Theorem ltmnq 10883

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 10860	. . 3 ⊢ ·Q :(Q × Q)⟶Q
2	1	fdmi 6673	. 2 ⊢ dom ·Q = (Q × Q)
3		ltrelnq 10837	. 2 ⊢ <Q ⊆ (Q × Q)
4		0nnq 10835	. 2 ⊢ ¬ ∅ ∈ Q
5		elpqn 10836	. . . . . . . . . 10 ⊢ (𝐶 ∈ Q → 𝐶 ∈ (N × N))
6	5	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 ∈ (N × N))
7		xp1st 7965	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
8	6, 7	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐶) ∈ N)
9		xp2nd 7966	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
10	6, 9	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐶) ∈ N)
11		mulclpi 10804	. . . . . . . 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 10815	. . . . . . 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 6847	. . . . . . . 8 ⊢ (1st ‘𝐶) ∈ V
16		fvex 6847	. . . . . . . 8 ⊢ (2nd ‘𝐶) ∈ V
17		fvex 6847	. . . . . . . 8 ⊢ (1st ‘𝐴) ∈ V
18		mulcompi 10807	. . . . . . . 8 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
19		mulasspi 10808	. . . . . . . 8 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
20		fvex 6847	. . . . . . . 8 ⊢ (2nd ‘𝐵) ∈ V
21	15, 16, 17, 18, 19, 20	caov4 7589	. . . . . . 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 6847	. . . . . . . 8 ⊢ (1st ‘𝐵) ∈ V
23		fvex 6847	. . . . . . . 8 ⊢ (2nd ‘𝐴) ∈ V
24	15, 16, 22, 18, 19, 23	caov4 7589	. . . . . . 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 5107	. . . . . 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 10853	. . . . 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 10836	. . . . . . 7 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
30	29	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 ∈ (N × N))
31		mulpipq2 10850	. . . . . 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 10836	. . . . . . 7 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
34	33	3ad2ant2 1134	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐵 ∈ (N × N))
35		mulpipq2 10850	. . . . . 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 5111	. . . 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 10854	. . . 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 10852	. . . . . . 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 10852	. . . . . . 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 5111	. . . 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 10881	. . . 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 7548	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 4586   class class class wbr 5098   × cxp 5622  ‘cfv 6492  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  Ncnpi 10755   ·_N cmi 10757   <_N clti 10758   ·_pQ cmpq 10760   <_pQ cltpq 10761  Qcnq 10763  [Q]cerq 10765   ·_Q cmq 10767   <_Q cltq 10769

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-omul 8402  df-er 8635  df-ni 10783  df-mi 10785  df-lti 10786  df-mpq 10820  df-ltpq 10821  df-enq 10822  df-nq 10823  df-erq 10824  df-mq 10826  df-1nq 10827  df-ltnq 10829

This theorem is referenced by:  ltaddnq  10885  ltrnq  10890  addclprlem1  10927  mulclprlem  10930  mulclpr  10931  distrlem4pr  10937  1idpr  10940  prlem934  10944  prlem936  10958  reclem3pr  10960  reclem4pr  10961