Theorem dvhopaddN 42172

Description: Sum of DVecH vectors expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
dvhopadd.a	⊢ 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓)𝑃(2nd ‘𝑔))⟩)

Assertion

Ref	Expression
dvhopaddN	⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (⟨𝐹, 𝑈⟩𝐴⟨𝐺, 𝑉⟩) = ⟨(𝐹 ∘ 𝐺), (𝑈𝑃𝑉)⟩)

Distinct variable groups:   𝑓,𝑔,𝐸   𝑃,𝑓,𝑔   𝑇,𝑓,𝑔

Allowed substitution hints:   𝐴(𝑓, 𝑔)   𝑈(𝑓, 𝑔)   𝐹(𝑓, 𝑔)   𝐺(𝑓, 𝑔)   𝑉(𝑓, 𝑔)

Proof of Theorem dvhopaddN

Step	Hyp	Ref	Expression
1		opelxpi 5687	. . 3 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸))
2		opelxpi 5687	. . 3 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸) → ⟨𝐺, 𝑉⟩ ∈ (𝑇 × 𝐸))
3		dvhopadd.a	. . . 4 ⊢ 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓)𝑃(2nd ‘𝑔))⟩)
4	3	dvhvaddval 42148	. . 3 ⊢ ((⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸) ∧ ⟨𝐺, 𝑉⟩ ∈ (𝑇 × 𝐸)) → (⟨𝐹, 𝑈⟩𝐴⟨𝐺, 𝑉⟩) = ⟨((1st ‘⟨𝐹, 𝑈⟩) ∘ (1st ‘⟨𝐺, 𝑉⟩)), ((2nd ‘⟨𝐹, 𝑈⟩)𝑃(2nd ‘⟨𝐺, 𝑉⟩))⟩)
5	1, 2, 4	syl2an 608	. 2 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (⟨𝐹, 𝑈⟩𝐴⟨𝐺, 𝑉⟩) = ⟨((1st ‘⟨𝐹, 𝑈⟩) ∘ (1st ‘⟨𝐺, 𝑉⟩)), ((2nd ‘⟨𝐹, 𝑈⟩)𝑃(2nd ‘⟨𝐺, 𝑉⟩))⟩)
6		op1stg 8013	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (1st ‘⟨𝐹, 𝑈⟩) = 𝐹)
7	6	adantr 486	. . . 4 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (1st ‘⟨𝐹, 𝑈⟩) = 𝐹)
8		op1stg 8013	. . . . 5 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸) → (1st ‘⟨𝐺, 𝑉⟩) = 𝐺)
9	8	adantl 487	. . . 4 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (1st ‘⟨𝐺, 𝑉⟩) = 𝐺)
10	7, 9	coeq12d 5841	. . 3 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → ((1st ‘⟨𝐹, 𝑈⟩) ∘ (1st ‘⟨𝐺, 𝑉⟩)) = (𝐹 ∘ 𝐺))
11		op2ndg 8014	. . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (2nd ‘⟨𝐹, 𝑈⟩) = 𝑈)
12		op2ndg 8014	. . . 4 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸) → (2nd ‘⟨𝐺, 𝑉⟩) = 𝑉)
13	11, 12	oveqan12d 7439	. . 3 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → ((2nd ‘⟨𝐹, 𝑈⟩)𝑃(2nd ‘⟨𝐺, 𝑉⟩)) = (𝑈𝑃𝑉))
14	10, 13	opeq12d 4840	. 2 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → ⟨((1st ‘⟨𝐹, 𝑈⟩) ∘ (1st ‘⟨𝐺, 𝑉⟩)), ((2nd ‘⟨𝐹, 𝑈⟩)𝑃(2nd ‘⟨𝐺, 𝑉⟩))⟩ = ⟨(𝐹 ∘ 𝐺), (𝑈𝑃𝑉)⟩)
15	5, 14	eqtrd 2795	1 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (⟨𝐹, 𝑈⟩𝐴⟨𝐺, 𝑉⟩) = ⟨(𝐹 ∘ 𝐺), (𝑈𝑃𝑉)⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ⟨cop 4589   × cxp 5648   ∘ ccom 5654  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002

This theorem is used by:  dvhopN  42174