Theorem ordpipq 10897

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 5430	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		opex 5430	. . 3 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
3		eleq1 2849	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 ∈ (N × N) ↔ ⟨𝐴, 𝐵⟩ ∈ (N × N)))
4	3	anbi1d 640	. . . . 5 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))))
5	4	anbi1d 640	. . . 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 6863	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
7		opelxp 5681	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) ↔ (𝐴 ∈ N ∧ 𝐵 ∈ N))
8		op1stg 7978	. . . . . . . . . 10 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
9	7, 8	sylbi 219	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
10	9	adantr 484	. . . . . . . 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 7407	. . . . . 6 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) = (𝐴 ·N (2nd ‘𝑦)))
13		fveq2 6863	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
14		op2ndg 7979	. . . . . . . . . 10 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
15	7, 14	sylbi 219	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
16	15	adantr 484	. . . . . . . 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 7408	. . . . . 6 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) = ((1st ‘𝑦) ·N 𝐵))
19	12, 18	breq12d 5112	. . . . 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 587	. . . 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 281	. . 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 639	. . . . 5 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))))
24	23	anbi1d 640	. . . 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 6863	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (2nd ‘𝑦) = (2nd ‘⟨𝐶, 𝐷⟩))
26		opelxp 5681	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐷⟩ ∈ (N × N) ↔ (𝐶 ∈ N ∧ 𝐷 ∈ N))
27		op2ndg 7979	. . . . . . . . . 10 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
28	26, 27	sylbi 219	. . . . . . . . 9 ⊢ (⟨𝐶, 𝐷⟩ ∈ (N × N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
29	28	adantl 485	. . . . . . . 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 7408	. . . . . 6 ⊢ ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (𝐴 ·N (2nd ‘𝑦)) = (𝐴 ·N 𝐷))
32		fveq2 6863	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (1st ‘𝑦) = (1st ‘⟨𝐶, 𝐷⟩))
33		op1stg 7978	. . . . . . . . . 10 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
34	26, 33	sylbi 219	. . . . . . . . 9 ⊢ (⟨𝐶, 𝐷⟩ ∈ (N × N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
35	34	adantl 485	. . . . . . . 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 7407	. . . . . 6 ⊢ ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → ((1st ‘𝑦) ·N 𝐵) = (𝐶 ·N 𝐵))
38	31, 37	breq12d 5112	. . . . 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 587	. . . 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 281	. . 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 10865	. . 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 5512	. 2 ⊢ (⟨𝐴, 𝐵⟩ <pQ ⟨𝐶, 𝐷⟩ ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)))
43		simpr 488	. . 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 10844	. . . . . 6 ⊢ <N ⊆ (N × N)
45	44	brel 5710	. . . . 5 ⊢ ((𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵) → ((𝐴 ·N 𝐷) ∈ N ∧ (𝐶 ·N 𝐵) ∈ N))
46		dmmulpi 10846	. . . . . . 7 ⊢ dom ·N = (N × N)
47		0npi 10837	. . . . . . 7 ⊢ ¬ ∅ ∈ N
48	46, 47	ndmovrcl 7578	. . . . . 6 ⊢ ((𝐴 ·N 𝐷) ∈ N → (𝐴 ∈ N ∧ 𝐷 ∈ N))
49	46, 47	ndmovrcl 7578	. . . . . 6 ⊢ ((𝐶 ·N 𝐵) ∈ N → (𝐶 ∈ N ∧ 𝐵 ∈ N))
50	48, 49	anim12i 622	. . . . 5 ⊢ (((𝐴 ·N 𝐷) ∈ N ∧ (𝐶 ·N 𝐵) ∈ N) → ((𝐴 ∈ N ∧ 𝐷 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐵 ∈ N)))
51		opelxpi 5682	. . . . . . 7 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
52	51	ad2ant2rl 759	. . . . . 6 ⊢ (((𝐴 ∈ N ∧ 𝐷 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐵 ∈ N)) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
53		simprl 780	. . . . . . 7 ⊢ (((𝐴 ∈ N ∧ 𝐷 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐵 ∈ N)) → 𝐶 ∈ N)
54		simplr 778	. . . . . . 7 ⊢ (((𝐴 ∈ N ∧ 𝐷 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐵 ∈ N)) → 𝐷 ∈ N)
55	53, 54	opelxpd 5684	. . . . . 6 ⊢ (((𝐴 ∈ N ∧ 𝐷 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐵 ∈ N)) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
56	52, 55	jca 519	. . . . 5 ⊢ (((𝐴 ∈ N ∧ 𝐷 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐵 ∈ N)) → (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)))
57	45, 50, 56	3syl 18	. . . 4 ⊢ ((𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵) → (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)))
58	57	ancri 557	. . 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 211	. 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 277	1 ⊢ (⟨𝐴, 𝐵⟩ <pQ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ⟨cop 4587   class class class wbr 5099   × cxp 5643  ‘cfv 6517  (class class class)co 7392  1^st c1st 7964  2^nd c2nd 7965  Ncnpi 10799   ·_N cmi 10801   <_N clti 10802   <_pQ cltpq 10805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-omul 8437  df-ni 10827  df-mi 10829  df-lti 10830  df-ltpq 10865

This theorem is referenced by:  ordpinq  10898  lterpq  10925  ltanq  10926  ltmnq  10927  1lt2nq  10928