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Theorem ltrnq 9746
Description: Ordering property of reciprocal for positive fractions. Proposition 9-2.6(iv) of [Gleason] p. 120. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltrnq (𝐴 <Q 𝐵 ↔ (*Q𝐵) <Q (*Q𝐴))

Proof of Theorem ltrnq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 9693 . . 3 <Q ⊆ (Q × Q)
21brel 5133 . 2 (𝐴 <Q 𝐵 → (𝐴Q𝐵Q))
31brel 5133 . . 3 ((*Q𝐵) <Q (*Q𝐴) → ((*Q𝐵) ∈ Q ∧ (*Q𝐴) ∈ Q))
4 dmrecnq 9735 . . . . 5 dom *Q = Q
5 0nnq 9691 . . . . 5 ¬ ∅ ∈ Q
64, 5ndmfvrcl 6177 . . . 4 ((*Q𝐵) ∈ Q𝐵Q)
74, 5ndmfvrcl 6177 . . . 4 ((*Q𝐴) ∈ Q𝐴Q)
86, 7anim12ci 590 . . 3 (((*Q𝐵) ∈ Q ∧ (*Q𝐴) ∈ Q) → (𝐴Q𝐵Q))
93, 8syl 17 . 2 ((*Q𝐵) <Q (*Q𝐴) → (𝐴Q𝐵Q))
10 breq1 4621 . . . 4 (𝑥 = 𝐴 → (𝑥 <Q 𝑦𝐴 <Q 𝑦))
11 fveq2 6150 . . . . 5 (𝑥 = 𝐴 → (*Q𝑥) = (*Q𝐴))
1211breq2d 4630 . . . 4 (𝑥 = 𝐴 → ((*Q𝑦) <Q (*Q𝑥) ↔ (*Q𝑦) <Q (*Q𝐴)))
1310, 12bibi12d 335 . . 3 (𝑥 = 𝐴 → ((𝑥 <Q 𝑦 ↔ (*Q𝑦) <Q (*Q𝑥)) ↔ (𝐴 <Q 𝑦 ↔ (*Q𝑦) <Q (*Q𝐴))))
14 breq2 4622 . . . 4 (𝑦 = 𝐵 → (𝐴 <Q 𝑦𝐴 <Q 𝐵))
15 fveq2 6150 . . . . 5 (𝑦 = 𝐵 → (*Q𝑦) = (*Q𝐵))
1615breq1d 4628 . . . 4 (𝑦 = 𝐵 → ((*Q𝑦) <Q (*Q𝐴) ↔ (*Q𝐵) <Q (*Q𝐴)))
1714, 16bibi12d 335 . . 3 (𝑦 = 𝐵 → ((𝐴 <Q 𝑦 ↔ (*Q𝑦) <Q (*Q𝐴)) ↔ (𝐴 <Q 𝐵 ↔ (*Q𝐵) <Q (*Q𝐴))))
18 recclnq 9733 . . . . . 6 (𝑥Q → (*Q𝑥) ∈ Q)
19 recclnq 9733 . . . . . 6 (𝑦Q → (*Q𝑦) ∈ Q)
20 mulclnq 9714 . . . . . 6 (((*Q𝑥) ∈ Q ∧ (*Q𝑦) ∈ Q) → ((*Q𝑥) ·Q (*Q𝑦)) ∈ Q)
2118, 19, 20syl2an 494 . . . . 5 ((𝑥Q𝑦Q) → ((*Q𝑥) ·Q (*Q𝑦)) ∈ Q)
22 ltmnq 9739 . . . . 5 (((*Q𝑥) ·Q (*Q𝑦)) ∈ Q → (𝑥 <Q 𝑦 ↔ (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑥) <Q (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑦)))
2321, 22syl 17 . . . 4 ((𝑥Q𝑦Q) → (𝑥 <Q 𝑦 ↔ (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑥) <Q (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑦)))
24 mulcomnq 9720 . . . . . . 7 (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑥) = (𝑥 ·Q ((*Q𝑥) ·Q (*Q𝑦)))
25 mulassnq 9726 . . . . . . 7 ((𝑥 ·Q (*Q𝑥)) ·Q (*Q𝑦)) = (𝑥 ·Q ((*Q𝑥) ·Q (*Q𝑦)))
26 mulcomnq 9720 . . . . . . 7 ((𝑥 ·Q (*Q𝑥)) ·Q (*Q𝑦)) = ((*Q𝑦) ·Q (𝑥 ·Q (*Q𝑥)))
2724, 25, 263eqtr2i 2654 . . . . . 6 (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑥) = ((*Q𝑦) ·Q (𝑥 ·Q (*Q𝑥)))
