Theorem dvhopaddN 38254

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 5595	. . 3 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸))
2		opelxpi 5595	. . 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 38230	. . 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 7704	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (1st ‘⟨𝐹, 𝑈⟩) = 𝐹)
7	6	adantr 483	. . . 4 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (1st ‘⟨𝐹, 𝑈⟩) = 𝐹)
8		op1stg 7704	. . . . 5 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸) → (1st ‘⟨𝐺, 𝑉⟩) = 𝐺)
9	8	adantl 484	. . . 4 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (1st ‘⟨𝐺, 𝑉⟩) = 𝐺)
10	7, 9	coeq12d 5738	. . 3 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → ((1st ‘⟨𝐹, 𝑈⟩) ∘ (1st ‘⟨𝐺, 𝑉⟩)) = (𝐹 ∘ 𝐺))
11		op2ndg 7705	. . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → (2nd ‘⟨𝐹, 𝑈⟩) = 𝑈)
12		op2ndg 7705	. . . 4 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸) → (2nd ‘⟨𝐺, 𝑉⟩) = 𝑉)
13	11, 12	oveqan12d 7178	. . 3 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → ((2nd ‘⟨𝐹, 𝑈⟩)𝑃(2nd ‘⟨𝐺, 𝑉⟩)) = (𝑈𝑃𝑉))
14	10, 13	opeq12d 4814	. 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 2859	1 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (⟨𝐹, 𝑈⟩𝐴⟨𝐺, 𝑉⟩) = ⟨(𝐹 ∘ 𝐺), (𝑈𝑃𝑉)⟩)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ⟨cop 4576   × cxp 5556   ∘ ccom 5562  ‘cfv 6358  (class class class)co 7159   ∈ cmpo 7161  1^st c1st 7690  2^nd c2nd 7691

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-iota 6317  df-fun 6360  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693

This theorem is referenced by:  dvhopN  38256