Theorem addpipq2 10976

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 6906	. . . . 5 ⊢ (𝑥 = 𝐴 → (1st ‘𝑥) = (1st ‘𝐴))
2	1	oveq1d 7446	. . . 4 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝐴) ·N (2nd ‘𝑦)))
3		fveq2 6906	. . . . 5 ⊢ (𝑥 = 𝐴 → (2nd ‘𝑥) = (2nd ‘𝐴))
4	3	oveq2d 7447	. . . 4 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑦) ·N (2nd ‘𝑥)) = ((1st ‘𝑦) ·N (2nd ‘𝐴)))
5	2, 4	oveq12d 7449	. . 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 7446	. . 3 ⊢ (𝑥 = 𝐴 → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) = ((2nd ‘𝐴) ·N (2nd ‘𝑦)))
7	5, 6	opeq12d 4881	. 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 6906	. . . . 5 ⊢ (𝑦 = 𝐵 → (2nd ‘𝑦) = (2nd ‘𝐵))
9	8	oveq2d 7447	. . . 4 ⊢ (𝑦 = 𝐵 → ((1st ‘𝐴) ·N (2nd ‘𝑦)) = ((1st ‘𝐴) ·N (2nd ‘𝐵)))
10		fveq2 6906	. . . . 5 ⊢ (𝑦 = 𝐵 → (1st ‘𝑦) = (1st ‘𝐵))
11	10	oveq1d 7446	. . . 4 ⊢ (𝑦 = 𝐵 → ((1st ‘𝑦) ·N (2nd ‘𝐴)) = ((1st ‘𝐵) ·N (2nd ‘𝐴)))
12	9, 11	oveq12d 7449	. . 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 7447	. . 3 ⊢ (𝑦 = 𝐵 → ((2nd ‘𝐴) ·N (2nd ‘𝑦)) = ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
14	12, 13	opeq12d 4881	. 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 10948	. 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 5469	. 2 ⊢ ⟨(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ ∈ V
17	7, 14, 15, 16	ovmpo 7593	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 395   = wceq 1540   ∈ wcel 2108  ⟨cop 4632   × cxp 5683  ‘cfv 6561  (class class class)co 7431  1^st c1st 8012  2^nd c2nd 8013  Ncnpi 10884   +_N cpli 10885   ·_N cmi 10886   +_pQ cplpq 10888

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-plpq 10948

This theorem is referenced by:  addpipq  10977  addcompq  10990  adderpqlem  10994  addassnq  10998  distrnq  11001  ltanq  11011