Theorem dvhopaddN 41519

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 5671	. . 3 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸))
2		opelxpi 5671	. . 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 41495	. . 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 597	. 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 7957	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (1st ‘⟨𝐹, 𝑈⟩) = 𝐹)
7	6	adantr 480	. . . 4 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (1st ‘⟨𝐹, 𝑈⟩) = 𝐹)
8		op1stg 7957	. . . . 5 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸) → (1st ‘⟨𝐺, 𝑉⟩) = 𝐺)
9	8	adantl 481	. . . 4 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (1st ‘⟨𝐺, 𝑉⟩) = 𝐺)
10	7, 9	coeq12d 5823	. . 3 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → ((1st ‘⟨𝐹, 𝑈⟩) ∘ (1st ‘⟨𝐺, 𝑉⟩)) = (𝐹 ∘ 𝐺))
11		op2ndg 7958	. . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (2nd ‘⟨𝐹, 𝑈⟩) = 𝑈)
12		op2ndg 7958	. . . 4 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸) → (2nd ‘⟨𝐺, 𝑉⟩) = 𝑉)
13	11, 12	oveqan12d 7389	. . 3 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → ((2nd ‘⟨𝐹, 𝑈⟩)𝑃(2nd ‘⟨𝐺, 𝑉⟩)) = (𝑈𝑃𝑉))
14	10, 13	opeq12d 4839	. 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 2772	1 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (⟨𝐹, 𝑈⟩𝐴⟨𝐺, 𝑉⟩) = ⟨(𝐹 ∘ 𝐺), (𝑈𝑃𝑉)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4588   × cxp 5632   ∘ ccom 5638  ‘cfv 6502  (class class class)co 7370   ∈ cmpo 7372  1^st c1st 7943  2^nd c2nd 7944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6458  df-fun 6504  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-1st 7945  df-2nd 7946

This theorem is referenced by:  dvhopN  41521