28 recidnq 9732 . . . . . . . 8 (𝑥Q → (𝑥 ·Q (*Q𝑥)) = 1Q)
2928oveq2d 6621 . . . . . . 7 (𝑥Q → ((*Q𝑦) ·Q (𝑥 ·Q (*Q𝑥))) = ((*Q𝑦) ·Q 1Q))
30 mulidnq 9730 . . . . . . . 8 ((*Q𝑦) ∈ Q → ((*Q𝑦) ·Q 1Q) = (*Q𝑦))
3119, 30syl 17 . . . . . . 7 (𝑦Q → ((*Q𝑦) ·Q 1Q) = (*Q𝑦))
3229, 31sylan9eq 2680 . . . . . 6 ((𝑥Q𝑦Q) → ((*Q𝑦) ·Q (𝑥 ·Q (*Q𝑥))) = (*Q𝑦))
3327, 32syl5eq 2672 . . . . 5 ((𝑥Q𝑦Q) → (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑥) = (*Q𝑦))
34 mulassnq 9726 . . . . . . 7 (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑦) = ((*Q𝑥) ·Q ((*Q𝑦) ·Q 𝑦))
35 mulcomnq 9720 . . . . . . . 8 ((*Q𝑦) ·Q 𝑦) = (𝑦 ·Q (*Q𝑦))
3635oveq2i 6616 . . . . . . 7 ((*Q𝑥) ·Q ((*Q𝑦) ·Q 𝑦)) = ((*Q𝑥) ·Q (𝑦 ·Q (*Q𝑦)))
3734, 36eqtri 2648 . . . . . 6 (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑦) = ((*Q𝑥) ·Q (𝑦 ·Q (*Q𝑦)))
38 recidnq 9732 . . . . . . . 8 (𝑦Q → (𝑦 ·Q (*Q𝑦)) = 1Q)
3938oveq2d 6621 . . . . . . 7 (𝑦Q → ((*Q𝑥) ·Q (𝑦 ·Q (*Q𝑦))) = ((*Q𝑥) ·Q 1Q))
40 mulidnq 9730 . . . . . . . 8 ((*Q𝑥) ∈ Q → ((*Q𝑥) ·Q 1Q) = (*Q𝑥))
4118, 40syl 17 . . . . . . 7 (𝑥Q → ((*Q𝑥) ·Q 1Q) = (*Q𝑥))
4239, 41sylan9eqr 2682 . . . . . 6 ((𝑥Q𝑦Q) → ((*Q𝑥) ·Q (𝑦 ·Q (*Q𝑦))) = (*Q𝑥))
4337, 42syl5eq 2672 . . . . 5 ((𝑥Q𝑦Q) → (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑦) = (*Q𝑥))
4433, 43breq12d 4631 . . . 4 ((𝑥Q𝑦Q) → ((((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑥) <Q (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑦) ↔ (*Q𝑦) <Q (*Q𝑥)))
4523, 44bitrd 268 . . 3 ((𝑥Q𝑦Q) → (𝑥 <Q 𝑦 ↔ (*Q𝑦) <Q (*Q𝑥)))
4613, 17, 45vtocl2ga 3265 . 2 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ (*Q𝐵) <Q (*Q𝐴)))
472, 9, 46pm5.21nii 368 1 (𝐴 <Q 𝐵 ↔ (*Q𝐵) <Q (*Q𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1992   class class class wbr 4618  cfv 5850  (class class class)co 6605  Qcnq 9619  1Qc1q 9620   ·Q cmq 9623  *Qcrq 9624   <Q cltq 9625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-omul 7511  df-er 7688  df-ni 9639  df-mi 9641  df-lti 9642  df-mpq 9676  df-ltpq 9677  df-enq 9678  df-nq 9679  df-erq 9680  df-mq 9682  df-1nq 9683  df-rq 9684  df-ltnq 9685
This theorem is referenced by:  addclprlem1  9783  reclem2pr  9815  reclem3pr  9816
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