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Theorem addpipq 10975
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)⟩)

Proof of Theorem addpipq
StepHypRef Expression
1 opelxpi 5726 . . 3 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
2 opelxpi 5726 . . 3 ((𝐶N𝐷N) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
3 addpipq2 10974 . . 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 596 . 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 8025 . . . . 5 ((𝐴N𝐵N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
6 op2ndg 8026 . . . . 5 ((𝐶N𝐷N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
75, 6oveqan12d 7450 . . . 4 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ((1st ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) = (𝐴 ·N 𝐷))
8 op1stg 8025 . . . . 5 ((𝐶N𝐷N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
9 op2ndg 8026 . . . . 5 ((𝐴N𝐵N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
108, 9oveqan12rd 7451 . . . 4 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ((1st ‘⟨𝐶, 𝐷⟩) ·N (2nd ‘⟨𝐴, 𝐵⟩)) = (𝐶 ·N 𝐵))
117, 10oveq12d 7449 . . 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 7450 . . 3 (((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → ((2nd ‘⟨𝐴, 𝐵⟩) ·N (2nd ‘⟨𝐶, 𝐷⟩)) = (𝐵 ·N 𝐷))
1311, 12opeq12d 4886 . 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 2775 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 395   = wceq 1537  wcel 2106  cop 4637   × cxp 5687  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  Ncnpi 10882   +N cpli 10883   ·N cmi 10884   +pQ cplpq 10886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-plpq 10946
This theorem is referenced by:  addassnq  10996  distrnq  10999  1lt2nq  11011  ltexnq  11013  prlem934  11071
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