Theorem lterpq 11036

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 10976	. . . 4 ⊢ <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)))}
2		opabssxp 5743	. . . 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 3977	. . 3 ⊢ <pQ ⊆ ((N × N) × (N × N))
4	3	brel 5716	. 2 ⊢ (𝐴 <pQ 𝐵 → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
5		ltrelnq 10992	. . . 4 ⊢ <Q ⊆ (Q × Q)
6	5	brel 5716	. . 3 ⊢ (([Q]‘𝐴) <Q ([Q]‘𝐵) → (([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q))
7		elpqn 10991	. . . 4 ⊢ (([Q]‘𝐴) ∈ Q → ([Q]‘𝐴) ∈ (N × N))
8		elpqn 10991	. . . 4 ⊢ (([Q]‘𝐵) ∈ Q → ([Q]‘𝐵) ∈ (N × N))
9		nqerf 10996	. . . . . . 7 ⊢ [Q]:(N × N)⟶Q
10	9	fdmi 6713	. . . . . 6 ⊢ dom [Q] = (N × N)
11		0nelxp 5685	. . . . . 6 ⊢ ¬ ∅ ∈ (N × N)
12	10, 11	ndmfvrcl 6910	. . . . 5 ⊢ (([Q]‘𝐴) ∈ (N × N) → 𝐴 ∈ (N × N))
13	10, 11	ndmfvrcl 6910	. . . . 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 8022	. . . . 5 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
18		xp2nd 8023	. . . . 5 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
19		mulclpi 10959	. . . . 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 10970	. . . 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 10997	. . . 4 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
24		nqercl 10997	. . . 4 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q)
25		ordpinq 11009	. . . 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 8029	. . . . . 6 ⊢ (𝐴 ∈ (N × N) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
28		1st2nd2 8029	. . . . . 6 ⊢ (𝐵 ∈ (N × N) → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
29	27, 28	breqan12d 5119	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ <pQ ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩))
30		ordpipq 11008	. . . . 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 8022	. . . . . . 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 8023	. . . . . . 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 10959	. . . . . 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 10970	. . . . 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 10962	. . . . . 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 10998	. . . . . . . . 9 ⊢ (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))
43	23, 7	syl 18	. . . . . . . . . 10 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ (N × N))
44		enqbreq2 10986	. . . . . . . . . 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 2767	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → ((1st ‘([Q]‘𝐴)) ·N (2nd ‘𝐴)) = ((1st ‘𝐴) ·N (2nd ‘([Q]‘𝐴))))
48		nqerrel 10998	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → 𝐵 ~Q ([Q]‘𝐵))
49	24, 8	syl 18	. . . . . . . . 9 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ (N × N))
50		enqbreq2 10986	. . . . . . . . 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 7431	. . . . . 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 10962	. . . . . . 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 6890	. . . . . . . 8 ⊢ (1st ‘𝐵) ∈ V
56		fvex 6890	. . . . . . . 8 ⊢ (2nd ‘𝐴) ∈ V
57		fvex 6890	. . . . . . . 8 ⊢ (1st ‘([Q]‘𝐴)) ∈ V
58		mulcompi 10962	. . . . . . . 8 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
59		mulasspi 10963	. . . . . . . 8 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
60		fvex 6890	. . . . . . . 8 ⊢ (2nd ‘([Q]‘𝐵)) ∈ V
61	55, 56, 57, 58, 59, 60	caov411 7645	. . . . . . 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 2784	. . . . . 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 10962	. . . . . . 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 6890	. . . . . . . 8 ⊢ (1st ‘([Q]‘𝐵)) ∈ V
65		fvex 6890	. . . . . . . 8 ⊢ (2nd ‘([Q]‘𝐴)) ∈ V
66		fvex 6890	. . . . . . . 8 ⊢ (1st ‘𝐴) ∈ V
67		fvex 6890	. . . . . . . 8 ⊢ (2nd ‘𝐵) ∈ V
68	64, 65, 66, 58, 59, 67	caov411 7645	. . . . . . 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 2784	. . . . . 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 2821	. . . . 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 5116	. . . 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 2145  ⟨cop 4590   class class class wbr 5103  {copab 5167   × 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   ~_Q ceq 10917  Qcnq 10918  [Q]cerq 10920   <_Q cltq 10924

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-3or 1104  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-rmo 3366  df-reu 3367  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-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  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-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  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-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-oadd 8464  df-omul 8465  df-er 8701  df-ni 10938  df-mi 10940  df-lti 10941  df-ltpq 10976  df-enq 10977  df-nq 10978  df-erq 10979  df-1nq 10982  df-ltnq 10984

This theorem is used by:  ltanq  11037  ltmnq  11038  1lt2nq  11039