Theorem dvhopspN 41134

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 5691	. . 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 41118	. . 3 ⊢ ((𝑅 ∈ 𝐸 ∧ ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸)) → (𝑅𝑆⟨𝐹, 𝑈⟩) = ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩)
4	1, 3	sylan2 593	. 2 ⊢ ((𝑅 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑅𝑆⟨𝐹, 𝑈⟩) = ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩)
5		op1stg 8000	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (1st ‘⟨𝐹, 𝑈⟩) = 𝐹)
6	5	fveq2d 6880	. . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (𝑅‘(1st ‘⟨𝐹, 𝑈⟩)) = (𝑅‘𝐹))
7		op2ndg 8001	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (2nd ‘⟨𝐹, 𝑈⟩) = 𝑈)
8	7	coeq2d 5842	. . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩)) = (𝑅 ∘ 𝑈))
9	6, 8	opeq12d 4857	. . 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 2770	1 ⊢ ((𝑅 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑅𝑆⟨𝐹, 𝑈⟩) = ⟨(𝑅‘𝐹), (𝑅 ∘ 𝑈)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ⟨cop 4607   × cxp 5652   ∘ ccom 5658  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  1^st c1st 7986  2^nd c2nd 7987

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989

This theorem is referenced by:  dvhopN  41135