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Theorem dvhopspN 36823
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 5257 . . 3 ((𝐹𝑇𝑈𝐸) → ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸))
2 dvhopsp.s . . . 4 𝑆 = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
32dvhvscaval 36807 . . 3 ((𝑅𝐸 ∧ ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸)) → (𝑅𝑆𝐹, 𝑈⟩) = ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩)
41, 3sylan2 492 . 2 ((𝑅𝐸 ∧ (𝐹𝑇𝑈𝐸)) → (𝑅𝑆𝐹, 𝑈⟩) = ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩)
5 op1stg 7297 . . . . 5 ((𝐹𝑇𝑈𝐸) → (1st ‘⟨𝐹, 𝑈⟩) = 𝐹)
65fveq2d 6308 . . . 4 ((𝐹𝑇𝑈𝐸) → (𝑅‘(1st ‘⟨𝐹, 𝑈⟩)) = (𝑅𝐹))
7 op2ndg 7298 . . . . 5 ((𝐹𝑇𝑈𝐸) → (2nd ‘⟨𝐹, 𝑈⟩) = 𝑈)
87coeq2d 5392 . . . 4 ((𝐹𝑇𝑈𝐸) → (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩)) = (𝑅𝑈))
96, 8opeq12d 4517 . . 3 ((𝐹𝑇𝑈𝐸) → ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩ = ⟨(𝑅𝐹), (𝑅𝑈)⟩)
109adantl 473 . 2 ((𝑅𝐸 ∧ (𝐹𝑇𝑈𝐸)) → ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩ = ⟨(𝑅𝐹), (𝑅𝑈)⟩)
114, 10eqtrd 2758 1 ((𝑅𝐸 ∧ (𝐹𝑇𝑈𝐸)) → (𝑅𝑆𝐹, 𝑈⟩) = ⟨(𝑅𝐹), (𝑅𝑈)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  cop 4291   × cxp 5216  ccom 5222  cfv 6001  (class class class)co 6765  cmpt2 6767  1st c1st 7283  2nd c2nd 7284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-iota 5964  df-fun 6003  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-1st 7285  df-2nd 7286
This theorem is referenced by:  dvhopN  36824
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