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Theorem lterpq 9980
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 9920 . . . 4 <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))}
2 opabssxp 5346 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))} ⊆ ((N × N) × (N × N))
31, 2eqsstri 3772 . . 3 <pQ ⊆ ((N × N) × (N × N))
43brel 5321 . 2 (𝐴 <pQ 𝐵 → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
5 ltrelnq 9936 . . . 4 <Q ⊆ (Q × Q)
65brel 5321 . . 3 (([Q]‘𝐴) <Q ([Q]‘𝐵) → (([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q))
7 elpqn 9935 . . . 4 (([Q]‘𝐴) ∈ Q → ([Q]‘𝐴) ∈ (N × N))
8 elpqn 9935 . . . 4 (([Q]‘𝐵) ∈ Q → ([Q]‘𝐵) ∈ (N × N))
9 nqerf 9940 . . . . . . 7 [Q]:(N × N)⟶Q
109fdmi 6209 . . . . . 6 dom [Q] = (N × N)
11 0nelxp 5296 . . . . . 6 ¬ ∅ ∈ (N × N)
1210, 11ndmfvrcl 6376 . . . . 5 (([Q]‘𝐴) ∈ (N × N) → 𝐴 ∈ (N × N))
1310, 11ndmfvrcl 6376 . . . . 5 (([Q]‘𝐵) ∈ (N × N) → 𝐵 ∈ (N × N))
1412, 13anim12i 591 . . . 4 ((([Q]‘𝐴) ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N)) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
157, 8, 14syl2an 495 . . 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 7361 . . . . 5 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
18 xp2nd 7362 . . . . 5 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
19 mulclpi 9903 . . . . 5 (((1st𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
2017, 18, 19syl2an 495 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
21 ltmpi 9914 . . . 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 9941 . . . 4 (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
24 nqercl 9941 . . . 4 (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q)
25 ordpinq 9953 . . . 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 495 . . 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 7368 . . . . . 6 (𝐴 ∈ (N × N) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
28 1st2nd2 7368 . . . . . 6 (𝐵 ∈ (N × N) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
2927, 28breqan12d 4816 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ <pQ ⟨(1st𝐵), (2nd𝐵)⟩))
30 ordpipq 9952 . . . . 5 (⟨(1st𝐴), (2nd𝐴)⟩ <pQ ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)))
3129, 30syl6bb 276 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
32 xp1st 7361 . . . . . . 7 (([Q]‘𝐴) ∈ (N × N) → (1st ‘([Q]‘𝐴)) ∈ N)
3323, 7, 323syl 18 . . . . . 6 (𝐴 ∈ (N × N) → (1st ‘([Q]‘𝐴)) ∈ N)
34 xp2nd 7362 . . . . . . 7 (([Q]‘𝐵) ∈ (N × N) → (2nd ‘([Q]‘𝐵)) ∈ N)
3524, 8, 343syl 18 . . . . . 6 (𝐵 ∈ (N × N) → (2nd ‘([Q]‘𝐵)) ∈ N)
36 mulclpi 9903 . . . . . 6 (((1st ‘([Q]‘𝐴)) ∈ N ∧ (2nd ‘([Q]‘𝐵)) ∈ N) → ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ∈ N)
3733, 35, 36syl2an 495 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ∈ N)
38 ltmpi 9914 . . . . 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 9906 . . . . . 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 9942 . . . . . . . . 9 (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))
4323, 7syl 17 . . . . . . . . . 10 (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ (N × N))
