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Theorem ltrnq 11020
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 10967 . . 3 <Q ⊆ (Q × Q)
21brel 5749 . 2 (𝐴 <Q 𝐵 → (𝐴Q𝐵Q))
31brel 5749 . . 3 ((*Q𝐵) <Q (*Q𝐴) → ((*Q𝐵) ∈ Q ∧ (*Q𝐴) ∈ Q))
4 dmrecnq 11009 . . . . 5 dom *Q = Q
5 0nnq 10965 . . . . 5 ¬ ∅ ∈ Q
64, 5ndmfvrcl 6941 . . . 4 ((*Q𝐵) ∈ Q𝐵Q)
74, 5ndmfvrcl 6941 . . . 4 ((*Q𝐴) ∈ Q𝐴Q)
86, 7anim12ci 614 . . 3 (((*Q𝐵) ∈ Q ∧ (*Q𝐴) ∈ Q) → (𝐴Q𝐵Q))
93, 8syl 17 . 2 ((*Q𝐵) <Q (*Q𝐴) → (𝐴Q𝐵Q))
10 breq1 5145 . . . 4 (𝑥 = 𝐴 → (𝑥 <Q 𝑦𝐴 <Q 𝑦))
11 fveq2 6905 . . . . 5 (𝑥 = 𝐴 → (*Q𝑥) = (*Q𝐴))
1211breq2d 5154 . . . 4 (𝑥 = 𝐴 → ((*Q𝑦) <Q (*Q𝑥) ↔ (*Q𝑦) <Q (*Q𝐴)))
1310, 12bibi12d 345 . . 3 (𝑥 = 𝐴 → ((𝑥 <Q 𝑦 ↔ (*Q𝑦) <Q (*Q𝑥)) ↔ (𝐴 <Q 𝑦 ↔ (*Q𝑦) <Q (*Q𝐴))))
14 breq2 5146 . . . 4 (𝑦 = 𝐵 → (𝐴 <Q 𝑦𝐴 <Q 𝐵))
15 fveq2 6905 . . . . 5 (𝑦 = 𝐵 → (*Q𝑦) = (*Q𝐵))
1615breq1d 5152 . . . 4 (𝑦 = 𝐵 → ((*Q𝑦) <Q (*Q𝐴) ↔ (*Q𝐵) <Q (*Q𝐴)))
1714, 16bibi12d 345 . . 3 (𝑦 = 𝐵 → ((𝐴 <Q 𝑦 ↔ (*Q𝑦) <Q (*Q𝐴)) ↔ (𝐴 <Q 𝐵 ↔ (*Q𝐵) <Q (*Q𝐴))))
18 recclnq 11007 . . . . . 6 (𝑥Q → (*Q𝑥) ∈ Q)
19 recclnq 11007 . . . . . 6 (𝑦Q → (*Q𝑦) ∈ Q)
20 mulclnq 10988 . . . . . 6 (((*Q𝑥) ∈ Q ∧ (*Q𝑦) ∈ Q) → ((*Q𝑥) ·Q (*Q𝑦)) ∈ Q)
2118, 19, 20syl2an 596 . . . . 5 ((𝑥Q𝑦Q) → ((*Q𝑥) ·Q (*Q𝑦)) ∈ Q)
22 ltmnq 11013 . . . . 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 10994 . . . . . . 7 (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑥) = (𝑥 ·Q ((*Q𝑥) ·Q (*Q𝑦)))
25 mulassnq 11000 . . . . . . 7 ((𝑥 ·Q (*Q𝑥)) ·Q (*Q𝑦)) = (𝑥 ·Q ((*Q𝑥) ·Q (*Q𝑦)))
26 mulcomnq 10994 . . . . . . 7 ((𝑥 ·Q (*Q𝑥)) ·Q (*Q𝑦)) = ((*Q𝑦) ·Q (𝑥 ·Q (*Q𝑥)))
2724, 25, 263eqtr2i 2770 . . . . . 6 (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑥) = ((*Q𝑦) ·Q (𝑥 ·Q (*Q𝑥)))
28 recidnq 11006 . . . . . . . 8 (𝑥Q → (𝑥 ·Q (*Q𝑥)) = 1Q)
2928oveq2d 7448 . . . . . . 7 (𝑥Q → ((*Q𝑦) ·Q (𝑥 ·Q (*Q𝑥))) = ((*Q𝑦) ·Q 1Q))
30 mulidnq 11004 . . . . . . . 8 ((*Q𝑦) ∈ Q → ((*Q𝑦) ·Q 1Q) = (*Q𝑦))
3119, 30syl 17 . . . . . . 7 (𝑦Q → ((*Q𝑦) ·Q 1Q) = (*Q𝑦))
