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Theorem lterpq 9671
Description: Compatibility of ordering on equivalent fractions. (Contributed by Mario Carneiro, 9-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
lterpq (𝐴 <pQ 𝐵 ↔ ([Q]‘𝐴) <Q ([Q]‘𝐵))

Proof of Theorem lterpq
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ltpq 9611 . . . 4 <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))}
2 opabssxp 5116 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))} ⊆ ((N × N) × (N × N))
31, 2eqsstri 3598 . . 3 <pQ ⊆ ((N × N) × (N × N))
43brel 5090 . 2 (𝐴 <pQ 𝐵 → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
5 ltrelnq 9627 . . . 4 <Q ⊆ (Q × Q)
65brel 5090 . . 3 (([Q]‘𝐴) <Q ([Q]‘𝐵) → (([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q))
7 elpqn 9626 . . . 4 (([Q]‘𝐴) ∈ Q → ([Q]‘𝐴) ∈ (N × N))
8 elpqn 9626 . . . 4 (([Q]‘𝐵) ∈ Q → ([Q]‘𝐵) ∈ (N × N))
9 nqerf 9631 . . . . . . 7 [Q]:(N × N)⟶Q
109fdmi 5965 . . . . . 6 dom [Q] = (N × N)
11 0nelxp 5067 . . . . . 6 ¬ ∅ ∈ (N × N)
1210, 11ndmfvrcl 6129 . . . . 5 (([Q]‘𝐴) ∈ (N × N) → 𝐴 ∈ (N × N))
1310, 11ndmfvrcl 6129 . . . . 5 (([Q]‘𝐵) ∈ (N × N) → 𝐵 ∈ (N × N))
1412, 13anim12i 588 . . . 4 ((([Q]‘𝐴) ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N)) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
157, 8, 14syl2an 493 . . 3 ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
166, 15syl 17 . 2 (([Q]‘𝐴) <Q ([Q]‘𝐵) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
17 xp1st 7089 . . . . 5 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
18 xp2nd 7090 . . . . 5 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
19 mulclpi 9594 . . . . 5 (((1st𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
2017, 18, 19syl2an 493 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
21 ltmpi 9605 . . . 4 (((1st𝐴) ·N (2nd𝐵)) ∈ N → (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) <N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))) ↔ (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵)))) <N (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))))))
2220, 21syl 17 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) <N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))) ↔ (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵)))) <N (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))))))
23 nqercl 9632 . . . 4 (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
24 nqercl 9632 . . . 4 (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q)
25 ordpinq 9644 . . . 4 ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (([Q]‘𝐴) <Q ([Q]‘𝐵) ↔ ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) <N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴)))))
2623, 24, 25syl2an 493 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) <Q ([Q]‘𝐵) ↔ ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) <N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴)))))
27 1st2nd2 7096 . . . . . 6 (𝐴 ∈ (N × N) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
28 1st2nd2 7096 . . . . . 6 (𝐵 ∈ (N × N) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
2927, 28breqan12d 4599 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ <pQ ⟨(1st𝐵), (2nd𝐵)⟩))
30 ordpipq 9643 . . . . 5 (⟨(1st𝐴), (2nd𝐴)⟩ <pQ ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)))
3129, 30syl6bb 275 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
32 xp1st 7089 . . . . . . 7 (([Q]‘𝐴) ∈ (N × N) → (1st ‘([Q]‘𝐴)) ∈ N)
3323, 7, 323syl 18 . . . . . 6 (𝐴 ∈ (N × N) → (1st ‘([Q]‘𝐴)) ∈ N)
34 xp2nd 7090 . . . . . . 7 (([Q]‘𝐵) ∈ (N × N) → (2nd ‘([Q]‘𝐵)) ∈ N)
3524, 8, 343syl 18 . . . . . 6 (𝐵 ∈ (N × N) → (2nd ‘([Q]‘𝐵)) ∈ N)
36 mulclpi 9594 . . . . . 6 (((1st ‘([Q]‘𝐴)) ∈ N ∧ (2nd ‘([Q]‘𝐵)) ∈ N) → ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ∈ N)
3733, 35, 36syl2an 493 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ∈ N)
38 ltmpi 9605 . . . . 5 (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ∈ N → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐵) ·N (2nd𝐴)))))
3937, 38syl 17 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐵) ·N (2nd𝐴)))))
40 mulcompi 9597 . . . . . 6 (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))))
4140a1i 11 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵)))))
