Theorem ordpipq 10902

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 5427	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		opex 5427	. . 3 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
3		eleq1 2817	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 ∈ (N × N) ↔ ⟨𝐴, 𝐵⟩ ∈ (N × N)))
4	3	anbi1d 631	. . . . 5 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))))
5	4	anbi1d 631	. . . 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 6861	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (1st ‘𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
7		opelxp 5677	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) ↔ (𝐴 ∈ N ∧ 𝐵 ∈ N))
8		op1stg 7983	. . . . . . . . . 10 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
9	7, 8	sylbi 217	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
10	9	adantr 480	. . . . . . . 8 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
11	6, 10	sylan9eq 2785	. . . . . . 7 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → (1st ‘𝑥) = 𝐴)
12	11	oveq1d 7405	. . . . . 6 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → ((1st ‘𝑥) ·N (2nd ‘𝑦)) = (𝐴 ·N (2nd ‘𝑦)))
13		fveq2 6861	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 𝐵⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
14		op2ndg 7984	. . . . . . . . . 10 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
15	7, 14	sylbi 217	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ ∈ (N × N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
16	15	adantr 480	. . . . . . . 8 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
17	13, 16	sylan9eq 2785	. . . . . . 7 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → (2nd ‘𝑥) = 𝐵)
18	17	oveq2d 7406	. . . . . 6 ⊢ ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → ((1st ‘𝑦) ·N (2nd ‘𝑥)) = ((1st ‘𝑦) ·N 𝐵))
19	12, 18	breq12d 5123	. . . . 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 579	. . . 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 279	. . 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 2817	. . . . . 6 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 ∈ (N × N) ↔ ⟨𝐶, 𝐷⟩ ∈ (N × N)))
23	22	anbi2d 630	. . . . 5 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))))
24	23	anbi1d 631	. . . 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 6861	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (2nd ‘𝑦) = (2nd ‘⟨𝐶, 𝐷⟩))
26		opelxp 5677	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐷⟩ ∈ (N × N) ↔ (𝐶 ∈ N ∧ 𝐷 ∈ N))
27		op2ndg 7984	. . . . . . . . . 10 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
28	26, 27	sylbi 217	. . . . . . . . 9 ⊢ (⟨𝐶, 𝐷⟩ ∈ (N × N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
29	28	adantl 481	. . . . . . . 8 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
30	25, 29	sylan9eq 2785	. . . . . . 7 ⊢ ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (2nd ‘𝑦) = 𝐷)
31	30	oveq2d 7406	. . . . . 6 ⊢ ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (𝐴 ·N (2nd ‘𝑦)) = (𝐴 ·N 𝐷))
32		fveq2 6861	. . . . . . . 8 ⊢ (𝑦 = ⟨𝐶, 𝐷⟩ → (1st ‘𝑦) = (1st ‘⟨𝐶, 𝐷⟩))
33		op1stg 7983	. . . . . . . . . 10 ⊢ ((𝐶 ∈ N ∧ 𝐷 ∈ N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
34	26, 33	sylbi 217	. . . . . . . . 9 ⊢ (⟨𝐶, 𝐷⟩ ∈ (N × N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
35	34	adantl 481	. . . . . . . 8 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
36	32, 35	sylan9eq 2785	. . . . . . 7 ⊢ ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (1st ‘𝑦) = 𝐶)
37	36	oveq1d 7405	. . . . . 6 ⊢ ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → ((1st ‘𝑦) ·N 𝐵) = (𝐶 ·N 𝐵))
38	31, 37	breq12d 5123	. . . . 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 579	. . . 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 279	. . 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 10870	. . 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 5506	. 2 ⊢ (⟨𝐴, 𝐵⟩ <pQ ⟨𝐶, 𝐷⟩ ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)))
43		simpr 484	. . 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 10849	. . . . . 6 ⊢ <N ⊆ (N × N)
45	44	brel 5706	. . . . 5 ⊢ ((𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵) → ((𝐴 ·N 𝐷) ∈ N ∧ (𝐶 ·N 𝐵) ∈ N))
46		dmmulpi 10851	. . . . . . 7 ⊢ dom ·N = (N × N)
47		0npi 10842	. . . . . . 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 613	. . . . 5 ⊢ (((𝐴 ·N 𝐷) ∈ N ∧ (𝐶 ·N 𝐵) ∈ N) → ((𝐴 ∈ N ∧ 𝐷 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐵 ∈ N)))
51		opelxpi 5678	. . . . . . 7 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
52	51	ad2ant2rl 749	. . . . . 6 ⊢ (((𝐴 ∈ N ∧ 𝐷 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐵 ∈ N)) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
53		simprl 770	. . . . . . 7 ⊢ (((𝐴 ∈ N ∧ 𝐷 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐵 ∈ N)) → 𝐶 ∈ N)
54		simplr 768	. . . . . . 7 ⊢ (((𝐴 ∈ N ∧ 𝐷 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐵 ∈ N)) → 𝐷 ∈ N)
55	53, 54	opelxpd 5680	. . . . . 6 ⊢ (((𝐴 ∈ N ∧ 𝐷 ∈ N) ∧ (𝐶 ∈ N ∧ 𝐵 ∈ N)) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
56	52, 55	jca 511	. . . . 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 549	. . 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 209	. 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 275	1 ⊢ (⟨𝐴, 𝐵⟩ <pQ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4598   class class class wbr 5110   × cxp 5639  ‘cfv 6514  (class class class)co 7390  1^st c1st 7969  2^nd c2nd 7970  Ncnpi 10804   ·_N cmi 10806   <_N clti 10807   <_pQ cltpq 10810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-omul 8442  df-ni 10832  df-mi 10834  df-lti 10835  df-ltpq 10870

This theorem is referenced by:  ordpinq  10903  lterpq  10930  ltanq  10931  ltmnq  10932  1lt2nq  10933