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 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 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → (⟨𝐴, 𝐵⟩ +pQ𝐶, 𝐷⟩) = ⟨((𝐴 ·N 𝐷) +N (𝐶 ·N 𝐵)), (𝐵 ·N 𝐷)⟩)

StepHypRef Expression
1 opelxpi 5564 . . 3 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2 opelxpi 5564 . . 3 ((𝐶N𝐷N) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
3 addpipq2 10401 . . 3 ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) → (⟨𝐴, 𝐵⟩ +pQ𝐶, 𝐷⟩) = ⟨(((1st ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) +N ((1st ‘⟨𝐶, 𝐷⟩) ·N (2nd ‘⟨𝐴, 𝐵⟩))), ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩))⟩)
41, 2, 3syl2an 598 . 2 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → (⟨𝐴, 𝐵⟩ +pQ𝐶, 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 7710 . . . . 5 ((𝐴N𝐵N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
6 op2ndg 7711 . . . . 5 ((𝐶N𝐷N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
75, 6oveqan12d 7174 . . . 4 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ((1st ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) = (𝐴 ·N 𝐷))
8 op1stg 7710 . . . . 5 ((𝐶N𝐷N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
9 op2ndg 7711 . . . . 5 ((𝐴N𝐵N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
108, 9oveqan12rd 7175 . . . 4 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ((1st ‘⟨𝐶, 𝐷⟩) ·N (2nd ‘⟨𝐴, 𝐵⟩)) = (𝐶 ·N 𝐵))
117, 10oveq12d 7173 . . 3 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → (((1st ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) +N ((1st ‘⟨𝐶, 𝐷⟩) ·N (2nd ‘⟨𝐴, 𝐵⟩))) = ((𝐴 ·N 𝐷) +N (𝐶 ·N 𝐵)))
129, 6oveqan12d 7174 . . 3 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) = (𝐵 ·N 𝐷))
1311, 12opeq12d 4774 . 2 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ⟨(((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)⟩)
144, 13eqtrd 2793 1 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → (⟨𝐴, 𝐵⟩ +pQ𝐶, 𝐷⟩) = ⟨((𝐴 ·N 𝐷) +N (𝐶 ·N 𝐵)), (𝐵 ·N 𝐷)⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ⟨cop 4531   × cxp 5525  ‘cfv 6339  (class class class)co 7155  1st c1st 7696  2nd c2nd 7697  Ncnpi 10309   +N cpli 10310   ·N cmi 10311   +pQ cplpq 10313 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-iota 6298  df-fun 6341  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7698  df-2nd 7699  df-plpq 10373 This theorem is referenced by:  addassnq  10423  distrnq  10426  1lt2nq  10438  ltexnq  10440  prlem934  10498
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