Theorem ordpipq 10955

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⟩ <pQ ⟨𝐶, 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.

Step	Hyp	Ref	Expression
1		opex 5443	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		opex 5443	. . 3 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
3		eleq1 2850	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 ∈ (N × N) ↔ ⟨𝐴, 𝐵⟩ ∈ (N × N)))
4	3	anbi1d 643	. . . . 5 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))))
5	4	anbi1d 643	. . . 4 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥))) ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)))))
6		fveq2 6882	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
7		opelxp 5695	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) ↔ (𝐴 ∈ N ∧ 𝐵 ∈ N))
8		op1stg 8002	. . . . . . . . . 10 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
9	7, 8	sylbi 220	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
10	9	adantr 486	. . . . . . . 8 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
11	6, 10	sylan9eq 2817	. . . . . . 7 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → (1st ‘𝑥) = 𝐴)
12	11	oveq1d 7432	. . . . . 6 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) = (𝐴 ·N (2nd ‘𝑦)))
13		fveq2 6882	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
14		op2ndg 8003	. . . . . . . . . 10 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
15	7, 14	sylbi 220	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
16	15	adantr 486	. . . . . . . 8 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
17	13, 16	sylan9eq 2817	. . . . . . 7 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → (2nd ‘𝑥) = 𝐵)
18	17	oveq2d 7433	. . . . . 6 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) = ((1st ‘𝑦) ·N 𝐵))
19	12, 18	breq12d 5120	. . . . 5 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → (((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)) ↔ (𝐴 ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N 𝐵)))
20	19	pm5.32da 590	. . . 4 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥))) ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝐴 ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N 𝐵))))
21	5, 20	bitrd 282	. . 3 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥))) ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝐴 ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N 𝐵))))
22		eleq1 2850	. . . . . 6 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 ∈ (N × N) ↔ ⟨𝐶, 𝐷⟩ ∈ (N × N)))
23	22	anbi2d 642	. . . . 5 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))))
24	23	anbi1d 643	. . . 4 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝐴 ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N 𝐵)) ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N 𝐵))))
25		fveq2 6882	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (2nd ‘𝑦) = (2nd ‘⟨𝐶, 𝐷⟩))
26		opelxp 5695	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐷⟩ ∈ (N × N) ↔ (𝐶 ∈ N ∧ 𝐷 ∈ N))
27		op2ndg 8003	. . . . . . . . . 10 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
28	26, 27	sylbi 220	. . . . . . . . 9 ⊢ (⟨𝐶, 𝐷⟩ ∈ (N × N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
29	28	adantl 487	. . . . . . . 8 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
30	25, 29	sylan9eq 2817	. . . . . . 7 ⊢ ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (2nd ‘𝑦) = 𝐷)
31	30	oveq2d 7433	. . . . . 6 ⊢ ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (𝐴 ·N (2nd ‘𝑦)) = (𝐴 ·N 𝐷))
32		fveq2 6882	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (1st ‘𝑦) = (1st ‘⟨𝐶, 𝐷⟩))
33		op1stg 8002	. . . . . . . . . 10 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
34	26, 33	sylbi 220	. . . . . . . . 9 ⊢ (⟨𝐶, 𝐷⟩ ∈ (N × N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
35	34	adantl 487	. . . . . . . 8 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
36	32, 35	sylan9eq 2817	. . . . . . 7 ⊢ ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (1st ‘𝑦) = 𝐶)
37	36	oveq1d 7432	. . . . . 6 ⊢ ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → ((1st ‘𝑦) ·N 𝐵) = (𝐶 ·N 𝐵))
38	31, 37	breq12d 5120	. . . . 5 ⊢ ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → ((𝐴 ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N 𝐵) ↔ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)))
39	38	pm5.32da 590	. . . 4 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N 𝐵)) ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))))
40	24, 39	bitrd 282	. . 3 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝐴 ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N 𝐵)) ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))))
41		df-ltpq 10923	. . 3 ⊢ <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)))}
42	1, 2, 21, 40, 41	brab 5526	. 2 ⊢ (⟨𝐴, 𝐵⟩ <pQ ⟨𝐶, 𝐷⟩ ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)))
43		simpr 490	. . 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 10902	. . . . . 6 ⊢ <N ⊆ (N × N)
45	44	brel 5724	. . . . 5 ⊢ ((𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵) → ((𝐴 ·N 𝐷) ∈ N ∧ (𝐶 ·N 𝐵) ∈ N))
46		dmmulpi 10904	. . . . . . 7 ⊢ dom ·N = (N × N)
47		0npi 10895	. . . . . . 7 ⊢ ¬ ∅ ∈ N
48	46, 47	ndmovrcl 7604	. . . . . 6 ⊢ ((𝐴 ·N 𝐷) ∈ N → (𝐴 ∈ N ∧ 𝐷 ∈ N))
49	46, 47	ndmovrcl 7604	. . . . . 6 ⊢ ((𝐶 ·N 𝐵) ∈ N → (𝐶 ∈ N ∧ 𝐵 ∈ N))
50	48, 49	anim12i 625	. . . . 5 ⊢ (((𝐴 ·N 𝐷) ∈ N ∧ (𝐶 ·N 𝐵) ∈ N) → ((𝐴 ∈ N ∧ 𝐷 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐵 ∈ N)))
51		opelxpi 5696	. . . . . . 7 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
52	51	ad2ant2rl 762	. . . . . 6 ⊢ (((𝐴 ∈ N ∧ 𝐷 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐵 ∈ N)) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
53		simprl 783	. . . . . . 7 ⊢ (((𝐴 ∈ N ∧ 𝐷 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐵 ∈ N)) → 𝐶 ∈ N)
54		simplr 781	. . . . . . 7 ⊢ (((𝐴 ∈ N ∧ 𝐷 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐵 ∈ N)) → 𝐷 ∈ N)
55	53, 54	opelxpd 5698	. . . . . 6 ⊢ (((𝐴 ∈ N ∧ 𝐷 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐵 ∈ N)) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
56	52, 55	jca 521	. . . . 5 ⊢ (((𝐴 ∈ N ∧ 𝐷 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐵 ∈ N)) → (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)))
57	45, 50, 56	3syl 19	. . . 4 ⊢ ((𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵) → (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)))
58	57	ancri 559	. . 3 ⊢ ((𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵) → ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)))
59	43, 58	impbii 212	. 2 ⊢ (((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)) ↔ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))
60	42, 59	bitri 278	1 ⊢ (⟨𝐴, 𝐵⟩ <pQ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ⟨cop 4593   class class class wbr 5107   × cxp 5657  ‘cfv 6537  (class class class)co 7417  1^st c1st 7988  2^nd c2nd 7989  Ncnpi 10857   ·_N cmi 10859   <_N clti 10860   <_pQ cltpq 10863

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-omul 8464  df-ni 10885  df-mi 10887  df-lti 10888  df-ltpq 10923

This theorem is used by:  ordpinq  10956  lterpq  10983  ltanq  10984  ltmnq  10985  1lt2nq  10986