Theorem lterpq 10943

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 10883	. . . 4 ⊢ <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)))}
2		opabssxp 5744	. . . 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 3985	. . 3 ⊢ <pQ ⊆ ((N × N) × (N × N))
4	3	brel 5717	. 2 ⊢ (𝐴 <pQ 𝐵 → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
5		ltrelnq 10899	. . . 4 ⊢ <Q ⊆ (Q × Q)
6	5	brel 5717	. . 3 ⊢ (([Q]‘𝐴) <Q ([Q]‘𝐵) → (([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q))
7		elpqn 10898	. . . 4 ⊢ (([Q]‘𝐴) ∈ Q → ([Q]‘𝐴) ∈ (N × N))
8		elpqn 10898	. . . 4 ⊢ (([Q]‘𝐵) ∈ Q → ([Q]‘𝐵) ∈ (N × N))
9		nqerf 10903	. . . . . . 7 ⊢ [Q]:(N × N)⟶Q
10	9	fdmi 6707	. . . . . 6 ⊢ dom [Q] = (N × N)
11		0nelxp 5686	. . . . . 6 ⊢ ¬ ∅ ∈ (N × N)
12	10, 11	ndmfvrcl 6904	. . . . 5 ⊢ (([Q]‘𝐴) ∈ (N × N) → 𝐴 ∈ (N × N))
13	10, 11	ndmfvrcl 6904	. . . . 5 ⊢ (([Q]‘𝐵) ∈ (N × N) → 𝐵 ∈ (N × N))
14	12, 13	anim12i 624	. . . 4 ⊢ ((([Q]‘𝐴) ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N)) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
15	7, 8, 14	syl2an 607	. . 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 8006	. . . . 5 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
18		xp2nd 8007	. . . . 5 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
19		mulclpi 10866	. . . . 5 ⊢ (((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
20	17, 18, 19	syl2an 607	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
21		ltmpi 10877	. . . 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 10904	. . . 4 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
24		nqercl 10904	. . . 4 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q)
25		ordpinq 10916	. . . 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 607	. . 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 8013	. . . . . 6 ⊢ (𝐴 ∈ (N × N) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
28		1st2nd2 8013	. . . . . 6 ⊢ (𝐵 ∈ (N × N) → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
29	27, 28	breqan12d 5121	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ <pQ ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩))
30		ordpipq 10915	. . . . 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 8006	. . . . . . 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 8007	. . . . . . 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 10866	. . . . . 6 ⊢ (((1st ‘([Q]‘𝐴)) ∈ N ∧ (2nd ‘([Q]‘𝐵)) ∈ N) → ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ∈ N)
37	33, 35, 36	syl2an 607	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ∈ N)
38		ltmpi 10877	. . . . 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 10869	. . . . . 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 10905	. . . . . . . . 9 ⊢ (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))
43	23, 7	syl 18	. . . . . . . . . 10 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ (N × N))
44		enqbreq2 10893	. . . . . . . . . 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 699	. . . . . . . . 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 2771	. . . . . . 7 ⊢ (𝐴 ∈ (N × N) → ((1st ‘([Q]‘𝐴)) ·N (2nd ‘𝐴)) = ((1st ‘𝐴) ·N (2nd ‘([Q]‘𝐴))))
48		nqerrel 10905	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → 𝐵 ~Q ([Q]‘𝐵))
49	24, 8	syl 18	. . . . . . . . 9 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ (N × N))
50		enqbreq2 10893	. . . . . . . . 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 699	. . . . . . . 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 7419	. . . . . 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 10869	. . . . . . 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 6884	. . . . . . . 8 ⊢ (1st ‘𝐵) ∈ V
56		fvex 6884	. . . . . . . 8 ⊢ (2nd ‘𝐴) ∈ V
57		fvex 6884	. . . . . . . 8 ⊢ (1st ‘([Q]‘𝐴)) ∈ V
58		mulcompi 10869	. . . . . . . 8 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
59		mulasspi 10870	. . . . . . . 8 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
60		fvex 6884	. . . . . . . 8 ⊢ (2nd ‘([Q]‘𝐵)) ∈ V
61	55, 56, 57, 58, 59, 60	caov411 7632	. . . . . . 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 2788	. . . . . 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 10869	. . . . . . 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 6884	. . . . . . . 8 ⊢ (1st ‘([Q]‘𝐵)) ∈ V
65		fvex 6884	. . . . . . . 8 ⊢ (2nd ‘([Q]‘𝐴)) ∈ V
66		fvex 6884	. . . . . . . 8 ⊢ (1st ‘𝐴) ∈ V
67		fvex 6884	. . . . . . . 8 ⊢ (2nd ‘𝐵) ∈ V
68	64, 65, 66, 58, 59, 67	caov411 7632	. . . . . . 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 2788	. . . . . 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 2825	. . . . 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 5118	. . . 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))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ⟨cop 4591   class class class wbr 5105  {copab 5167   × cxp 5650  ‘cfv 6525  (class class class)co 7400  1^st c1st 7972  2^nd c2nd 7973  Ncnpi 10817   ·_N cmi 10819   <_N clti 10820   <_pQ cltpq 10823   ~_Q ceq 10824  Qcnq 10825  [Q]cerq 10827   <_Q cltq 10831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-oadd 8445  df-omul 8446  df-er 8682  df-ni 10845  df-mi 10847  df-lti 10848  df-ltpq 10883  df-enq 10884  df-nq 10885  df-erq 10886  df-1nq 10889  df-ltnq 10891

This theorem is referenced by:  ltanq  10944  ltmnq  10945  1lt2nq  10946