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Theorem ordpipq 9724
Description: Ordering of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ordpipq (⟨𝐴, 𝐵⟩ <pQ𝐶, 𝐷⟩ ↔ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))

Proof of Theorem ordpipq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 4903 . . 3 𝐴, 𝐵⟩ ∈ V
2 opex 4903 . . 3 𝐶, 𝐷⟩ ∈ V
3 eleq1 2686 . . . . . 6 (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 ∈ (N × N) ↔ ⟨𝐴, 𝐵⟩ ∈ (N × N)))
43anbi1d 740 . . . . 5 (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))))
54anbi1d 740 . . . 4 (𝑥 = ⟨𝐴, 𝐵⟩ → (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥))) ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))))
6 fveq2 6158 . . . . . . . 8 (𝑥 = ⟨𝐴, 𝐵⟩ → (1st𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
7 opelxp 5116 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩ ∈ (N × N) ↔ (𝐴N𝐵N))
8 op1stg 7140 . . . . . . . . . 10 ((𝐴N𝐵N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
97, 8sylbi 207 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
109adantr 481 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
116, 10sylan9eq 2675 . . . . . . 7 ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → (1st𝑥) = 𝐴)
1211oveq1d 6630 . . . . . 6 ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → ((1st𝑥) ·N (2nd𝑦)) = (𝐴 ·N (2nd𝑦)))
13 fveq2 6158 . . . . . . . 8 (𝑥 = ⟨𝐴, 𝐵⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
14 op2ndg 7141 . . . . . . . . . 10 ((𝐴N𝐵N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
157, 14sylbi 207 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
1615adantr 481 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
1713, 16sylan9eq 2675 . . . . . . 7 ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → (2nd𝑥) = 𝐵)
1817oveq2d 6631 . . . . . 6 ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → ((1st𝑦) ·N (2nd𝑥)) = ((1st𝑦) ·N 𝐵))
1912, 18breq12d 4636 . . . . 5 ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → (((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)) ↔ (𝐴 ·N (2nd𝑦)) <N ((1st𝑦) ·N 𝐵)))
2019pm5.32da 672 . . . 4 (𝑥 = ⟨𝐴, 𝐵⟩ → (((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥))) ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝐴 ·N (2nd𝑦)) <N ((1st𝑦) ·N 𝐵))))
215, 20bitrd 268 . . 3 (𝑥 = ⟨𝐴, 𝐵⟩ → (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥))) ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝐴 ·N (2nd𝑦)) <N ((1st𝑦) ·N 𝐵))))
22 eleq1 2686 . . . . . 6 (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 ∈ (N × N) ↔ ⟨𝐶, 𝐷⟩ ∈ (N × N)))
2322anbi2d 739 . . . . 5 (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))))
2423anbi1d 740 . . . 4 (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝐴 ·N (2nd𝑦)) <N ((1st𝑦) ·N 𝐵)) ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N (2nd𝑦)) <N ((1st𝑦) ·N 𝐵))))
25 fveq2 6158 . . . . . . . 8 (𝑦 = ⟨𝐶, 𝐷⟩ → (2nd𝑦) = (2nd ‘⟨𝐶, 𝐷⟩))
26 opelxp 5116 . . . . . . . . . 10 (⟨𝐶, 𝐷⟩ ∈ (N × N) ↔ (𝐶N𝐷N))
27 op2ndg 7141 . . . . . . . . . 10 ((𝐶N𝐷N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
2826, 27sylbi 207 . . . . . . . . 9 (⟨𝐶, 𝐷⟩ ∈ (N × N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
2928adantl 482 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
3025, 29sylan9eq 2675 . . . . . . 7 ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (2nd𝑦) = 𝐷)
3130oveq2d 6631 . . . . . 6 ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (𝐴 ·N (2nd𝑦)) = (𝐴 ·N 𝐷))
32 fveq2 6158 . . . . . . . 8 (𝑦 = ⟨𝐶, 𝐷⟩ → (1st𝑦) = (1st ‘⟨𝐶, 𝐷⟩))
33 op1stg 7140 . . . . . . . . . 10 ((𝐶N𝐷N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
3426, 33sylbi 207 . . . . . . . . 9 (⟨𝐶, 𝐷⟩ ∈ (N × N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
3534adantl 482 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
3632, 35sylan9eq 2675 . . . . . . 7 ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (1st𝑦) = 𝐶)
3736oveq1d 6630 . . . . . 6 ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → ((1st𝑦) ·N 𝐵) = (𝐶 ·N 𝐵))
3831, 37breq12d 4636 . . . . 5 ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → ((𝐴 ·N (2nd𝑦)) <N ((1st𝑦) ·N 𝐵) ↔ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)))
3938pm5.32da 672 . . . 4 (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N (2nd𝑦)) <N ((1st𝑦) ·N 𝐵)) ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))))
4024, 39bitrd 268 . . 3 (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝐴 ·N (2nd𝑦)) <N ((1st𝑦) ·N 𝐵)) ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))))
41 df-ltpq 9692 . . 3 <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))}
421, 2, 21, 40, 41brab 4968 . 2 (⟨𝐴, 𝐵⟩ <pQ𝐶, 𝐷⟩ ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)))
43 simpr 477 . . 3 (((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)) → (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))
44 ltrelpi 9671 . . . . . 6 <N ⊆ (N × N)
4544brel 5138 . . . . 5 ((𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵) → ((𝐴 ·N 𝐷) ∈ N ∧ (𝐶 ·N 𝐵) ∈ N))
46 dmmulpi 9673 . . . . . . 7 dom ·N = (N × N)
47 0npi 9664 . . . . . . 7 ¬ ∅ ∈ N
4846, 47ndmovrcl 6785 . . . . . 6 ((𝐴 ·N 𝐷) ∈ N → (𝐴N𝐷N))
4946, 47ndmovrcl 6785 . . . . . 6 ((𝐶 ·N 𝐵) ∈ N → (𝐶N𝐵N))
5048, 49anim12i 589 . . . . 5 (((𝐴 ·N 𝐷) ∈ N ∧ (𝐶 ·N 𝐵) ∈ N) → ((𝐴N𝐷N) ∧ (𝐶N𝐵N)))
51 opelxpi 5118 . . . . . . 7 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
5251ad2ant2rl 784 . . . . . 6 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
53 simprl 793 . . . . . . 7 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → 𝐶N)
54 simplr 791 . . . . . . 7 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → 𝐷N)
55 opelxpi 5118 . . . . . . 7 ((𝐶N𝐷N) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
5653, 54, 55syl2anc 692 . . . . . 6 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
5752, 56jca 554 . . . . 5 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)))
5845, 50, 573syl 18 . . . 4 ((𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵) → (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)))
5958ancri 574 . . 3 ((𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵) → ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)))
6043, 59impbii 199 . 2 (((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)) ↔ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))
6142, 60bitri 264 1 (⟨𝐴, 𝐵⟩ <pQ𝐶, 𝐷⟩ ↔ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  cop 4161   class class class wbr 4623   × cxp 5082  cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  Ncnpi 9626   ·N cmi 9628   <N clti 9629   <pQ cltpq 9632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-omul 7525  df-ni 9654  df-mi 9656  df-lti 9657  df-ltpq 9692
This theorem is referenced by:  ordpinq  9725  lterpq  9752  ltanq  9753  ltmnq  9754  1lt2nq  9755
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