Theorem ltmnq 10717

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 10694	. . 3 ⊢ ·Q :(Q × Q)⟶Q
2	1	fdmi 6606	. 2 ⊢ dom ·Q = (Q × Q)
3		ltrelnq 10671	. 2 ⊢ <Q ⊆ (Q × Q)
4		0nnq 10669	. 2 ⊢ ¬ ∅ ∈ Q
5		elpqn 10670	. . . . . . . . . 10 ⊢ (𝐶 ∈ Q → 𝐶 ∈ (N × N))
6	5	3ad2ant3 1134	. . . . . . . . 9 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐶 ∈ (N × N))
7		xp1st 7854	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (1st ‘𝐶) ∈ N)
8	6, 7	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (1st ‘𝐶) ∈ N)
9		xp2nd 7855	. . . . . . . . 9 ⊢ (𝐶 ∈ (N × N) → (2nd ‘𝐶) ∈ N)
10	6, 9	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (2nd ‘𝐶) ∈ N)
11		mulclpi 10638	. . . . . . . 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 10649	. . . . . . 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 6781	. . . . . . . 8 ⊢ (1st ‘𝐶) ∈ V
16		fvex 6781	. . . . . . . 8 ⊢ (2nd ‘𝐶) ∈ V
17		fvex 6781	. . . . . . . 8 ⊢ (1st ‘𝐴) ∈ V
18		mulcompi 10641	. . . . . . . 8 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
19		mulasspi 10642	. . . . . . . 8 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
20		fvex 6781	. . . . . . . 8 ⊢ (2nd ‘𝐵) ∈ V
21	15, 16, 17, 18, 19, 20	caov4 7495	. . . . . . 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 6781	. . . . . . . 8 ⊢ (1st ‘𝐵) ∈ V
23		fvex 6781	. . . . . . . 8 ⊢ (2nd ‘𝐴) ∈ V
24	15, 16, 22, 18, 19, 23	caov4 7495	. . . . . . 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 5084	. . . . . 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 10687	. . . . 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 10670	. . . . . . 7 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
30	29	3ad2ant1 1132	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐴 ∈ (N × N))
31		mulpipq2 10684	. . . . . 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 10670	. . . . . . 7 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
34	33	3ad2ant2 1133	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → 𝐵 ∈ (N × N))
35		mulpipq2 10684	. . . . . 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 5088	. . . 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 281	. . 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 10688	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
40	39	3adant3 1131	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
41		mulpqnq 10686	. . . . . . 7 ⊢ ((𝐶 ∈ Q ∧ 𝐴 ∈ Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
42	41	ancoms 459	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
43	42	3adant2 1130	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
44		mulpqnq 10686	. . . . . . 7 ⊢ ((𝐶 ∈ Q ∧ 𝐵 ∈ Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
45	44	ancoms 459	. . . . . 6 ⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
46	45	3adant1 1129	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐶 ∈ Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
47	43, 46	breq12d 5088	. . . 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 10715	. . . 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 7454	1 ⊢ (𝐶 ∈ Q → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ⟨cop 4569   class class class wbr 5075   × cxp 5584  ‘cfv 6428  (class class class)co 7269  1^st c1st 7820  2^nd c2nd 7821  Ncnpi 10589   ·_N cmi 10591   <_N clti 10592   ·_pQ cmpq 10594   <_pQ cltpq 10595  Qcnq 10597  [Q]cerq 10599   ·_Q cmq 10601   <_Q cltq 10603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7580

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5486  df-eprel 5492  df-po 5500  df-so 5501  df-fr 5541  df-we 5543  df-xp 5592  df-rel 5593  df-cnv 5594  df-co 5595  df-dm 5596  df-rn 5597  df-res 5598  df-ima 5599  df-pred 6197  df-ord 6264  df-on 6265  df-lim 6266  df-suc 6267  df-iota 6386  df-fun 6430  df-fn 6431  df-f 6432  df-f1 6433  df-fo 6434  df-f1o 6435  df-fv 6436  df-ov 7272  df-oprab 7273  df-mpo 7274  df-om 7705  df-1st 7822  df-2nd 7823  df-frecs 8086  df-wrecs 8117  df-recs 8191  df-rdg 8230  df-1o 8286  df-oadd 8290  df-omul 8291  df-er 8487  df-ni 10617  df-mi 10619  df-lti 10620  df-mpq 10654  df-ltpq 10655  df-enq 10656  df-nq 10657  df-erq 10658  df-mq 10660  df-1nq 10661  df-ltnq 10663

This theorem is referenced by:  ltaddnq  10719  ltrnq  10724  addclprlem1  10761  mulclprlem  10764  mulclpr  10765  distrlem4pr  10771  1idpr  10774  prlem934  10778  prlem936  10792  reclem3pr  10794  reclem4pr  10795