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Theorem ltrnq 10390
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 10337 . . 3 <Q ⊆ (Q × Q)
21brel 5594 . 2 (𝐴 <Q 𝐵 → (𝐴Q𝐵Q))
31brel 5594 . . 3 ((*Q𝐵) <Q (*Q𝐴) → ((*Q𝐵) ∈ Q ∧ (*Q𝐴) ∈ Q))
4 dmrecnq 10379 . . . . 5 dom *Q = Q
5 0nnq 10335 . . . . 5 ¬ ∅ ∈ Q
64, 5ndmfvrcl 6683 . . . 4 ((*Q𝐵) ∈ Q𝐵Q)
74, 5ndmfvrcl 6683 . . . 4 ((*Q𝐴) ∈ Q𝐴Q)
86, 7anim12ci 616 . . 3 (((*Q𝐵) ∈ Q ∧ (*Q𝐴) ∈ Q) → (𝐴Q𝐵Q))
93, 8syl 17 . 2 ((*Q𝐵) <Q (*Q𝐴) → (𝐴Q𝐵Q))
10 breq1 5045 . . . 4 (𝑥 = 𝐴 → (𝑥 <Q 𝑦𝐴 <Q 𝑦))
11 fveq2 6652 . . . . 5 (𝑥 = 𝐴 → (*Q𝑥) = (*Q𝐴))
1211breq2d 5054 . . . 4 (𝑥 = 𝐴 → ((*Q𝑦) <Q (*Q𝑥) ↔ (*Q𝑦) <Q (*Q𝐴)))
1310, 12bibi12d 349 . . 3 (𝑥 = 𝐴 → ((𝑥 <Q 𝑦 ↔ (*Q𝑦) <Q (*Q𝑥)) ↔ (𝐴 <Q 𝑦 ↔ (*Q𝑦) <Q (*Q𝐴))))
14 breq2 5046 . . . 4 (𝑦 = 𝐵 → (𝐴 <Q 𝑦𝐴 <Q 𝐵))
15 fveq2 6652 . . . . 5 (𝑦 = 𝐵 → (*Q𝑦) = (*Q𝐵))
1615breq1d 5052 . . . 4 (𝑦 = 𝐵 → ((*Q𝑦) <Q (*Q𝐴) ↔ (*Q𝐵) <Q (*Q𝐴)))
1714, 16bibi12d 349 . . 3 (𝑦 = 𝐵 → ((𝐴 <Q 𝑦 ↔ (*Q𝑦) <Q (*Q𝐴)) ↔ (𝐴 <Q 𝐵 ↔ (*Q𝐵) <Q (*Q𝐴))))
18 recclnq 10377 . . . . . 6 (𝑥Q → (*Q𝑥) ∈ Q)
19 recclnq 10377 . . . . . 6 (𝑦Q → (*Q𝑦) ∈ Q)
20 mulclnq 10358 . . . . . 6 (((*Q𝑥) ∈ Q ∧ (*Q𝑦) ∈ Q) → ((*Q𝑥) ·Q (*Q𝑦)) ∈ Q)
2118, 19, 20syl2an 598 . . . . 5 ((𝑥Q𝑦Q) → ((*Q𝑥) ·Q (*Q𝑦)) ∈ Q)
22 ltmnq 10383 . . . . 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 10364 . . . . . . 7 (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑥) = (𝑥 ·Q ((*Q𝑥) ·Q (*Q𝑦)))
25 mulassnq 10370 . . . . . . 7 ((𝑥 ·Q (*Q𝑥)) ·Q (*Q𝑦)) = (𝑥 ·Q ((*Q𝑥) ·Q (*Q𝑦)))
26 mulcomnq 10364 . . . . . . 7 ((𝑥 ·Q (*Q𝑥)) ·Q (*Q𝑦)) = ((*Q𝑦) ·Q (𝑥 ·Q (*Q𝑥)))
2724, 25, 263eqtr2i 2851 . . . . . 6 (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑥) = ((*Q𝑦) ·Q (𝑥 ·Q (*Q𝑥)))
28 recidnq 10376 . . . . . . . 8 (𝑥Q → (𝑥 ·Q (*Q𝑥)) = 1Q)
2928oveq2d 7156 . . . . . . 7 (𝑥Q → ((*Q𝑦) ·Q (𝑥 ·Q (*Q𝑥))) = ((*Q𝑦) ·Q 1Q))
30 mulidnq 10374 . . . . . . . 8 ((*Q𝑦) ∈ Q → ((*Q𝑦) ·Q 1Q) = (*Q𝑦))
3119, 30syl 17 . . . . . . 7 (𝑦Q → ((*Q𝑦) ·Q 1Q) = (*Q𝑦))
