Theorem ltmnq 10925

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 10902	. . 3 ⊢ ·Q :(Q × Q)⟶Q
2	1	fdmi 6699	. 2 ⊢ dom ·Q = (Q × Q)
3		ltrelnq 10879	. 2 ⊢ <Q ⊆ (Q × Q)
4		0nnq 10877	. 2 ⊢ ¬ ∅ ∈ Q
5		elpqn 10878	. . . . . . . . . 10 ⊢ (𝐶 ∈ Q → 𝐶 ∈ (N × N))
6	5	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 ∈ (N × N))
7		xp1st 8000	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
8	6, 7	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐶) ∈ N)
9		xp2nd 8001	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
10	6, 9	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐶) ∈ N)
11		mulclpi 10846	. . . . . . . 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 10857	. . . . . . 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 6871	. . . . . . . 8 ⊢ (1st ‘𝐶) ∈ V
16		fvex 6871	. . . . . . . 8 ⊢ (2nd ‘𝐶) ∈ V
17		fvex 6871	. . . . . . . 8 ⊢ (1st ‘𝐴) ∈ V
18		mulcompi 10849	. . . . . . . 8 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
19		mulasspi 10850	. . . . . . . 8 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
20		fvex 6871	. . . . . . . 8 ⊢ (2nd ‘𝐵) ∈ V
21	15, 16, 17, 18, 19, 20	caov4 7620	. . . . . . 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 6871	. . . . . . . 8 ⊢ (1st ‘𝐵) ∈ V
23		fvex 6871	. . . . . . . 8 ⊢ (2nd ‘𝐴) ∈ V
24	15, 16, 22, 18, 19, 23	caov4 7620	. . . . . . 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 5116	. . . . . 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 10895	. . . . 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 10878	. . . . . . 7 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
30	29	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 ∈ (N × N))
31		mulpipq2 10892	. . . . . 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 10878	. . . . . . 7 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
34	33	3ad2ant2 1134	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐵 ∈ (N × N))
35		mulpipq2 10892	. . . . . 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 5120	. . . 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 10896	. . . 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 10894	. . . . . . 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 10894	. . . . . . 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 5120	. . . 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 10923	. . . 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 7579	1 ⊢ (𝐶 ∈ Q → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ⟨cop 4595   class class class wbr 5107   × cxp 5636  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  Ncnpi 10797   ·_N cmi 10799   <_N clti 10800   ·_pQ cmpq 10802   <_pQ cltpq 10803  Qcnq 10805  [Q]cerq 10807   ·_Q cmq 10809   <_Q cltq 10811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-oadd 8438  df-omul 8439  df-er 8671  df-ni 10825  df-mi 10827  df-lti 10828  df-mpq 10862  df-ltpq 10863  df-enq 10864  df-nq 10865  df-erq 10866  df-mq 10868  df-1nq 10869  df-ltnq 10871

This theorem is referenced by:  ltaddnq  10927  ltrnq  10932  addclprlem1  10969  mulclprlem  10972  mulclpr  10973  distrlem4pr  10979  1idpr  10982  prlem934  10986  prlem936  11000  reclem3pr  11002  reclem4pr  11003