Theorem ltmnq 10881

Description: Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ltmnq	⊢ (𝐶 ∈ Q → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))

Proof of Theorem ltmnq

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

Step	Hyp	Ref	Expression
1		mulnqf 10858	. . 3 ⊢ ·Q :(Q × Q)⟶Q
2	1	fdmi 6671	. 2 ⊢ dom ·Q = (Q × Q)
3		ltrelnq 10835	. 2 ⊢ <Q ⊆ (Q × Q)
4		0nnq 10833	. 2 ⊢ ¬ ∅ ∈ Q
5		elpqn 10834	. . . . . . . . . 10 ⊢ (𝐶 ∈ Q → 𝐶 ∈ (N × N))
6	5	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 ∈ (N × N))
7		xp1st 7963	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
8	6, 7	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐶) ∈ N)
9		xp2nd 7964	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
10	6, 9	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐶) ∈ N)
11		mulclpi 10802	. . . . . . . 8 ⊢ (((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((1st ‘𝐶) ·N (2nd ‘𝐶)) ∈ N)
12	8, 10, 11	syl2anc 584	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1st ‘𝐶) ·N (2nd ‘𝐶)) ∈ N)
13		ltmpi 10813	. . . . . . 7 ⊢ (((1st ‘𝐶) ·N (2nd ‘𝐶)) ∈ N → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ (((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) <N (((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))))
14	12, 13	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ (((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) <N (((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))))
15		fvex 6845	. . . . . . . 8 ⊢ (1st ‘𝐶) ∈ V
16		fvex 6845	. . . . . . . 8 ⊢ (2nd ‘𝐶) ∈ V
17		fvex 6845	. . . . . . . 8 ⊢ (1st ‘𝐴) ∈ V
18		mulcompi 10805	. . . . . . . 8 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
19		mulasspi 10806	. . . . . . . 8 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
20		fvex 6845	. . . . . . . 8 ⊢ (2nd ‘𝐵) ∈ V
21	15, 16, 17, 18, 19, 20	caov4 7587	. . . . . . 7 ⊢ (((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = (((1st ‘𝐶) ·N (1st ‘𝐴)) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐵)))
22		fvex 6845	. . . . . . . 8 ⊢ (1st ‘𝐵) ∈ V
23		fvex 6845	. . . . . . . 8 ⊢ (2nd ‘𝐴) ∈ V
24	15, 16, 22, 18, 19, 23	caov4 7587	. . . . . . 7 ⊢ (((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) = (((1st ‘𝐶) ·N (1st ‘𝐵)) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐴)))
25	21, 24	breq12i 5105	. . . . . 6 ⊢ ((((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) <N (((1st ‘𝐶) ·N (2nd ‘𝐶)) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))) ↔ (((1st ‘𝐶) ·N (1st ‘𝐴)) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐵))) <N (((1st ‘𝐶) ·N (1st ‘𝐵)) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐴))))
26	14, 25	bitrdi 287	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ (((1st ‘𝐶) ·N (1st ‘𝐴)) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐵))) <N (((1st ‘𝐶) ·N (1st ‘𝐵)) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐴)))))
27		ordpipq 10851	. . . . 5 ⊢ (⟨((1st ‘𝐶) ·N (1st ‘𝐴)), ((2nd ‘𝐶) ·N (2nd ‘𝐴))⟩ <pQ ⟨((1st ‘𝐶) ·N (1st ‘𝐵)), ((2nd ‘𝐶) ·N (2nd ‘𝐵))⟩ ↔ (((1st ‘𝐶) ·N (1st ‘𝐴)) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐵))) <N (((1st ‘𝐶) ·N (1st ‘𝐵)) ·N ((2nd ‘𝐶) ·N (2nd ‘𝐴))))