3229, 31sylan9eq 2796 . . . . . 6 ((𝑥Q𝑦Q) → ((*Q𝑦) ·Q (𝑥 ·Q (*Q𝑥))) = (*Q𝑦))
3327, 32eqtrid 2788 . . . . 5 ((𝑥Q𝑦Q) → (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑥) = (*Q𝑦))
34 mulassnq 11000 . . . . . . 7 (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑦) = ((*Q𝑥) ·Q ((*Q𝑦) ·Q 𝑦))
35 mulcomnq 10994 . . . . . . . 8 ((*Q𝑦) ·Q 𝑦) = (𝑦 ·Q (*Q𝑦))
3635oveq2i 7443 . . . . . . 7 ((*Q𝑥) ·Q ((*Q𝑦) ·Q 𝑦)) = ((*Q𝑥) ·Q (𝑦 ·Q (*Q𝑦)))
3734, 36eqtri 2764 . . . . . 6 (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑦) = ((*Q𝑥) ·Q (𝑦 ·Q (*Q𝑦)))
38 recidnq 11006 . . . . . . . 8 (𝑦Q → (𝑦 ·Q (*Q𝑦)) = 1Q)
3938oveq2d 7448 . . . . . . 7 (𝑦Q → ((*Q𝑥) ·Q (𝑦 ·Q (*Q𝑦))) = ((*Q𝑥) ·Q 1Q))
40 mulidnq 11004 . . . . . . . 8 ((*Q𝑥) ∈ Q → ((*Q𝑥) ·Q 1Q) = (*Q𝑥))
4118, 40syl 17 . . . . . . 7 (𝑥Q → ((*Q𝑥) ·Q 1Q) = (*Q𝑥))
4239, 41sylan9eqr 2798 . . . . . 6 ((𝑥Q𝑦Q) → ((*Q𝑥) ·Q (𝑦 ·Q (*Q𝑦))) = (*Q𝑥))
4337, 42eqtrid 2788 . . . . 5 ((𝑥Q𝑦Q) → (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑦) = (*Q𝑥))
4433, 43breq12d 5155 . . . 4 ((𝑥Q𝑦Q) → ((((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑥) <Q (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑦) ↔ (*Q𝑦) <Q (*Q𝑥)))
4523, 44bitrd 279 . . 3 ((𝑥Q𝑦Q) → (𝑥 <Q 𝑦 ↔ (*Q𝑦) <Q (*Q𝑥)))
4613, 17, 45vtocl2ga 3577 . 2 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ (*Q𝐵) <Q (*Q𝐴)))
472, 9, 46pm5.21nii 378 1 (𝐴 <Q 𝐵 ↔ (*Q𝐵) <Q (*Q𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539  wcel 2107   class class class wbr 5142  cfv 6560  (class class class)co 7432  Qcnq 10893  1Qc1q 10894   ·Q cmq 10897  *Qcrq 10898   <Q cltq 10899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-omul 8512  df-er 8746  df-ni 10913  df-mi 10915  df-lti 10916  df-mpq 10950  df-ltpq 10951  df-enq 10952  df-nq 10953  df-erq 10954  df-mq 10956  df-1nq 10957  df-rq 10958  df-ltnq 10959
This theorem is referenced by:  addclprlem1  11057  reclem2pr  11089  reclem3pr  11090
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