44 enqbreq2 9930 . . . . . . . . . 10 ((𝐴 ∈ (N × N) ∧ ([Q]‘𝐴) ∈ (N × N)) → (𝐴 ~Q ([Q]‘𝐴) ↔ ((1st𝐴) ·N (2nd ‘([Q]‘𝐴))) = ((1st ‘([Q]‘𝐴)) ·N (2nd𝐴))))
4543, 44mpdan 705 . . . . . . . . 9 (𝐴 ∈ (N × N) → (𝐴 ~Q ([Q]‘𝐴) ↔ ((1st𝐴) ·N (2nd ‘([Q]‘𝐴))) = ((1st ‘([Q]‘𝐴)) ·N (2nd𝐴))))
4642, 45mpbid 222 . . . . . . . 8 (𝐴 ∈ (N × N) → ((1st𝐴) ·N (2nd ‘([Q]‘𝐴))) = ((1st ‘([Q]‘𝐴)) ·N (2nd𝐴)))
4746eqcomd 2762 . . . . . . 7 (𝐴 ∈ (N × N) → ((1st ‘([Q]‘𝐴)) ·N (2nd𝐴)) = ((1st𝐴) ·N (2nd ‘([Q]‘𝐴))))
48 nqerrel 9942 . . . . . . . 8 (𝐵 ∈ (N × N) → 𝐵 ~Q ([Q]‘𝐵))
4924, 8syl 17 . . . . . . . . 9 (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ (N × N))
50 enqbreq2 9930 . . . . . . . . 9 ((𝐵 ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N)) → (𝐵 ~Q ([Q]‘𝐵) ↔ ((1st𝐵) ·N (2nd ‘([Q]‘𝐵))) = ((1st ‘([Q]‘𝐵)) ·N (2nd𝐵))))
5149, 50mpdan 705 . . . . . . . 8 (𝐵 ∈ (N × N) → (𝐵 ~Q ([Q]‘𝐵) ↔ ((1st𝐵) ·N (2nd ‘([Q]‘𝐵))) = ((1st ‘([Q]‘𝐵)) ·N (2nd𝐵))))
5248, 51mpbid 222 . . . . . . 7 (𝐵 ∈ (N × N) → ((1st𝐵) ·N (2nd ‘([Q]‘𝐵))) = ((1st ‘([Q]‘𝐵)) ·N (2nd𝐵)))
5347, 52oveqan12d 6828 . . . . . 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 9906 . . . . . . 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 6358 . . . . . . . 8 (1st𝐵) ∈ V
56 fvex 6358 . . . . . . . 8 (2nd𝐴) ∈ V
57 fvex 6358 . . . . . . . 8 (1st ‘([Q]‘𝐴)) ∈ V
58 mulcompi 9906 . . . . . . . 8 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
59 mulasspi 9907 . . . . . . . 8 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
60 fvex 6358 . . . . . . . 8 (2nd ‘([Q]‘𝐵)) ∈ V
6155, 56, 57, 58, 59, 60caov411 7027 . . . . . . 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 2778 . . . . . 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 9906 . . . . . . 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 6358 . . . . . . . 8 (1st ‘([Q]‘𝐵)) ∈ V
65 fvex 6358 . . . . . . . 8 (2nd ‘([Q]‘𝐴)) ∈ V
66 fvex 6358 . . . . . . . 8 (1st𝐴) ∈ V
67 fvex 6358 . . . . . . . 8 (2nd𝐵) ∈ V
6864, 65, 66, 58, 59, 67caov411 7027 . . . . . . 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 2778 . . . . . 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 2815 . . . . 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 4813 . . . 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 294 . . 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 301 . 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 196  wa 383   = wceq 1628  wcel 2135  cop 4323   class class class wbr 4800  {copab 4860   × cxp 5260  cfv 6045  (class class class)co 6809  1st c1st 7327  2nd c2nd 7328  Ncnpi 9854   ·N cmi 9856   <N clti 9857   <pQ cltpq 9860   ~Q ceq 9861  Qcnq 9862  [Q]cerq 9864   <Q cltq 9868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rmo 3054  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-om 7227  df-1st 7329  df-2nd 7330  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-1o 7725  df-oadd 7729  df-omul 7730  df-er 7907  df-ni 9882  df-mi 9884  df-lti 9885  df-ltpq 9920  df-enq 9921  df-nq 9922  df-erq 9923  df-1nq 9926  df-ltnq 9928
This theorem is referenced by:  ltanq  9981  ltmnq  9982  1lt2nq  9983
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