Theorem dvhopspN 38250

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 5591	. . 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 38234	. . 3 ⊢ ((𝑅 ∈ 𝐸 ∧ ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸)) → (𝑅𝑆⟨𝐹, 𝑈⟩) = ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩)
4	1, 3	sylan2 594	. 2 ⊢ ((𝑅 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑅𝑆⟨𝐹, 𝑈⟩) = ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩)
5		op1stg 7700	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (1st ‘⟨𝐹, 𝑈⟩) = 𝐹)
6	5	fveq2d 6673	. . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (𝑅‘(1st ‘⟨𝐹, 𝑈⟩)) = (𝑅‘𝐹))
7		op2ndg 7701	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (2nd ‘⟨𝐹, 𝑈⟩) = 𝑈)
8	7	coeq2d 5732	. . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩)) = (𝑅 ∘ 𝑈))
9	6, 8	opeq12d 4810	. . 3 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩ = ⟨(𝑅‘𝐹), (𝑅 ∘ 𝑈)⟩)
10	9	adantl 484	. 2 ⊢ ((𝑅 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → ⟨(𝑅‘(1st ‘⟨𝐹, 𝑈⟩)), (𝑅 ∘ (2nd ‘⟨𝐹, 𝑈⟩))⟩ = ⟨(𝑅‘𝐹), (𝑅 ∘ 𝑈)⟩)
11	4, 10	eqtrd 2856	1 ⊢ ((𝑅 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑅𝑆⟨𝐹, 𝑈⟩) = ⟨(𝑅‘𝐹), (𝑅 ∘ 𝑈)⟩)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ⟨cop 4572   × cxp 5552   ∘ ccom 5558  ‘cfv 6354  (class class class)co 7155   ∈ cmpo 7157  1^st c1st 7686  2^nd c2nd 7687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689

This theorem is referenced by:  dvhopN  38251