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Theorem ordpinq 10981
Description: Ordering of positive fractions in terms of positive integers. (Contributed by NM, 13-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ordpinq ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))

Proof of Theorem ordpinq
StepHypRef Expression
1 brinxp 5767 . . 3 ((𝐴Q𝐵Q) → (𝐴 <pQ 𝐵𝐴( <pQ ∩ (Q × Q))𝐵))
2 df-ltnq 10956 . . . 4 <Q = ( <pQ ∩ (Q × Q))
32breqi 5154 . . 3 (𝐴 <Q 𝐵𝐴( <pQ ∩ (Q × Q))𝐵)
41, 3bitr4di 289 . 2 ((𝐴Q𝐵Q) → (𝐴 <pQ 𝐵𝐴 <Q 𝐵))
5 relxp 5707 . . . . 5 Rel (N × N)
6 elpqn 10963 . . . . 5 (𝐴Q𝐴 ∈ (N × N))
7 1st2nd 8063 . . . . 5 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
85, 6, 7sylancr 587 . . . 4 (𝐴Q𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
9 elpqn 10963 . . . . 5 (𝐵Q𝐵 ∈ (N × N))
10 1st2nd 8063 . . . . 5 ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
115, 9, 10sylancr 587 . . . 4 (𝐵Q𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
128, 11breqan12d 5164 . . 3 ((𝐴Q𝐵Q) → (𝐴 <pQ 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ <pQ ⟨(1st𝐵), (2nd𝐵)⟩))
13 ordpipq 10980 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ <pQ ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)))
1412, 13bitrdi 287 . 2 ((𝐴Q𝐵Q) → (𝐴 <pQ 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
154, 14bitr3d 281 1 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  cin 3962  cop 4637   class class class wbr 5148   × cxp 5687  Rel wrel 5694  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  Ncnpi 10882   ·N cmi 10884   <N clti 10885   <pQ cltpq 10888  Qcnq 10890   <Q cltq 10896
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-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  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-iun 4998  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-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-omul 8510  df-ni 10910  df-mi 10912  df-lti 10913  df-ltpq 10948  df-nq 10950  df-ltnq 10956
This theorem is referenced by:  ltsonq  11007  lterpq  11008  ltanq  11009  ltmnq  11010  ltexnq  11013  archnq  11018
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