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Theorem ordpinq 10054
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 5386 . . 3 ((𝐴Q𝐵Q) → (𝐴 <pQ 𝐵𝐴( <pQ ∩ (Q × Q))𝐵))
2 df-ltnq 10029 . . . 4 <Q = ( <pQ ∩ (Q × Q))
32breqi 4850 . . 3 (𝐴 <Q 𝐵𝐴( <pQ ∩ (Q × Q))𝐵)
41, 3syl6bbr 281 . 2 ((𝐴Q𝐵Q) → (𝐴 <pQ 𝐵𝐴 <Q 𝐵))
5 relxp 5331 . . . . 5 Rel (N × N)
6 elpqn 10036 . . . . 5 (𝐴Q𝐴 ∈ (N × N))
7 1st2nd 7450 . . . . 5 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
85, 6, 7sylancr 582 . . . 4 (𝐴Q𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
9 elpqn 10036 . . . . 5 (𝐵Q𝐵 ∈ (N × N))
10 1st2nd 7450 . . . . 5 ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
115, 9, 10sylancr 582 . . . 4 (𝐵Q𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
128, 11breqan12d 4860 . . 3 ((𝐴Q𝐵Q) → (𝐴 <pQ 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ <pQ ⟨(1st𝐵), (2nd𝐵)⟩))
13 ordpipq 10053 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ <pQ ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)))
1412, 13syl6bb 279 . 2 ((𝐴Q𝐵Q) → (𝐴 <pQ 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
154, 14bitr3d 273 1 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  cin 3769  cop 4375   class class class wbr 4844   × cxp 5311  Rel wrel 5318  cfv 6102  (class class class)co 6879  1st c1st 7400  2nd c2nd 7401  Ncnpi 9955   ·N cmi 9957   <N clti 9958   <pQ cltpq 9961  Qcnq 9963   <Q cltq 9969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-tp 4374  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5221  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-we 5274  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-ord 5945  df-on 5946  df-lim 5947  df-suc 5948  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-om 7301  df-1st 7402  df-2nd 7403  df-omul 7805  df-ni 9983  df-mi 9985  df-lti 9986  df-ltpq 10021  df-nq 10023  df-ltnq 10029
This theorem is referenced by:  ltsonq  10080  lterpq  10081  ltanq  10082  ltmnq  10083  ltexnq  10086  archnq  10091
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