹𝑇𝑈𝐸) → ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩ = ⟨(𝑅𝐹), (𝑅𝑈)⟩)
109adantl 485 . 2 ((𝑅𝐸 ∧ (𝐹𝑇𝑈𝐸)) → ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩ = ⟨(𝑅𝐹), (𝑅𝑈)⟩)
114, 10eqtrd 2859 1 ((𝑅𝐸 ∧ (𝐹𝑇𝑈𝐸)) → (𝑅𝑆𝐹, 𝑈⟩) = ⟨(𝑅𝐹), (𝑅𝑈)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  cop 4554   × cxp 5534  ccom 5540  cfv 6336  (class class class)co 7138  cmpo 7140  1st c1st 7670  2nd c2nd 7671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-iota 6295  df-fun 6338  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-1st 7672  df-2nd 7673
This theorem is referenced by:  dvhopN  38312
  Copyright terms: Public domain W3C validator