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Theorem ltmnq 9673
Description: Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltmnq (𝐶Q → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))

Proof of Theorem ltmnq
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulnqf 9650 . . 3 ·Q :(Q × Q)⟶Q
21fdmi 5965 . 2 dom ·Q = (Q × Q)
3 ltrelnq 9627 . 2 <Q ⊆ (Q × Q)
4 0nnq 9625 . 2 ¬ ∅ ∈ Q
5 elpqn 9626 . . . . . . . . . 10 (𝐶Q𝐶 ∈ (N × N))
653ad2ant3 1077 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐶 ∈ (N × N))
7 xp1st 7089 . . . . . . . . 9 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
86, 7syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (1st𝐶) ∈ N)
9 xp2nd 7090 . . . . . . . . 9 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
106, 9syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐶) ∈ N)
11 mulclpi 9594 . . . . . . . 8 (((1st𝐶) ∈ N ∧ (2nd𝐶) ∈ N) → ((1st𝐶) ·N (2nd𝐶)) ∈ N)
128, 10, 11syl2anc 691 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐶) ·N (2nd𝐶)) ∈ N)
13 ltmpi 9605 . . . . . . 7 (((1st𝐶) ·N (2nd𝐶)) ∈ N → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))))
1412, 13syl 17 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))))
15 fvex 6113 . . . . . . . 8 (1st𝐶) ∈ V
16 fvex 6113 . . . . . . . 8 (2nd𝐶) ∈ V
17 fvex 6113 . . . . . . . 8 (1st𝐴) ∈ V
18 mulcompi 9597 . . . . . . . 8 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
19 mulasspi 9598 . . . . . . . 8 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
20 fvex 6113 . . . . . . . 8 (2nd𝐵) ∈ V
2115, 16, 17, 18, 19, 20caov4 6763 . . . . . . 7 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐶) ·N (1st𝐴)) ·N ((2nd𝐶) ·N (2nd𝐵)))
22 fvex 6113 . . . . . . . 8 (1st𝐵) ∈ V
23 fvex 6113 . . . . . . . 8 (2nd𝐴) ∈ V
2415, 16, 22, 18, 19, 23caov4 6763 . . . . . . 7 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((1st𝐶) ·N (1st𝐵)) ·N ((2nd𝐶) ·N (2nd𝐴)))
2521, 24breq12i 4592 . . . . . 6 ((((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ (((1st𝐶) ·N (1st𝐴)) ·N ((2nd𝐶) ·N (2nd𝐵))) <N (((1st𝐶) ·N (1st𝐵)) ·N ((2nd𝐶) ·N (2nd𝐴))))
2614, 25syl6bb 275 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (((1st𝐶) ·N (1st𝐴)) ·N ((2nd𝐶) ·N (2nd𝐵))) <N (((1st𝐶) ·N (1st𝐵)) ·N ((2nd𝐶) ·N (2nd𝐴)))))
27 ordpipq 9643 . . . . 5 (⟨((1st𝐶) ·N (1st𝐴)), ((2nd𝐶) ·N (2nd𝐴))⟩ <pQ ⟨((1st𝐶) ·N (1st𝐵)), ((2nd𝐶) ·N (2nd𝐵))⟩ ↔ (((1st𝐶) ·N (1st𝐴)) ·N ((2nd𝐶) ·N (2nd𝐵))) <N (((1st𝐶) ·N (1st𝐵)) ·N ((2nd𝐶) ·N (2nd𝐴))))
2826, 27syl6bbr 277 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ ⟨((1st𝐶) ·N (1st𝐴)), ((2nd𝐶) ·N (2nd𝐴))⟩ <pQ ⟨((1st𝐶) ·N (1st𝐵)), ((2nd𝐶) ·N (2nd𝐵))⟩))
29 elpqn 9626 . . . . . . 7 (𝐴Q𝐴 ∈ (N × N))
30293ad2ant1 1075 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → 𝐴 ∈ (N × N))
31 mulpipq2 9640 . . . . . 6 ((𝐶 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐶 ·pQ 𝐴) = ⟨((1st𝐶) ·N (1st𝐴)), ((2nd𝐶) ·N (2nd𝐴))⟩)
326, 30, 31syl2anc 691 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 ·pQ 𝐴) = ⟨((1st𝐶) ·N (1st𝐴)), ((2nd𝐶) ·N (2nd𝐴))⟩)
33 elpqn 9626 . . . . . . 7 (𝐵Q𝐵 ∈ (N × N))
34333ad2ant2 1076 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → 𝐵 ∈ (N × N))
35 mulpipq2 9640 . . . . . 6 ((𝐶 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐶 ·pQ 𝐵) = ⟨((1st𝐶) ·N (1st𝐵)), ((2nd𝐶) ·N (2nd𝐵))⟩)
366, 34, 35syl2anc 691 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 ·pQ 𝐵) = ⟨((1st𝐶) ·N (1st𝐵)), ((2nd𝐶) ·N (2nd𝐵))⟩)
3732, 36breq12d 4596 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ((𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵) ↔ ⟨((1st𝐶) ·N (1st𝐴)), ((2nd𝐶) ·N (2nd𝐴))⟩ <pQ ⟨((1st𝐶) ·N (1st𝐵)), ((2nd𝐶) ·N (2nd𝐵))⟩))
3828, 37bitr4d 270 . . 3 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵)))
39 ordpinq 9644 . . . 4 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
40393adant3 1074 . . 3 ((𝐴Q𝐵Q𝐶Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
41 mulpqnq 9642 . . . . . . 7 ((𝐶Q𝐴Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
4241ancoms 468 . . . . . 6 ((𝐴Q𝐶Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
43423adant2 1073 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
44 mulpqnq 9642 . . . . . . 7 ((𝐶Q𝐵Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
4544ancoms 468 . . . . . 6 ((𝐵Q𝐶Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
46453adant1 1072 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
4743, 46breq12d 4596 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ((𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵) ↔ ([Q]‘(𝐶 ·pQ 𝐴)) <Q ([Q]‘(𝐶 ·pQ 𝐵))))
48 lterpq 9671 . . . 4 ((𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵) ↔ ([Q]‘(𝐶 ·pQ 𝐴)) <Q ([Q]‘(𝐶 ·pQ 𝐵)))
4947, 48syl6bbr 277 . . 3 ((𝐴Q𝐵Q𝐶Q) → ((𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵) ↔ (𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵)))
5038, 40, 493bitr4d 299 . 2 ((𝐴Q𝐵Q𝐶Q) → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))
512, 3, 4, 50ndmovord 6722 1 (𝐶Q → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  w3a 1031   = wceq 1475  wcel 1977  cop 4131   class class class wbr 4583   × cxp 5036  cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Ncnpi 9545   ·N cmi 9547   <N clti 9548   ·pQ cmpq 9550   <pQ cltpq 9551  Qcnq 9553  [Q]cerq 9555   ·Q cmq 9557   <Q cltq 9559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ni 9573  df-mi 9575  df-lti 9576  df-mpq 9610  df-ltpq 9611  df-enq 9612  df-nq 9613  df-erq 9614  df-mq 9616  df-1nq 9617  df-ltnq 9619
This theorem is referenced by:  ltaddnq  9675  ltrnq  9680  addclprlem1  9717  mulclprlem  9720  mulclpr  9721  distrlem4pr  9727  1idpr  9730  prlem934  9734  prlem936  9748  reclem3pr  9750  reclem4pr  9751
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