Theorem lterpq 11082

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 11022	. . . 4 ⊢ <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)))}
2		opabssxp 5748	. . . 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 3959	. . 3 ⊢ <pQ ⊆ ((N × N) × (N × N))
4	3	brel 5720	. 2 ⊢ (𝐴 <pQ 𝐵 → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
5		ltrelnq 11038	. . . 4 ⊢ <Q ⊆ (Q × Q)
6	5	brel 5720	. . 3 ⊢ (([Q]‘𝐴) <Q ([Q]‘𝐵) → (([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q))
7		elpqn 11037	. . . 4 ⊢ (([Q]‘𝐴) ∈ Q → ([Q]‘𝐴) ∈ (N × N))
8		elpqn 11037	. . . 4 ⊢ (([Q]‘𝐵) ∈ Q → ([Q]‘𝐵) ∈ (N × N))
9		nqerf 11042	. . . . . . 7 ⊢ [Q]:(N × N)⟶Q
10	9	fdmi 6743	. . . . . 6 ⊢ dom [Q] = (N × N)
11		0nelxp 5689	. . . . . 6 ⊢ ¬ ∅ ∈ (N × N)
12	10, 11	ndmfvrcl 6941	. . . . 5 ⊢ (([Q]‘𝐴) ∈ (N × N) → 𝐴 ∈ (N × N))
13	10, 11	ndmfvrcl 6941	. . . . 5 ⊢ (([Q]‘𝐵) ∈ (N × N) → 𝐵 ∈ (N × N))
14	12, 13	anim12i 625	. . . 4 ⊢ ((([Q]‘𝐴) ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N)) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
15	7, 8, 14	syl2an 608	. . 3 ⊢ ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
16	6, 15	syl 18	. 2 ⊢ (([Q]‘𝐴) <Q ([Q]‘𝐵) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
17		xp1st 8057	. . . . 5 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
18		xp2nd 8058	. . . . 5 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
19		mulclpi 11005	. . . . 5 ⊢ (((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
20	17, 18, 19	syl2an 608	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
21		ltmpi 11016	. . . 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 18	. . 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 11043	. . . 4 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
24		nqercl 11043	. . . 4 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q)
25		ordpinq 11055	. . . 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 608	. . 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 8064	. . . . . 6 ⊢ (𝐴 ∈ (N × N) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
28		1st2nd2 8064	. . . . . 6 ⊢ (𝐵 ∈ (N × N) → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
29	27, 28	breqan12d 5122	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ <pQ ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩))
30		ordpipq 11054	. . . . 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 290	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
32		xp1st 8057	. . . . . . 7 ⊢ (([Q]‘𝐴) ∈ (N × N) → (1st ‘([Q]‘𝐴)) ∈ N)
33	23, 7, 32	3syl 19	. . . . . 6 ⊢ (𝐴 ∈ (N × N) → (1st ‘([Q]‘𝐴)) ∈ N)
34		xp2nd 8058	. . . . . . 7 ⊢ (([Q]‘𝐵) ∈ (N × N) → (2nd ‘([Q]‘𝐵)) ∈ N)
35	24, 8, 34	3syl 19	. . . . . 6 ⊢ (𝐵 ∈ (N × N) → (2nd ‘([Q]‘𝐵)) ∈ N)
36		mulclpi 11005	. . . . . 6 ⊢ (((1st ‘([Q]‘𝐴)) ∈ N ∧ (2nd ‘([Q]‘𝐵)) ∈ N) → ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ∈ N)
37	33, 35, 36	syl2an 608	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ∈ N)
38		ltmpi 11016	. . . . 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 18	. . . 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 11008	. . . . . 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 11044	. . . . . . . . 9 ⊢ (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))
