Theorem lterpq 10964

Description: Compatibility of ordering on equivalent fractions. (Contributed by Mario Carneiro, 9-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
lterpq	⊢ (𝐴 <pQ 𝐵 ↔ ([Q]‘𝐴) <Q ([Q]‘𝐵))

Proof of Theorem lterpq

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ltpq 10904	. . . 4 ⊢ <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)))}
2		opabssxp 5761	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)))} ⊆ ((N × N) × (N × N))
3	1, 2	eqsstri 4011	. . 3 ⊢ <pQ ⊆ ((N × N) × (N × N))
4	3	brel 5734	. 2 ⊢ (𝐴 <pQ 𝐵 → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
5		ltrelnq 10920	. . . 4 ⊢ <Q ⊆ (Q × Q)
6	5	brel 5734	. . 3 ⊢ (([Q]‘𝐴) <Q ([Q]‘𝐵) → (([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q))
7		elpqn 10919	. . . 4 ⊢ (([Q]‘𝐴) ∈ Q → ([Q]‘𝐴) ∈ (N × N))
8		elpqn 10919	. . . 4 ⊢ (([Q]‘𝐵) ∈ Q → ([Q]‘𝐵) ∈ (N × N))
9		nqerf 10924	. . . . . . 7 ⊢ [Q]:(N × N)⟶Q
10	9	fdmi 6722	. . . . . 6 ⊢ dom [Q] = (N × N)
11		0nelxp 5703	. . . . . 6 ⊢ ¬ ∅ ∈ (N × N)
12	10, 11	ndmfvrcl 6920	. . . . 5 ⊢ (([Q]‘𝐴) ∈ (N × N) → 𝐴 ∈ (N × N))
13	10, 11	ndmfvrcl 6920	. . . . 5 ⊢ (([Q]‘𝐵) ∈ (N × N) → 𝐵 ∈ (N × N))
14	12, 13	anim12i 612	. . . 4 ⊢ ((([Q]‘𝐴) ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N)) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
15	7, 8, 14	syl2an 595	. . 3 ⊢ ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
16	6, 15	syl 17	. 2 ⊢ (([Q]‘𝐴) <Q ([Q]‘𝐵) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
17		xp1st 8003	. . . . 5 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
18		xp2nd 8004	. . . . 5 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
19		mulclpi 10887	. . . . 5 ⊢ (((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
20	17, 18, 19	syl2an 595	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
21		ltmpi 10898	. . . 4 ⊢ (((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N → (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) <N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))) ↔ (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵)))) <N (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))))))
22	20, 21	syl 17	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) <N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))) ↔ (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵)))) <N (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))))))
23		nqercl 10925	. . . 4 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
24		nqercl 10925	. . . 4 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q)
25		ordpinq 10937	. . . 4 ⊢ ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (([Q]‘𝐴) <Q ([Q]‘𝐵) ↔ ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) <N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴)))))
26	23, 24, 25	syl2an 595	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) <Q ([Q]‘𝐵) ↔ ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) <N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴)))))
27		1st2nd2 8010	. . . . . 6 ⊢ (𝐴 ∈ (N × N) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
28		1st2nd2 8010	. . . . . 6 ⊢ (𝐵 ∈ (N × N) → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
29	27, 28	breqan12d 5157	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ <pQ ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩))
30		ordpipq 10936	. . . . 5 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ <pQ ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)))
31	29, 30	bitrdi 287	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
32		xp1st 8003	. . . . . . 7 ⊢ (([Q]‘𝐴) ∈ (N × N) → (1st ‘([Q]‘𝐴)) ∈ N)
33	23, 7, 32	3syl 18	. . . . . 6 ⊢ (𝐴 ∈ (N × N) → (1st ‘([Q]‘𝐴)) ∈ N)
34		xp2nd 8004	. . . . . . 7 ⊢ (([Q]‘𝐵) ∈ (N × N) → (2nd ‘([Q]‘𝐵)) ∈ N)
35	24, 8, 34	3syl 18	. . . . . 6 ⊢ (𝐵 ∈ (N × N) → (2nd ‘([Q]‘𝐵)) ∈ N)
36		mulclpi 10887	. . . . . 6 ⊢ (((1st ‘([Q]‘𝐴)) ∈ N ∧ (2nd ‘([Q]‘𝐵)) ∈ N) → ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ∈ N)
37	33, 35, 36	syl2an 595	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ∈ N)
38		ltmpi 10898	. . . . 5 ⊢ (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ∈ N → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) <N (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))))
39	37, 38	syl 17	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) <N (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))))
40		mulcompi 10890	. . . . . 6 ⊢ (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))))
41	40	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵)))))