3229, 31sylan9eq 2877 . . . . . 6 ((𝑥Q𝑦Q) → ((*Q𝑦) ·Q (𝑥 ·Q (*Q𝑥))) = (*Q𝑦))
3327, 32syl5eq 2869 . . . . 5 ((𝑥Q𝑦Q) → (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑥) = (*Q𝑦))
34 mulassnq 10370 . . . . . . 7 (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑦) = ((*Q𝑥) ·Q ((*Q𝑦) ·Q 𝑦))
35 mulcomnq 10364 . . . . . . . 8 ((*Q𝑦) ·Q 𝑦) = (𝑦 ·Q (*Q𝑦))
3635oveq2i 7151 . . . . . . 7 ((*Q𝑥) ·Q ((*Q𝑦) ·Q 𝑦)) = ((*Q𝑥) ·Q (𝑦 ·Q (*Q𝑦)))
3734, 36eqtri 2845 . . . . . 6 (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑦) = ((*Q𝑥) ·Q (𝑦 ·Q (*Q𝑦)))
38 recidnq 10376 . . . . . . . 8 (𝑦Q → (𝑦 ·Q (*Q𝑦)) = 1Q)
3938oveq2d 7156 . . . . . . 7 (𝑦Q → ((*Q𝑥) ·Q (𝑦 ·Q (*Q𝑦))) = ((*Q𝑥) ·Q 1Q))
40 mulidnq 10374 . . . . . . . 8 ((*Q𝑥) ∈ Q → ((*Q𝑥) ·Q 1Q) = (*Q𝑥))
4118, 40syl 17 . . . . . . 7 (𝑥Q → ((*Q𝑥) ·Q 1Q) = (*Q𝑥))
4239, 41sylan9eqr 2879 . . . . . 6 ((𝑥Q𝑦Q) → ((*Q𝑥) ·Q (𝑦 ·Q (*Q𝑦))) = (*Q𝑥))
4337, 42syl5eq 2869 . . . . 5 ((𝑥Q𝑦Q) → (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑦) = (*Q𝑥))
4433, 43breq12d 5055 . . . 4 ((𝑥Q𝑦Q) → ((((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑥) <Q (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑦) ↔ (*Q𝑦) <Q (*Q𝑥)))
4523, 44bitrd 282 . . 3 ((𝑥Q𝑦Q) → (𝑥 <Q 𝑦 ↔ (*Q𝑦) <Q (*Q𝑥)))
4613, 17, 45vtocl2ga 3550 . 2 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ (*Q𝐵) <Q (*Q𝐴)))
472, 9, 46pm5.21nii 383 1 (𝐴 <Q 𝐵 ↔ (*Q𝐵) <Q (*Q𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2114   class class class wbr 5042  cfv 6334  (class class class)co 7140  Qcnq 10263  1Qc1q 10264   ·Q cmq 10267  *Qcrq 10268   <Q cltq 10269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-omul 8094  df-er 8276  df-ni 10283  df-mi 10285  df-lti 10286  df-mpq 10320  df-ltpq 10321  df-enq 10322  df-nq 10323  df-erq 10324  df-mq 10326  df-1nq 10327  df-rq 10328  df-ltnq 10329
This theorem is referenced by:  addclprlem1  10427  reclem2pr  10459  reclem3pr  10460
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