43	23, 7	syl 18	. . . . . . . . . 10 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ (N × N))
44		enqbreq2 11032	. . . . . . . . . 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 700	. . . . . . . . 9 ⊢ (𝐴 ∈ (N × N) → (𝐴 ~Q ([Q]‘𝐴) ↔ ((1st ‘𝐴) ·N (2nd ‘([Q]‘𝐴))) = ((1st ‘([Q]‘𝐴)) ·N (2nd ‘𝐴))))
46	42, 45	mpbid 235	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → ((1st ‘𝐴) ·N (2nd ‘([Q]‘𝐴))) = ((1st ‘([Q]‘𝐴)) ·N (2nd ‘𝐴)))
47	46	eqcomd 2769	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → ((1st ‘([Q]‘𝐴)) ·N (2nd ‘𝐴)) = ((1st ‘𝐴) ·N (2nd ‘([Q]‘𝐴))))
48		nqerrel 11044	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → 𝐵 ~Q ([Q]‘𝐵))
49	24, 8	syl 18	. . . . . . . . 9 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ (N × N))
50		enqbreq2 11032	. . . . . . . . 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 700	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → (𝐵 ~Q ([Q]‘𝐵) ↔ ((1st ‘𝐵) ·N (2nd ‘([Q]‘𝐵))) = ((1st ‘([Q]‘𝐵)) ·N (2nd ‘𝐵))))
52	48, 51	mpbid 235	. . . . . . 7 ⊢ (𝐵 ∈ (N × N) → ((1st ‘𝐵) ·N (2nd ‘([Q]‘𝐵))) = ((1st ‘([Q]‘𝐵)) ·N (2nd ‘𝐵)))
53	47, 52	oveqan12d 7462	. . . . . 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 11008	. . . . . . 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 6921	. . . . . . . 8 ⊢ (1st ‘𝐵) ∈ V
56		fvex 6921	. . . . . . . 8 ⊢ (2nd ‘𝐴) ∈ V
57		fvex 6921	. . . . . . . 8 ⊢ (1st ‘([Q]‘𝐴)) ∈ V
58		mulcompi 11008	. . . . . . . 8 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
59		mulasspi 11009	. . . . . . . 8 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
60		fvex 6921	. . . . . . . 8 ⊢ (2nd ‘([Q]‘𝐵)) ∈ V
61	55, 56, 57, 58, 59, 60	caov411 7677	. . . . . . 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 2786	. . . . . 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 11008	. . . . . . 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 6921	. . . . . . . 8 ⊢ (1st ‘([Q]‘𝐵)) ∈ V
65		fvex 6921	. . . . . . . 8 ⊢ (2nd ‘([Q]‘𝐴)) ∈ V
66		fvex 6921	. . . . . . . 8 ⊢ (1st ‘𝐴) ∈ V
67		fvex 6921	. . . . . . . 8 ⊢ (2nd ‘𝐵) ∈ V
68	64, 65, 66, 58, 59, 67	caov411 7677	. . . . . . 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 2786	. . . . . 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 2823	. . . . 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 5119	. . . 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 308	. . 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 315	. 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ ([Q]‘𝐴) <Q ([Q]‘𝐵)))
74	4, 16, 73	pm5.21nii 381	1 ⊢ (𝐴 <pQ 𝐵 ↔ ([Q]‘𝐴) <Q ([Q]‘𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ⟨cop 4838   class class class wbr 5106  {copab 5170   × cxp 5653  ‘cfv 6558  (class class class)co 7443  1^st c1st 8023  2^nd c2nd 8024  Ncnpi 10956   ·_N cmi 10958   <_N clti 10959   <_pQ cltpq 10962   ~_Q ceq 10963  Qcnq 10964  [Q]cerq 10966   <_Q cltq 10970

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-oadd 8508  df-omul 8509  df-er 8745  df-ni 10984  df-mi 10986  df-lti 10987  df-ltpq 11022  df-enq 11023  df-nq 11024  df-erq 11025  df-1nq 11028  df-ltnq 11030

This theorem is used by:  ltanq  11083  ltmnq  11084  1lt2nq  11085