Theorem addpipq2 10849

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 6826	. . . . 5 ⊢ (𝑥 = 𝐴 → (1st ‘𝑥) = (1st ‘𝐴))
2	1	oveq1d 7368	. . . 4 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑥) ·N (2nd ‘𝑦)) = ((1st ‘𝐴) ·N (2nd ‘𝑦)))
3		fveq2 6826	. . . . 5 ⊢ (𝑥 = 𝐴 → (2nd ‘𝑥) = (2nd ‘𝐴))
4	3	oveq2d 7369	. . . 4 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑦) ·N (2nd ‘𝑥)) = ((1st ‘𝑦) ·N (2nd ‘𝐴)))
5	2, 4	oveq12d 7371	. . 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 7368	. . 3 ⊢ (𝑥 = 𝐴 → ((2nd ‘𝑥) ·N (2nd ‘𝑦)) = ((2nd ‘𝐴) ·N (2nd ‘𝑦)))
7	5, 6	opeq12d 4835	. 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 6826	. . . . 5 ⊢ (𝑦 = 𝐵 → (2nd ‘𝑦) = (2nd ‘𝐵))
9	8	oveq2d 7369	. . . 4 ⊢ (𝑦 = 𝐵 → ((1st ‘𝐴) ·N (2nd ‘𝑦)) = ((1st ‘𝐴) ·N (2nd ‘𝐵)))
10		fveq2 6826	. . . . 5 ⊢ (𝑦 = 𝐵 → (1st ‘𝑦) = (1st ‘𝐵))
11	10	oveq1d 7368	. . . 4 ⊢ (𝑦 = 𝐵 → ((1st ‘𝑦) ·N (2nd ‘𝐴)) = ((1st ‘𝐵) ·N (2nd ‘𝐴)))
12	9, 11	oveq12d 7371	. . 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 7369	. . 3 ⊢ (𝑦 = 𝐵 → ((2nd ‘𝐴) ·N (2nd ‘𝑦)) = ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
14	12, 13	opeq12d 4835	. 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 10821	. 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 5411	. 2 ⊢ ⟨(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩ ∈ V
17	7, 14, 15, 16	ovmpo 7513	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 2109  ⟨cop 4585   × cxp 5621  ‘cfv 6486  (class class class)co 7353  1^st c1st 7929  2^nd c2nd 7930  Ncnpi 10757   +_N cpli 10758   ·_N cmi 10759   +_pQ cplpq 10761

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  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 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-plpq 10821

This theorem is referenced by:  addpipq  10850  addcompq  10863  adderpqlem  10867  addassnq  10871  distrnq  10874  ltanq  10884