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Theorem ltrnq 9555
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 9502 . . 3 <Q ⊆ (Q × Q)
21brel 4984 . 2 (𝐴 <Q 𝐵 → (𝐴Q𝐵Q))
31brel 4984 . . 3 ((*Q𝐵) <Q (*Q𝐴) → ((*Q𝐵) ∈ Q ∧ (*Q𝐴) ∈ Q))
4 dmrecnq 9544 . . . . 5 dom *Q = Q
5 0nnq 9500 . . . . 5 ¬ ∅ ∈ Q
64, 5ndmfvrcl 6012 . . . 4 ((*Q𝐵) ∈ Q𝐵Q)
74, 5ndmfvrcl 6012 . . . 4 ((*Q𝐴) ∈ Q𝐴Q)
86, 7anim12ci 588 . . 3 (((*Q𝐵) ∈ Q ∧ (*Q𝐴) ∈ Q) → (𝐴Q𝐵Q))
93, 8syl 17 . 2 ((*Q𝐵) <Q (*Q𝐴) → (𝐴Q𝐵Q))
10 breq1 4484 . . . 4 (𝑥 = 𝐴 → (𝑥 <Q 𝑦𝐴 <Q 𝑦))
11 fveq2 5986 . . . . 5 (𝑥 = 𝐴 → (*Q𝑥) = (*Q𝐴))
1211breq2d 4493 . . . 4 (𝑥 = 𝐴 → ((*Q𝑦) <Q (*Q𝑥) ↔ (*Q𝑦) <Q (*Q𝐴)))
1310, 12bibi12d 333 . . 3 (𝑥 = 𝐴 → ((𝑥 <Q 𝑦 ↔ (*Q𝑦) <Q (*Q𝑥)) ↔ (𝐴 <Q 𝑦 ↔ (*Q𝑦) <Q (*Q𝐴))))
14 breq2 4485 . . . 4 (𝑦 = 𝐵 → (𝐴 <Q 𝑦𝐴 <Q 𝐵))
15 fveq2 5986 . . . . 5 (𝑦 = 𝐵 → (*Q𝑦) = (*Q𝐵))
1615breq1d 4491 . . . 4 (𝑦 = 𝐵 → ((*Q𝑦) <Q (*Q𝐴) ↔ (*Q𝐵) <Q (*Q𝐴)))
1714, 16bibi12d 333 . . 3 (𝑦 = 𝐵 → ((𝐴 <Q 𝑦 ↔ (*Q𝑦) <Q (*Q𝐴)) ↔ (𝐴 <Q 𝐵 ↔ (*Q𝐵) <Q (*Q𝐴))))
18 recclnq 9542 . . . . . 6 (𝑥Q → (*Q𝑥) ∈ Q)
19 recclnq 9542 . . . . . 6 (𝑦Q → (*Q𝑦) ∈ Q)
20 mulclnq 9523 . . . . . 6 (((*Q𝑥) ∈ Q ∧ (*Q𝑦) ∈ Q) → ((*Q𝑥) ·Q (*Q𝑦)) ∈ Q)
2118, 19, 20syl2an 492 . . . . 5 ((𝑥Q𝑦Q) → ((*Q𝑥) ·Q (*Q𝑦)) ∈ Q)
22 ltmnq 9548 . . . . 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 9529 . . . . . . 7 (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑥) = (𝑥 ·Q ((*Q𝑥) ·Q (*Q𝑦)))
25 mulassnq 9535 . . . . . . 7 ((𝑥 ·Q (*Q𝑥)) ·Q (*Q𝑦)) = (𝑥 ·Q ((*Q𝑥) ·Q (*Q𝑦)))
26 mulcomnq 9529 . . . . . . 7 ((𝑥 ·Q (*Q𝑥)) ·Q (*Q𝑦)) = ((*Q𝑦) ·Q (𝑥 ·Q (*Q𝑥)))
2724, 25, 263eqtr2i 2542 . . . . . 6 (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑥) = ((*Q𝑦) ·Q (𝑥 ·Q (*Q𝑥)))
28 recidnq 9541 . . . . . . . 8 (𝑥Q → (𝑥 ·Q (*Q𝑥)) = 1Q)
2928oveq2d 6441 . . . . . . 7 (𝑥Q → ((*Q𝑦) ·Q (𝑥 ·Q (*Q𝑥))) = ((*Q𝑦) ·Q 1Q))
30 mulidnq 9539 . . . . . . . 8 ((*Q𝑦) ∈ Q → ((*Q𝑦) ·Q 1Q) = (*Q𝑦))
3119, 30syl 17 . . . . . . 7 (𝑦Q → ((*Q𝑦) ·Q 1Q) = (*Q𝑦))
3229, 31sylan9eq 2568 . . . . . 6 ((𝑥Q𝑦Q) → ((*Q𝑦) ·Q (𝑥 ·Q (*Q𝑥))) = (*Q𝑦))
