Theorem addpipq2 10692

Description: Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
addpipq2	⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = ⟨(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)

Proof of Theorem addpipq2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6774	. . . . 5 ⊢ (𝑥 = 𝐴 → (1st ‘𝑥) = (1st ‘𝐴))
2	1	oveq1d 7290	. . . 4 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝐴) ·N (2nd ‘𝑦)))
3		fveq2 6774	. . . . 5 ⊢ (𝑥 = 𝐴 → (2nd ‘𝑥) = (2nd ‘𝐴))
4	3	oveq2d 7291	. . . 4 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑦) ·N (2nd ‘𝑥)) = ((1st ‘𝑦) ·N (2nd ‘𝐴)))
5	2, 4	oveq12d 7293	. . 3 ⊢ (𝑥 = 𝐴 → (((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))) = (((1st ‘𝐴) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝐴))))
6	3	oveq1d 7290	. . 3 ⊢ (𝑥 = 𝐴 → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) = ((2nd ‘𝐴) ·N (2nd ‘𝑦)))
7	5, 6	opeq12d 4812	. 2 ⊢ (𝑥 = 𝐴 → ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩ = ⟨(((1st ‘𝐴) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝑦))⟩)
8		fveq2 6774	. . . . 5 ⊢ (𝑦 = 𝐵 → (2nd ‘𝑦) = (2nd ‘𝐵))
9	8	oveq2d 7291	. . . 4 ⊢ (𝑦 = 𝐵 → ((1st ‘𝐴) ·N (2nd ‘𝑦)) = ((1st ‘𝐴) ·N (2nd ‘𝐵)))
10		fveq2 6774	. . . . 5 ⊢ (𝑦 = 𝐵 → (1st ‘𝑦) = (1st ‘𝐵))
11	10	oveq1d 7290	. . . 4 ⊢ (𝑦 = 𝐵 → ((1st ‘𝑦) ·N (2nd ‘𝐴)) = ((1st ‘𝐵) ·N (2nd ‘𝐴)))
12	9, 11	oveq12d 7293	. . 3 ⊢ (𝑦 = 𝐵 → (((1st ‘𝐴) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝐴))) = (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
13	8	oveq2d 7291	. . 3 ⊢ (𝑦 = 𝐵 → ((2nd ‘𝐴) ·N (2nd ‘𝑦)) = ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
14	12, 13	opeq12d 4812	. 2 ⊢ (𝑦 = 𝐵 → ⟨(((1st ‘𝐴) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝑦))⟩ = ⟨(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)
15		df-plpq 10664	. 2 ⊢ +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
16		opex 5379	. 2 ⊢ ⟨(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ ∈ V
17	7, 14, 15, 16	ovmpo 7433	1 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = ⟨(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ⟨cop 4567   × cxp 5587  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  Ncnpi 10600   +_N cpli 10601   ·_N cmi 10602   +_pQ cplpq 10604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-plpq 10664

This theorem is referenced by:  addpipq  10693  addcompq  10706  adderpqlem  10710  addassnq  10714  distrnq  10717  ltanq  10727