Theorem addpipq 10860

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 5669	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2		opelxpi 5669	. . 3 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
3		addpipq2 10859	. . 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 597	. 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 7955	. . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
6		op2ndg 7956	. . . . 5 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
7	5, 6	oveqan12d 7387	. . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((1st ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) = (𝐴 ·N 𝐷))
8		op1stg 7955	. . . . 5 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
9		op2ndg 7956	. . . . 5 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
10	8, 9	oveqan12rd 7388	. . . 4 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((1st ‘⟨𝐶, 𝐷⟩) ·N (2nd ‘⟨𝐴, 𝐵⟩)) = (𝐶 ·N 𝐵))
11	7, 10	oveq12d 7386	. . 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 7387	. . 3 ⊢ (((𝐴 ∈ N ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐷 ∈ N)) → ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) = (𝐵 ·N 𝐷))
13	11, 12	opeq12d 4839	. 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 2772	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 1542   ∈ wcel 2114  ⟨cop 4588   × cxp 5630  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942  Ncnpi 10767   +_N cpli 10768   ·_N cmi 10769   +_pQ cplpq 10771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-plpq 10831

This theorem is referenced by:  addassnq  10881  distrnq  10884  1lt2nq  10896  ltexnq  10898  prlem934  10956