3327, 32syl5eq 2560 . . . . 5 ((𝑥Q𝑦Q) → (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑥) = (*Q𝑦))
34 mulassnq 9535 . . . . . . 7 (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑦) = ((*Q𝑥) ·Q ((*Q𝑦) ·Q 𝑦))
35 mulcomnq 9529 . . . . . . . 8 ((*Q𝑦) ·Q 𝑦) = (𝑦 ·Q (*Q𝑦))
3635oveq2i 6436 . . . . . . 7 ((*Q𝑥) ·Q ((*Q𝑦) ·Q 𝑦)) = ((*Q𝑥) ·Q (𝑦 ·Q (*Q𝑦)))
3734, 36eqtri 2536 . . . . . 6 (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑦) = ((*Q𝑥) ·Q (𝑦 ·Q (*Q𝑦)))
38 recidnq 9541 . . . . . . . 8 (𝑦Q → (𝑦 ·Q (*Q𝑦)) = 1Q)
3938oveq2d 6441 . . . . . . 7 (𝑦Q → ((*Q𝑥) ·Q (𝑦 ·Q (*Q𝑦))) = ((*Q𝑥) ·Q 1Q))
40 mulidnq 9539 . . . . . . . 8 ((*Q𝑥) ∈ Q → ((*Q𝑥) ·Q 1Q) = (*Q𝑥))
4118, 40syl 17 . . . . . . 7 (𝑥Q → ((*Q𝑥) ·Q 1Q) = (*Q𝑥))
4239, 41sylan9eqr 2570 . . . . . 6 ((𝑥Q𝑦Q) → ((*Q𝑥) ·Q (𝑦 ·Q (*Q𝑦))) = (*Q𝑥))
4337, 42syl5eq 2560 . . . . 5 ((𝑥Q𝑦Q) → (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑦) = (*Q𝑥))
4433, 43breq12d 4494 . . . 4 ((𝑥Q𝑦Q) → ((((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑥) <Q (((*Q𝑥) ·Q (*Q𝑦)) ·Q 𝑦) ↔ (*Q𝑦) <Q (*Q𝑥)))
4523, 44bitrd 266 . . 3 ((𝑥Q𝑦Q) → (𝑥 <Q 𝑦 ↔ (*Q𝑦) <Q (*Q𝑥)))
4613, 17, 45vtocl2ga 3151 . 2 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ (*Q𝐵) <Q (*Q𝐴)))
472, 9, 46pm5.21nii 366 1 (𝐴 <Q 𝐵 ↔ (*Q𝐵) <Q (*Q𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wcel 1938   class class class wbr 4481  cfv 5689  (class class class)co 6425  Qcnq 9428  1Qc1q 9429   ·Q cmq 9432  *Qcrq 9433   <Q cltq 9434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6722
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-om 6833  df-1st 6933  df-2nd 6934  df-wrecs 7168  df-recs 7230  df-rdg 7268  df-1o 7322  df-oadd 7326  df-omul 7327  df-er 7504  df-ni 9448  df-mi 9450  df-lti 9451  df-mpq 9485  df-ltpq 9486  df-enq 9487  df-nq 9488  df-erq 9489  df-mq 9491  df-1nq 9492  df-rq 9493  df-ltnq 9494
This theorem is referenced by:  addclprlem1  9592  reclem2pr  9624  reclem3pr  9625
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