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Theorem ordpinq 9762
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 5179 . . 3 ((𝐴Q𝐵Q) → (𝐴 <pQ 𝐵𝐴( <pQ ∩ (Q × Q))𝐵))
2 df-ltnq 9737 . . . 4 <Q = ( <pQ ∩ (Q × Q))
32breqi 4657 . . 3 (𝐴 <Q 𝐵𝐴( <pQ ∩ (Q × Q))𝐵)
41, 3syl6bbr 278 . 2 ((𝐴Q𝐵Q) → (𝐴 <pQ 𝐵𝐴 <Q 𝐵))
5 relxp 5225 . . . . 5 Rel (N × N)
6 elpqn 9744 . . . . 5 (𝐴Q𝐴 ∈ (N × N))
7 1st2nd 7211 . . . . 5 ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
85, 6, 7sylancr 695 . . . 4 (𝐴Q𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
9 elpqn 9744 . . . . 5 (𝐵Q𝐵 ∈ (N × N))
10 1st2nd 7211 . . . . 5 ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
115, 9, 10sylancr 695 . . . 4 (𝐵Q𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
128, 11breqan12d 4667 . . 3 ((𝐴Q𝐵Q) → (𝐴 <pQ 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ <pQ ⟨(1st𝐵), (2nd𝐵)⟩))
13 ordpipq 9761 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ <pQ ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)))
1412, 13syl6bb 276 . 2 ((𝐴Q𝐵Q) → (𝐴 <pQ 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
154, 14bitr3d 270 1 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  cin 3571  cop 4181   class class class wbr 4651   × cxp 5110  Rel wrel 5117  cfv 5886  (class class class)co 6647  1st c1st 7163  2nd c2nd 7164  Ncnpi 9663   ·N cmi 9665   <N clti 9666   <pQ cltpq 9669  Qcnq 9671   <Q cltq 9677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-omul 7562  df-ni 9691  df-mi 9693  df-lti 9694  df-ltpq 9729  df-nq 9731  df-ltnq 9737
This theorem is referenced by:  ltsonq  9788  lterpq  9789  ltanq  9790  ltmnq  9791  ltexnq  9794  archnq  9799
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