42 nqerrel 9633 . . . . . . . . 9 (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))
4323, 7syl 17 . . . . . . . . . 10 (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ (N × N))
44 enqbreq2 9621 . . . . . . . . . 10 ((𝐴 ∈ (N × N) ∧ ([Q]‘𝐴) ∈ (N × N)) → (𝐴 ~Q ([Q]‘𝐴) ↔ ((1st𝐴) ·N (2nd ‘([Q]‘𝐴))) = ((1st ‘([Q]‘𝐴)) ·N (2nd𝐴))))
4543, 44mpdan 699 . . . . . . . . 9 (𝐴 ∈ (N × N) → (𝐴 ~Q ([Q]‘𝐴) ↔ ((1st𝐴) ·N (2nd ‘([Q]‘𝐴))) = ((1st ‘([Q]‘𝐴)) ·N (2nd𝐴))))
4642, 45mpbid 221 . . . . . . . 8 (𝐴 ∈ (N × N) → ((1st𝐴) ·N (2nd ‘([Q]‘𝐴))) = ((1st ‘([Q]‘𝐴)) ·N (2nd𝐴)))
4746eqcomd 2616 . . . . . . 7 (𝐴 ∈ (N × N) → ((1st ‘([Q]‘𝐴)) ·N (2nd𝐴)) = ((1st𝐴) ·N (2nd ‘([Q]‘𝐴))))
48 nqerrel 9633 . . . . . . . 8 (𝐵 ∈ (N × N) → 𝐵 ~Q ([Q]‘𝐵))
4924, 8syl 17 . . . . . . . . 9 (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ (N × N))
50 enqbreq2 9621 . . . . . . . . 9 ((𝐵 ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N)) → (𝐵 ~Q ([Q]‘𝐵) ↔ ((1st𝐵) ·N (2nd ‘([Q]‘𝐵))) = ((1st ‘([Q]‘𝐵)) ·N (2nd𝐵))))
5149, 50mpdan 699 . . . . . . . 8 (𝐵 ∈ (N × N) → (𝐵 ~Q ([Q]‘𝐵) ↔ ((1st𝐵) ·N (2nd ‘([Q]‘𝐵))) = ((1st ‘([Q]‘𝐵)) ·N (2nd𝐵))))
5248, 51mpbid 221 . . . . . . 7 (𝐵 ∈ (N × N) → ((1st𝐵) ·N (2nd ‘([Q]‘𝐵))) = ((1st ‘([Q]‘𝐵)) ·N (2nd𝐵)))
5347, 52oveqan12d 6568 . . . . . 6 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (((1st ‘([Q]‘𝐴)) ·N (2nd𝐴)) ·N ((1st𝐵) ·N (2nd ‘([Q]‘𝐵)))) = (((1st𝐴) ·N (2nd ‘([Q]‘𝐴))) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd𝐵))))
54 mulcompi 9597 . . . . . . 7 (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐵) ·N (2nd𝐴))) = (((1st𝐵) ·N (2nd𝐴)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))))
55 fvex 6113 . . . . . . . 8 (1st𝐵) ∈ V
56 fvex 6113 . . . . . . . 8 (2nd𝐴) ∈ V
57 fvex 6113 . . . . . . . 8 (1st ‘([Q]‘𝐴)) ∈ V
58 mulcompi 9597 . . . . . . . 8 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
59 mulasspi 9598 . . . . . . . 8 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
60 fvex 6113 . . . . . . . 8 (2nd ‘([Q]‘𝐵)) ∈ V
6155, 56, 57, 58, 59, 60caov411 6764 . . . . . . 7 (((1st𝐵) ·N (2nd𝐴)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵)))) = (((1st ‘([Q]‘𝐴)) ·N (2nd𝐴)) ·N ((1st𝐵) ·N (2nd ‘([Q]‘𝐵))))
6254, 61eqtri 2632 . . . . . 6 (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐵) ·N (2nd𝐴))) = (((1st ‘([Q]‘𝐴)) ·N (2nd𝐴)) ·N ((1st𝐵) ·N (2nd ‘([Q]‘𝐵))))
63 mulcompi 9597 . . . . . . 7 (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴)))) = (((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))) ·N ((1st𝐴) ·N (2nd𝐵)))
64 fvex 6113 . . . . . . . 8 (1st ‘([Q]‘𝐵)) ∈ V
65 fvex 6113 . . . . . . . 8 (2nd ‘([Q]‘𝐴)) ∈ V
66 fvex 6113 . . . . . . . 8 (1st𝐴) ∈ V
67 fvex 6113 . . . . . . . 8 (2nd𝐵) ∈ V
6864, 65, 66, 58, 59, 67caov411 6764 . . . . . . 7 (((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd ‘([Q]‘𝐴))) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd𝐵)))
6963, 68eqtri 2632 . . . . . 6 (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴)))) = (((1st𝐴) ·N (2nd ‘([Q]‘𝐴))) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd𝐵)))
7053, 62, 693eqtr4g 2669 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐵) ·N (2nd𝐴))) = (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴)))))
7141, 70breq12d 4596 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵)))) <N (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))))))
7231, 39, 713bitrd 293 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵)))) <N (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))))))
7322, 26, 723bitr4rd 300 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ ([Q]‘𝐴) <Q ([Q]‘𝐵)))
744, 16, 73pm5.21nii 367 1 (𝐴 <pQ 𝐵 ↔ ([Q]‘𝐴) <Q ([Q]‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383   = wceq 1475  wcel 1977  cop 4131   class class class wbr 4583  {copab 4642   × cxp 5036  cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Ncnpi 9545   ·N cmi 9547   <N clti 9548   <pQ cltpq 9551   ~Q ceq 9552  Qcnq 9553  [Q]cerq 9555   <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-ltpq 9611  df-enq 9612  df-nq 9613  df-erq 9614  df-1nq 9617  df-ltnq 9619
This theorem is referenced by:  ltanq  9672  ltmnq  9673  1lt2nq  9674
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