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Theorem ordpipq 10878
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 5421 . . 3 𝐴, 𝐵⟩ ∈ V
2 opex 5421 . . 3 𝐶, 𝐷⟩ ∈ V
3 eleq1 2825 . . . . . 6 (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 ∈ (N × N) ↔ ⟨𝐴, 𝐵⟩ ∈ (N × N)))
43anbi1d 630 . . . . 5 (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))))
54anbi1d 630 . . . 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 6842 . . . . . . . 8 (𝑥 = ⟨𝐴, 𝐵⟩ → (1st𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
7 opelxp 5669 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩ ∈ (N × N) ↔ (𝐴N𝐵N))
8 op1stg 7933 . . . . . . . . . 10 ((𝐴N𝐵N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
97, 8sylbi 216 . . . . . . . . 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 2796 . . . . . . 7 ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → (1st𝑥) = 𝐴)
1211oveq1d 7372 . . . . . 6 ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → ((1st𝑥) ·N (2nd𝑦)) = (𝐴 ·N (2nd𝑦)))
13 fveq2 6842 . . . . . . . 8 (𝑥 = ⟨𝐴, 𝐵⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
14 op2ndg 7934 . . . . . . . . . 10 ((𝐴N𝐵N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
157, 14sylbi 216 . . . . . . . . 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 2796 . . . . . . 7 ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → (2nd𝑥) = 𝐵)
1817oveq2d 7373 . . . . . 6 ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → ((1st𝑦) ·N (2nd𝑥)) = ((1st𝑦) ·N 𝐵))
1912, 18breq12d 5118 . . . . 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 579 . . . 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 278 . . 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 2825 . . . . . 6 (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 ∈ (N × N) ↔ ⟨𝐶, 𝐷⟩ ∈ (N × N)))
2322anbi2d 629 . . . . 5 (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))))
2423anbi1d 630 . . . 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 6842 . . . . . . . 8 (𝑦 = ⟨𝐶, 𝐷⟩ → (2nd𝑦) = (2nd ‘⟨𝐶, 𝐷⟩))
26 opelxp 5669 . . . . . . . . . 10 (⟨𝐶, 𝐷⟩ ∈ (N × N) ↔ (𝐶N𝐷N))
27 op2ndg 7934 . . . . . . . . . 10 ((𝐶N𝐷N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
2826, 27sylbi 216 . . . . . . . . 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 2796 . . . . . . 7 ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (2nd𝑦) = 𝐷)
3130oveq2d 7373 . . . . . 6 ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (𝐴 ·N (2nd𝑦)) = (𝐴 ·N 𝐷))
32 fveq2 6842 . . . . . . . 8 (𝑦 = ⟨𝐶, 𝐷⟩ → (1st𝑦) = (1st ‘⟨𝐶, 𝐷⟩))
33 op1stg 7933 . . . . . . . . . 10 ((𝐶N𝐷N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
3426, 33sylbi 216 . . . . . . . . 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 2796 . . . . . . 7 ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (1st𝑦) = 𝐶)
3736oveq1d 7372 . . . . . 6 ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → ((1st𝑦) ·N 𝐵) = (𝐶 ·N 𝐵))
3831, 37breq12d 5118 . . . . 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 579 . . . 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 278 . . 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 10846 . . 3 <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))}
421, 2, 21, 40, 41brab 5500 . 2 (⟨𝐴, 𝐵⟩ <pQ𝐶, 𝐷⟩ ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)))
43 simpr 485 . . 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 10825 . . . . . 6 <N ⊆ (N × N)
4544brel 5697 . . . . 5 ((𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵) → ((𝐴 ·N 𝐷) ∈ N ∧ (𝐶 ·N 𝐵) ∈ N))
46 dmmulpi 10827 . . . . . . 7 dom ·N = (N × N)
47 0npi 10818 . . . . . . 7 ¬ ∅ ∈ N
4846, 47ndmovrcl 7540 . . . . . 6 ((𝐴 ·N 𝐷) ∈ N → (𝐴N𝐷N))
4946, 47ndmovrcl 7540 . . . . . 6 ((𝐶 ·N 𝐵) ∈ N → (𝐶N𝐵N))
5048, 49anim12i 613 . . . . 5 (((𝐴 ·N 𝐷) ∈ N ∧ (𝐶 ·N 𝐵) ∈ N) → ((𝐴N𝐷N) ∧ (𝐶N𝐵N)))
51 opelxpi 5670 . . . . . . 7 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
5251ad2ant2rl 747 . . . . . 6 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
53 simprl 769 . . . . . . 7 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → 𝐶N)
54 simplr 767 . . . . . . 7 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → 𝐷N)
5553, 54opelxpd 5671 . . . . . 6 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
5652, 55jca 512 . . . . 5 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)))
5745, 50, 563syl 18 . . . 4 ((𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵) → (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)))
5857ancri 550 . . 3 ((𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵) → ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)))
5943, 58impbii 208 . 2 (((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)) ↔ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))
6042, 59bitri 274 1 (⟨𝐴, 𝐵⟩ <pQ𝐶, 𝐷⟩ ↔ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  cop 4592   class class class wbr 5105   × cxp 5631  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  Ncnpi 10780   ·N cmi 10782   <N clti 10783   <pQ cltpq 10786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-omul 8417  df-ni 10808  df-mi 10810  df-lti 10811  df-ltpq 10846
This theorem is referenced by:  ordpinq  10879  lterpq  10906  ltanq  10907  ltmnq  10908  1lt2nq  10909
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