Theorem ltmnq 10131

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 10108	. . 3 ⊢ ·Q :(Q × Q)⟶Q
2	1	fdmi 6303	. 2 ⊢ dom ·Q = (Q × Q)
3		ltrelnq 10085	. 2 ⊢ <Q ⊆ (Q × Q)
4		0nnq 10083	. 2 ⊢ ¬ ∅ ∈ Q
5		elpqn 10084	. . . . . . . . . 10 ⊢ (𝐶 ∈ Q → 𝐶 ∈ (N × N))
6	5	3ad2ant3 1126	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 ∈ (N × N))
7		xp1st 7479	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
8	6, 7	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐶) ∈ N)
9		xp2nd 7480	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
10	6, 9	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐶) ∈ N)
11		mulclpi 10052	. . . . . . . 8 ⊢ (((1st ‘𝐶) ∈ N ∧ (2nd ‘𝐶) ∈ N) → ((1st ‘𝐶) ·N (2nd ‘𝐶)) ∈ N)
12	8, 10, 11	syl2anc 579	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ((1st ‘𝐶) ·N (2nd ‘𝐶)) ∈ N)
13		ltmpi 10063	. . . . . . 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 6461	. . . . . . . 8 ⊢ (1st ‘𝐶) ∈ V
16		fvex 6461	. . . . . . . 8 ⊢ (2nd ‘𝐶) ∈ V
17		fvex 6461	. . . . . . . 8 ⊢ (1st ‘𝐴) ∈ V
18		mulcompi 10055	. . . . . . . 8 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
19		mulasspi 10056	. . . . . . . 8 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
20		fvex 6461	. . . . . . . 8 ⊢ (2nd ‘𝐵) ∈ V
21	15, 16, 17, 18, 19, 20	caov4 7144	. . . . . . 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 6461	. . . . . . . 8 ⊢ (1st ‘𝐵) ∈ V
23		fvex 6461	. . . . . . . 8 ⊢ (2nd ‘𝐴) ∈ V
24	15, 16, 22, 18, 19, 23	caov4 7144	. . . . . . 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 4897	. . . . . 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	syl6bb 279	. . . . 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 10101	. . . . 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	syl6bbr 281	. . . 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 10084	. . . . . . 7 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
30	29	3ad2ant1 1124	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 ∈ (N × N))
31		mulpipq2 10098	. . . . . 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 579	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 ·pQ 𝐴) = ⟨((1st ‘𝐶) ·N (1st ‘𝐴)), ((2nd ‘𝐶) ·N (2nd ‘𝐴))⟩)
33		elpqn 10084	. . . . . . 7 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
34	33	3ad2ant2 1125	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐵 ∈ (N × N))
35		mulpipq2 10098	. . . . . 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 579	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 ·pQ 𝐵) = ⟨((1st ‘𝐶) ·N (1st ‘𝐵)), ((2nd ‘𝐶) ·N (2nd ‘𝐵))⟩)
37	32, 36	breq12d 4901	. . . 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 274	. . 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 10102	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
40	39	3adant3 1123	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
41		mulpqnq 10100	. . . . . . 7 ⊢ ((𝐶 ∈ Q ∧ 𝐴 ∈ Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
42	41	ancoms 452	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
43	42	3adant2 1122	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
44		mulpqnq 10100	. . . . . . 7 ⊢ ((𝐶 ∈ Q ∧ 𝐵 ∈ Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
45	44	ancoms 452	. . . . . 6 ⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
46	45	3adant1 1121	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
47	43, 46	breq12d 4901	. . . 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 10129	. . . 4 ⊢ ((𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵) ↔ ([Q]‘(𝐶 ·pQ 𝐴)) <Q ([Q]‘(𝐶 ·pQ 𝐵)))
49	47, 48	syl6bbr 281	. . 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 303	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))
51	2, 3, 4, 50	ndmovord 7103	1 ⊢ (𝐶 ∈ Q → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ⟨cop 4404   class class class wbr 4888   × cxp 5355  ‘cfv 6137  (class class class)co 6924  1^st c1st 7445  2^nd c2nd 7446  Ncnpi 10003   ·_N cmi 10005   <_N clti 10006   ·_pQ cmpq 10008   <_pQ cltpq 10009  Qcnq 10011  [Q]cerq 10013   ·_Q cmq 10015   <_Q cltq 10017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-omul 7850  df-er 8028  df-ni 10031  df-mi 10033  df-lti 10034  df-mpq 10068  df-ltpq 10069  df-enq 10070  df-nq 10071  df-erq 10072  df-mq 10074  df-1nq 10075  df-ltnq 10077

This theorem is referenced by:  ltaddnq  10133  ltrnq  10138  addclprlem1  10175  mulclprlem  10178  mulclpr  10179  distrlem4pr  10185  1idpr  10188  prlem934  10192  prlem936  10206  reclem3pr  10208  reclem4pr  10209