Theorem addpipq2 10837

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 6831	. . . . 5 ⊢ (𝑥 = 𝐴 → (1st ‘𝑥) = (1st ‘𝐴))
2	1	oveq1d 7370	. . . 4 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝐴) ·N (2nd ‘𝑦)))
3		fveq2 6831	. . . . 5 ⊢ (𝑥 = 𝐴 → (2nd ‘𝑥) = (2nd ‘𝐴))
4	3	oveq2d 7371	. . . 4 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑦) ·N (2nd ‘𝑥)) = ((1st ‘𝑦) ·N (2nd ‘𝐴)))
5	2, 4	oveq12d 7373	. . 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 7370	. . 3 ⊢ (𝑥 = 𝐴 → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) = ((2nd ‘𝐴) ·N (2nd ‘𝑦)))
7	5, 6	opeq12d 4834	. 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 6831	. . . . 5 ⊢ (𝑦 = 𝐵 → (2nd ‘𝑦) = (2nd ‘𝐵))
9	8	oveq2d 7371	. . . 4 ⊢ (𝑦 = 𝐵 → ((1st ‘𝐴) ·N (2nd ‘𝑦)) = ((1st ‘𝐴) ·N (2nd ‘𝐵)))
10		fveq2 6831	. . . . 5 ⊢ (𝑦 = 𝐵 → (1st ‘𝑦) = (1st ‘𝐵))
11	10	oveq1d 7370	. . . 4 ⊢ (𝑦 = 𝐵 → ((1st ‘𝑦) ·N (2nd ‘𝐴)) = ((1st ‘𝐵) ·N (2nd ‘𝐴)))
12	9, 11	oveq12d 7373	. . 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 7371	. . 3 ⊢ (𝑦 = 𝐵 → ((2nd ‘𝐴) ·N (2nd ‘𝑦)) = ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
14	12, 13	opeq12d 4834	. 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 10809	. 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 5409	. 2 ⊢ ⟨(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ ∈ V
17	7, 14, 15, 16	ovmpo 7515	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 1541   ∈ wcel 2113  ⟨cop 4583   × cxp 5619  ‘cfv 6489  (class class class)co 7355  1^st c1st 7928  2^nd c2nd 7929  Ncnpi 10745   +_N cpli 10746   ·_N cmi 10747   +_pQ cplpq 10749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-plpq 10809

This theorem is referenced by:  addpipq  10838  addcompq  10851  adderpqlem  10855  addassnq  10859  distrnq  10862  ltanq  10872