Theorem dvhopspN 41072

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 = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)

Assertion

Ref	Expression
dvhopspN	⊢ ((𝑅 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑅𝑆⟨𝐹, 𝑈⟩) = ⟨(𝑅‘𝐹), (𝑅 ∘ 𝑈)⟩)

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

Step	Hyp	Ref	Expression
1		opelxpi 5737	. . 3 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸))
2		dvhopsp.s	. . . 4 ⊢ 𝑆 = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)
3	2	dvhvscaval 41056	. . 3 ⊢ ((𝑅 ∈ 𝐸 ∧ ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸)) → (𝑅𝑆⟨𝐹, 𝑈⟩) = ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩)
4	1, 3	sylan2 592	. 2 ⊢ ((𝑅 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑅𝑆⟨𝐹, 𝑈⟩) = ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩)
5		op1stg 8042	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (1st ‘⟨𝐹, 𝑈⟩) = 𝐹)
6	5	fveq2d 6924	. . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (𝑅‘(1st ‘⟨𝐹, 𝑈⟩)) = (𝑅‘𝐹))
7		op2ndg 8043	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (2nd ‘⟨𝐹, 𝑈⟩) = 𝑈)
8	7	coeq2d 5887	. . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩)) = (𝑅 ∘ 𝑈))
9	6, 8	opeq12d 4905	. . 3 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩ = ⟨(𝑅‘𝐹), (𝑅 ∘ 𝑈)⟩)
10	9	adantl 481	. 2 ⊢ ((𝑅 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩ = ⟨(𝑅‘𝐹), (𝑅 ∘ 𝑈)⟩)
11	4, 10	eqtrd 2780	1 ⊢ ((𝑅 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑅𝑆⟨𝐹, 𝑈⟩) = ⟨(𝑅‘𝐹), (𝑅 ∘ 𝑈)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ⟨cop 4654   × cxp 5698   ∘ ccom 5704  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  1^st c1st 8028  2^nd c2nd 8029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031

This theorem is referenced by:  dvhopN  41073