Theorem ordpipq 11008

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 5432	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		opex 5432	. . 3 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
3		eleq1 2849	. . . . . 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 6877	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
7		opelxp 5687	. . . . . . . . . 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 2816	. . . . . . 7 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → (1st ‘𝑥) = 𝐴)
12	11	oveq1d 7427	. . . . . 6 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) = (𝐴 ·N (2nd ‘𝑦)))
13		fveq2 6877	. . . . . . . 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 2816	. . . . . . 7 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → (2nd ‘𝑥) = 𝐵)
18	17	oveq2d 7428	. . . . . 6 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) = ((1st ‘𝑦) ·N 𝐵))
19	12, 18	breq12d 5116	. . . . 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 2849	. . . . . 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 6877	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (2nd ‘𝑦) = (2nd ‘⟨𝐶, 𝐷⟩))
26		opelxp 5687	. . . . . . . . . 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 2816	. . . . . . 7 ⊢ ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (2nd ‘𝑦) = 𝐷)
31	30	oveq2d 7428	. . . . . 6 ⊢ ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (𝐴 ·N (2nd ‘𝑦)) = (𝐴 ·N 𝐷))
32		fveq2 6877	. . . . . . . 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 2816	. . . . . . 7 ⊢ ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (1st ‘𝑦) = 𝐶)
37	36	oveq1d 7427	. . . . . 6 ⊢ ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → ((1st ‘𝑦) ·N 𝐵) = (𝐶 ·N 𝐵))
38	31, 37	breq12d 5116	. . . . 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 10976	. . 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 5518	. 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 10955	. . . . . 6 ⊢ <N ⊆ (N × N)
45	44	brel 5716	. . . . 5 ⊢ ((𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵) → ((𝐴 ·N 𝐷) ∈ N ∧ (𝐶 ·N 𝐵) ∈ N))
46		dmmulpi 10957	. . . . . . 7 ⊢ dom ·N = (N × N)
47		0npi 10948	. . . . . . 7 ⊢ ¬ ∅ ∈ N
48	46, 47	ndmovrcl 7599	. . . . . 6 ⊢ ((𝐴 ·N 𝐷) ∈ N → (𝐴 ∈ N ∧ 𝐷 ∈ N))
49	46, 47	ndmovrcl 7599	. . . . . 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 5688	. . . . . . 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 5690	. . . . . 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 4590   class class class wbr 5103   × cxp 5649  ‘cfv 6531  (class class class)co 7412  1^st c1st 7988  2^nd c2nd 7989  Ncnpi 10910   ·_N cmi 10912   <_N clti 10913   <_pQ cltpq 10916

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 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-omul 8465  df-ni 10938  df-mi 10940  df-lti 10941  df-ltpq 10976

This theorem is used by:  ordpinq  11009  lterpq  11036  ltanq  11037  ltmnq  11038  1lt2nq  11039