28	26, 27	bitr4di 289	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ ⟨((1st ‘𝐶) ·N (1st ‘𝐴)), ((2nd ‘𝐶) ·N (2nd ‘𝐴))⟩ <pQ ⟨((1st ‘𝐶) ·N (1st ‘𝐵)), ((2nd ‘𝐶) ·N (2nd ‘𝐵))⟩))
29		elpqn 10834	. . . . . . 7 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
30	29	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 ∈ (N × N))
31		mulpipq2 10848	. . . . . 6 ⊢ ((𝐶 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐶 ·pQ 𝐴) = ⟨((1st ‘𝐶) ·N (1st ‘𝐴)), ((2nd ‘𝐶) ·N (2nd ‘𝐴))⟩)
32	6, 30, 31	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 ·pQ 𝐴) = ⟨((1st ‘𝐶) ·N (1st ‘𝐴)), ((2nd ‘𝐶) ·N (2nd ‘𝐴))⟩)
33		elpqn 10834	. . . . . . 7 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
34	33	3ad2ant2 1134	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐵 ∈ (N × N))
35		mulpipq2 10848	. . . . . 6 ⊢ ((𝐶 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐶 ·pQ 𝐵) = ⟨((1st ‘𝐶) ·N (1st ‘𝐵)), ((2nd ‘𝐶) ·N (2nd ‘𝐵))⟩)
36	6, 34, 35	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 ·pQ 𝐵) = ⟨((1st ‘𝐶) ·N (1st ‘𝐵)), ((2nd ‘𝐶) ·N (2nd ‘𝐵))⟩)
37	32, 36	breq12d 5109	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵) ↔ ⟨((1st ‘𝐶) ·N (1st ‘𝐴)), ((2nd ‘𝐶) ·N (2nd ‘𝐴))⟩ <pQ ⟨((1st ‘𝐶) ·N (1st ‘𝐵)), ((2nd ‘𝐶) ·N (2nd ‘𝐵))⟩))
38	28, 37	bitr4d 282	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ (𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵)))
39		ordpinq 10852	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
40	39	3adant3 1132	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
41		mulpqnq 10850	. . . . . . 7 ⊢ ((𝐶 ∈ Q ∧ 𝐴 ∈ Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
42	41	ancoms 458	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
43	42	3adant2 1131	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
44		mulpqnq 10850	. . . . . . 7 ⊢ ((𝐶 ∈ Q ∧ 𝐵 ∈ Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
45	44	ancoms 458	. . . . . 6 ⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
46	45	3adant1 1130	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
47	43, 46	breq12d 5109	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵) ↔ ([Q]‘(𝐶 ·pQ 𝐴)) <Q ([Q]‘(𝐶 ·pQ 𝐵))))
48		lterpq 10879	. . . 4 ⊢ ((𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵) ↔ ([Q]‘(𝐶 ·pQ 𝐴)) <Q ([Q]‘(𝐶 ·pQ 𝐵)))
49	47, 48	bitr4di 289	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵) ↔ (𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵)))
50	38, 40, 49	3bitr4d 311	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))
51	2, 3, 4, 50	ndmovord 7546	1 ⊢ (𝐶 ∈ Q → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ⟨cop 4584   class class class wbr 5096   × cxp 5620  ‘cfv 6490  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  Ncnpi 10753   ·_N cmi 10755   <_N clti 10756   ·_pQ cmpq 10758   <_pQ cltpq 10759  Qcnq 10761  [Q]cerq 10763   ·_Q cmq 10765   <_Q cltq 10767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-omul 8400  df-er 8633  df-ni 10781  df-mi 10783  df-lti 10784  df-mpq 10818  df-ltpq 10819  df-enq 10820  df-nq 10821  df-erq 10822  df-mq 10824  df-1nq 10825  df-ltnq 10827

This theorem is referenced by:  ltaddnq  10883  ltrnq  10888  addclprlem1  10925  mulclprlem  10928  mulclpr  10929  distrlem4pr  10935  1idpr  10938  prlem934  10942  prlem936  10956  reclem3pr  10958  reclem4pr  10959