Theorem addpipq 10677

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

Assertion

Ref	Expression
addpipq	⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (⟨𝐴, 𝐵⟩ +pQ ⟨𝐶, 𝐷⟩) = ⟨((𝐴 ·N 𝐷) +N (𝐶 ·N 𝐵)), (𝐵 ·N 𝐷)⟩)

Proof of Theorem addpipq

Step	Hyp	Ref	Expression
1		opelxpi 5625	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2		opelxpi 5625	. . 3 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
3		addpipq2 10676	. . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) → (⟨𝐴, 𝐵⟩ +pQ ⟨𝐶, 𝐷⟩) = ⟨(((1st ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) +N ((1st ‘⟨𝐶, 𝐷⟩) ·N (2nd ‘⟨𝐴, 𝐵⟩))), ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩))⟩)
4	1, 2, 3	syl2an 595	. 2 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (⟨𝐴, 𝐵⟩ +pQ ⟨𝐶, 𝐷⟩) = ⟨(((1st ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) +N ((1st ‘⟨𝐶, 𝐷⟩) ·N (2nd ‘⟨𝐴, 𝐵⟩))), ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩))⟩)
5		op1stg 7829	. . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
6		op2ndg 7830	. . . . 5 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
7	5, 6	oveqan12d 7287	. . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((1st ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) = (𝐴 ·N 𝐷))
8		op1stg 7829	. . . . 5 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
9		op2ndg 7830	. . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
10	8, 9	oveqan12rd 7288	. . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((1st ‘⟨𝐶, 𝐷⟩) ·N (2nd ‘⟨𝐴, 𝐵⟩)) = (𝐶 ·N 𝐵))
11	7, 10	oveq12d 7286	. . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (((1st ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) +N ((1st ‘⟨𝐶, 𝐷⟩) ·N (2nd ‘⟨𝐴, 𝐵⟩))) = ((𝐴 ·N 𝐷) +N (𝐶 ·N 𝐵)))
12	9, 6	oveqan12d 7287	. . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) = (𝐵 ·N 𝐷))
13	11, 12	opeq12d 4817	. 2 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ⟨(((1st ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) +N ((1st ‘⟨𝐶, 𝐷⟩) ·N (2nd ‘⟨𝐴, 𝐵⟩))), ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩))⟩ = ⟨((𝐴 ·N 𝐷) +N (𝐶 ·N 𝐵)), (𝐵 ·N 𝐷)⟩)
14	4, 13	eqtrd 2779	1 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → (⟨𝐴, 𝐵⟩ +pQ ⟨𝐶, 𝐷⟩) = ⟨((𝐴 ·N 𝐷) +N (𝐶 ·N 𝐵)), (𝐵 ·N 𝐷)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109  ⟨cop 4572   × cxp 5586  ‘cfv 6430  (class class class)co 7268  1^st c1st 7815  2^nd c2nd 7816  Ncnpi 10584   +_N cpli 10585   ·_N cmi 10586   +_pQ cplpq 10588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-iota 6388  df-fun 6432  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-plpq 10648

This theorem is referenced by:  addassnq  10698  distrnq  10701  1lt2nq  10713  ltexnq  10715  prlem934  10773