42		nqerrel 10926	. . . . . . . . 9 ⊢ (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))
43	23, 7	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ (N × N))
44		enqbreq2 10914	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (N × N) ∧ ([Q]‘𝐴) ∈ (N × N)) → (𝐴 ~Q ([Q]‘𝐴) ↔ ((1st ‘𝐴) ·N (2nd ‘([Q]‘𝐴))) = ((1st ‘([Q]‘𝐴)) ·N (2nd ‘𝐴))))
45	43, 44	mpdan 684	. . . . . . . . 9 ⊢ (𝐴 ∈ (N × N) → (𝐴 ~Q ([Q]‘𝐴) ↔ ((1st ‘𝐴) ·N (2nd ‘([Q]‘𝐴))) = ((1st ‘([Q]‘𝐴)) ·N (2nd ‘𝐴))))
46	42, 45	mpbid 231	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → ((1st ‘𝐴) ·N (2nd ‘([Q]‘𝐴))) = ((1st ‘([Q]‘𝐴)) ·N (2nd ‘𝐴)))
47	46	eqcomd 2732	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → ((1st ‘([Q]‘𝐴)) ·N (2nd ‘𝐴)) = ((1st ‘𝐴) ·N (2nd ‘([Q]‘𝐴))))
48		nqerrel 10926	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → 𝐵 ~Q ([Q]‘𝐵))
49	24, 8	syl 17	. . . . . . . . 9 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ (N × N))
50		enqbreq2 10914	. . . . . . . . 9 ⊢ ((𝐵 ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N)) → (𝐵 ~Q ([Q]‘𝐵) ↔ ((1st ‘𝐵) ·N (2nd ‘([Q]‘𝐵))) = ((1st ‘([Q]‘𝐵)) ·N (2nd ‘𝐵))))
51	49, 50	mpdan 684	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → (𝐵 ~Q ([Q]‘𝐵) ↔ ((1st ‘𝐵) ·N (2nd ‘([Q]‘𝐵))) = ((1st ‘([Q]‘𝐵)) ·N (2nd ‘𝐵))))
52	48, 51	mpbid 231	. . . . . . 7 ⊢ (𝐵 ∈ (N × N) → ((1st ‘𝐵) ·N (2nd ‘([Q]‘𝐵))) = ((1st ‘([Q]‘𝐵)) ·N (2nd ‘𝐵)))
53	47, 52	oveqan12d 7423	. . . . . 6 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (((1st ‘([Q]‘𝐴)) ·N (2nd ‘𝐴)) ·N ((1st ‘𝐵) ·N (2nd ‘([Q]‘𝐵)))) = (((1st ‘𝐴) ·N (2nd ‘([Q]‘𝐴))) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘𝐵))))
54		mulcompi 10890	. . . . . . 7 ⊢ (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) = (((1st ‘𝐵) ·N (2nd ‘𝐴)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))))
55		fvex 6897	. . . . . . . 8 ⊢ (1st ‘𝐵) ∈ V
56		fvex 6897	. . . . . . . 8 ⊢ (2nd ‘𝐴) ∈ V
57		fvex 6897	. . . . . . . 8 ⊢ (1st ‘([Q]‘𝐴)) ∈ V
58		mulcompi 10890	. . . . . . . 8 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
59		mulasspi 10891	. . . . . . . 8 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
60		fvex 6897	. . . . . . . 8 ⊢ (2nd ‘([Q]‘𝐵)) ∈ V
61	55, 56, 57, 58, 59, 60	caov411 7635	. . . . . . 7 ⊢ (((1st ‘𝐵) ·N (2nd ‘𝐴)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵)))) = (((1st ‘([Q]‘𝐴)) ·N (2nd ‘𝐴)) ·N ((1st ‘𝐵) ·N (2nd ‘([Q]‘𝐵))))
62	54, 61	eqtri 2754	. . . . . 6 ⊢ (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) = (((1st ‘([Q]‘𝐴)) ·N (2nd ‘𝐴)) ·N ((1st ‘𝐵) ·N (2nd ‘([Q]‘𝐵))))
63		mulcompi 10890	. . . . . . 7 ⊢ (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴)))) = (((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵)))
64		fvex 6897	. . . . . . . 8 ⊢ (1st ‘([Q]‘𝐵)) ∈ V
65		fvex 6897	. . . . . . . 8 ⊢ (2nd ‘([Q]‘𝐴)) ∈ V
66		fvex 6897	. . . . . . . 8 ⊢ (1st ‘𝐴) ∈ V
67		fvex 6897	. . . . . . . 8 ⊢ (2nd ‘𝐵) ∈ V
68	64, 65, 66, 58, 59, 67	caov411 7635	. . . . . . 7 ⊢ (((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((1st ‘𝐴) ·N (2nd ‘([Q]‘𝐴))) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘𝐵)))
69	63, 68	eqtri 2754	. . . . . 6 ⊢ (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴)))) = (((1st ‘𝐴) ·N (2nd ‘([Q]‘𝐴))) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘𝐵)))
70	53, 62, 69	3eqtr4g 2791	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) = (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴)))))
71	41, 70	breq12d 5154	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) <N (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ↔ (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵)))) <N (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))))))
72	31, 39, 71	3bitrd 305	. . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵)))) <N (((1st ‘𝐴) ·N (2nd ‘𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))))))
73	22, 26, 72	3bitr4rd 312	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ ([Q]‘𝐴) <Q ([Q]‘𝐵)))
74	4, 16, 73	pm5.21nii 378	1 ⊢ (𝐴 <pQ 𝐵 ↔ ([Q]‘𝐴) <Q ([Q]‘𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ⟨cop 4629   class class class wbr 5141  {copab 5203   × cxp 5667  ‘cfv 6536  (class class class)co 7404  1^st c1st 7969  2^nd c2nd 7970  Ncnpi 10838   ·_N cmi 10840   <_N clti 10841   <_pQ cltpq 10844   ~_Q ceq 10845  Qcnq 10846  [Q]cerq 10848   <_Q cltq 10852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-oadd 8468  df-omul 8469  df-er 8702  df-ni 10866  df-mi 10868  df-lti 10869  df-ltpq 10904  df-enq 10905  df-nq 10906  df-erq 10907  df-1nq 10910  df-ltnq 10912

This theorem is referenced by:  ltanq  10965  ltmnq  10966  1lt2nq  10967