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Theorem ordpipq 9976
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 5081 . . 3 𝐴, 𝐵⟩ ∈ V
2 opex 5081 . . 3 𝐶, 𝐷⟩ ∈ V
3 eleq1 2827 . . . . . 6 (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 ∈ (N × N) ↔ ⟨𝐴, 𝐵⟩ ∈ (N × N)))
43anbi1d 743 . . . . 5 (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))))
54anbi1d 743 . . . 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 6353 . . . . . . . 8 (𝑥 = ⟨𝐴, 𝐵⟩ → (1st𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
7 opelxp 5303 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩ ∈ (N × N) ↔ (𝐴N𝐵N))
8 op1stg 7346 . . . . . . . . . 10 ((𝐴N𝐵N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
97, 8sylbi 207 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
109adantr 472 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
116, 10sylan9eq 2814 . . . . . . 7 ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → (1st𝑥) = 𝐴)
1211oveq1d 6829 . . . . . 6 ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → ((1st𝑥) ·N (2nd𝑦)) = (𝐴 ·N (2nd𝑦)))
13 fveq2 6353 . . . . . . . 8 (𝑥 = ⟨𝐴, 𝐵⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
14 op2ndg 7347 . . . . . . . . . 10 ((𝐴N𝐵N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
157, 14sylbi 207 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
1615adantr 472 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
1713, 16sylan9eq 2814 . . . . . . 7 ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → (2nd𝑥) = 𝐵)
1817oveq2d 6830 . . . . . 6 ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → ((1st𝑦) ·N (2nd𝑥)) = ((1st𝑦) ·N 𝐵))
1912, 18breq12d 4817 . . . . 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 676 . . . 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 2827 . . . . . 6 (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 ∈ (N × N) ↔ ⟨𝐶, 𝐷⟩ ∈ (N × N)))
2322anbi2d 742 . . . . 5 (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))))
2423anbi1d 743 . . . 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 6353 . . . . . . . 8 (𝑦 = ⟨𝐶, 𝐷⟩ → (2nd𝑦) = (2nd ‘⟨𝐶, 𝐷⟩))
26 opelxp 5303 . . . . . . . . . 10 (⟨𝐶, 𝐷⟩ ∈ (N × N) ↔ (𝐶N𝐷N))
27 op2ndg 7347 . . . . . . . . . 10 ((𝐶N𝐷N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
2826, 27sylbi 207 . . . . . . . . 9 (⟨𝐶, 𝐷⟩ ∈ (N × N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
2928adantl 473 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
3025, 29sylan9eq 2814 . . . . . . 7 ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (2nd𝑦) = 𝐷)
3130oveq2d 6830 . . . . . 6 ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (𝐴 ·N (2nd𝑦)) = (𝐴 ·N 𝐷))
32 fveq2 6353 . . . . . . . 8 (𝑦 = ⟨𝐶, 𝐷⟩ → (1st𝑦) = (1st ‘⟨𝐶, 𝐷⟩))
33 op1stg 7346 . . . . . . . . . 10 ((𝐶N𝐷N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
3426, 33sylbi 207 . . . . . . . . 9 (⟨𝐶, 𝐷⟩ ∈ (N × N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
3534adantl 473 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
3632, 35sylan9eq 2814 . . . . . . 7 ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (1st𝑦) = 𝐶)
3736oveq1d 6829 . . . . . 6 ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → ((1st𝑦) ·N 𝐵) = (𝐶 ·N 𝐵))
3831, 37breq12d 4817 . . . . 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 676 . . . 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 9944 . . 3 <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))}
421, 2, 21, 40, 41brab 5148 . 2 (⟨𝐴, 𝐵⟩ <pQ𝐶, 𝐷⟩ ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)))
43 simpr 479 . . 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 9923 . . . . . 6 <N ⊆ (N × N)
4544brel 5325 . . . . 5 ((𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵) → ((𝐴 ·N 𝐷) ∈ N ∧ (𝐶 ·N 𝐵) ∈ N))
46 dmmulpi 9925 . . . . . . 7 dom ·N = (N × N)
47 0npi 9916 . . . . . . 7 ¬ ∅ ∈ N
4846, 47ndmovrcl 6986 . . . . . 6 ((𝐴 ·N 𝐷) ∈ N → (𝐴N𝐷N))
4946, 47ndmovrcl 6986 . . . . . 6 ((𝐶 ·N 𝐵) ∈ N → (𝐶N𝐵N))
5048, 49anim12i 591 . . . . 5 (((𝐴 ·N 𝐷) ∈ N ∧ (𝐶 ·N 𝐵) ∈ N) → ((𝐴N𝐷N) ∧ (𝐶N𝐵N)))
51 opelxpi 5305 . . . . . . 7 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
5251ad2ant2rl 802 . . . . . 6 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
53 simprl 811 . . . . . . 7 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → 𝐶N)
54 simplr 809 . . . . . . 7 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → 𝐷N)
55 opelxpi 5305 . . . . . . 7 ((𝐶N𝐷N) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
5653, 54, 55syl2anc 696 . . . . . 6 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
5752, 56jca 555 . . . . 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 576 . . 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 383   = wceq 1632  wcel 2139  cop 4327   class class class wbr 4804   × cxp 5264  cfv 6049  (class class class)co 6814  1st c1st 7332  2nd c2nd 7333  Ncnpi 9878   ·N cmi 9880   <N clti 9881   <pQ cltpq 9884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-omul 7735  df-ni 9906  df-mi 9908  df-lti 9909  df-ltpq 9944
This theorem is referenced by:  ordpinq  9977  lterpq  10004  ltanq  10005  ltmnq  10006  1lt2nq  10007
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