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Theorem ltmnq 11012
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 (𝐶Q → (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.
StepHypRef Expression
1 mulnqf 10989 . . 3 ·Q :(Q × Q)⟶Q
21fdmi 6747 . 2 dom ·Q = (Q × Q)
3 ltrelnq 10966 . 2 <Q ⊆ (Q × Q)
4 0nnq 10964 . 2 ¬ ∅ ∈ Q
5 elpqn 10965 . . . . . . . . . 10 (𝐶Q𝐶 ∈ (N × N))
653ad2ant3 1136 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐶 ∈ (N × N))
7 xp1st 8046 . . . . . . . . 9 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
86, 7syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (1st𝐶) ∈ N)
9 xp2nd 8047 . . . . . . . . 9 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
106, 9syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐶) ∈ N)
11 mulclpi 10933 . . . . . . . 8 (((1st𝐶) ∈ N ∧ (2nd𝐶) ∈ N) → ((1st𝐶) ·N (2nd𝐶)) ∈ N)
128, 10, 11syl2anc 584 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐶) ·N (2nd𝐶)) ∈ N)
13 ltmpi 10944 . . . . . . 7 (((1st𝐶) ·N (2nd𝐶)) ∈ N → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))))
1412, 13syl 17 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴)))))
15 fvex 6919 . . . . . . . 8 (1st𝐶) ∈ V
16 fvex 6919 . . . . . . . 8 (2nd𝐶) ∈ V
17 fvex 6919 . . . . . . . 8 (1st𝐴) ∈ V
18 mulcompi 10936 . . . . . . . 8 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
19 mulasspi 10937 . . . . . . . 8 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
20 fvex 6919 . . . . . . . 8 (2nd𝐵) ∈ V
2115, 16, 17, 18, 19, 20caov4 7664 . . . . . . 7 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐶) ·N (1st𝐴)) ·N ((2nd𝐶) ·N (2nd𝐵)))
22 fvex 6919 . . . . . . . 8 (1st𝐵) ∈ V
23 fvex 6919 . . . . . . . 8 (2nd𝐴) ∈ V
2415, 16, 22, 18, 19, 23caov4 7664 . . . . . . 7 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((1st𝐶) ·N (1st𝐵)) ·N ((2nd𝐶) ·N (2nd𝐴)))
2521, 24breq12i 5152 . . . . . 6 ((((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ (((1st𝐶) ·N (1st𝐴)) ·N ((2nd𝐶) ·N (2nd𝐵))) <N (((1st𝐶) ·N (1st𝐵)) ·N ((2nd𝐶) ·N (2nd𝐴))))
2614, 25bitrdi 287 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (((1st𝐶) ·N (1st𝐴)) ·N ((2nd𝐶) ·N (2nd𝐵))) <N (((1st𝐶) ·N (1st𝐵)) ·N ((2nd𝐶) ·N (2nd𝐴)))))
27 ordpipq 10982 . . . . 5 (⟨((1st𝐶) ·N (1st𝐴)), ((2nd𝐶) ·N (2nd𝐴))⟩ <pQ ⟨((1st𝐶) ·N (1st𝐵)), ((2nd𝐶) ·N (2nd𝐵))⟩ ↔ (((1st𝐶) ·N (1st𝐴)) ·N ((2nd𝐶) ·N (2nd𝐵))) <N (((1st𝐶) ·N (1st𝐵)) ·N ((2nd𝐶) ·N (2nd𝐴))))
2826, 27bitr4di 289 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ ⟨((1st𝐶) ·N (1st𝐴)), ((2nd𝐶) ·N (2nd𝐴))⟩ <pQ ⟨((1st𝐶) ·N (1st𝐵)), ((2nd𝐶) ·N (2nd𝐵))⟩))
29 elpqn 10965 . . . . . . 7 (𝐴Q𝐴 ∈ (N × N))
30293ad2ant1 1134 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → 𝐴 ∈ (N × N))
31 mulpipq2 10979 . . . . . 6 ((𝐶 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐶 ·pQ 𝐴) = ⟨((1st𝐶) ·N (1st𝐴)), ((2nd𝐶) ·N (2nd𝐴))⟩)
326, 30, 31syl2anc 584 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 ·pQ 𝐴) = ⟨((1st𝐶) ·N (1st𝐴)), ((2nd𝐶) ·N (2nd𝐴))⟩)
33 elpqn 10965 . . . . . . 7 (𝐵Q𝐵 ∈ (N × N))
34333ad2ant2 1135 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → 𝐵 ∈ (N × N))
35 mulpipq2 10979 . . . . . 6 ((𝐶 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐶 ·pQ 𝐵) = ⟨((1st𝐶) ·N (1st𝐵)), ((2nd𝐶) ·N (2nd𝐵))⟩)
366, 34, 35syl2anc 584 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 ·pQ 𝐵) = ⟨((1st𝐶) ·N (1st𝐵)), ((2nd𝐶) ·N (2nd𝐵))⟩)
3732, 36breq12d 5156 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ((𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵) ↔ ⟨((1st𝐶) ·N (1st𝐴)), ((2nd𝐶) ·N (2nd𝐴))⟩ <pQ ⟨((1st𝐶) ·N (1st𝐵)), ((2nd𝐶) ·N (2nd𝐵))⟩))
3828, 37bitr4d 282 . . 3 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵)))
39 ordpinq 10983 . . . 4 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
40393adant3 1133 . . 3 ((𝐴Q𝐵Q𝐶Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
41 mulpqnq 10981 . . . . . . 7 ((𝐶Q𝐴Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
4241ancoms 458 . . . . . 6 ((𝐴Q𝐶Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
43423adant2 1132 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
44 mulpqnq 10981 . . . . . . 7 ((𝐶Q𝐵Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
4544ancoms 458 . . . . . 6 ((𝐵Q𝐶Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
46453adant1 1131 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
4743, 46breq12d 5156 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ((𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵) ↔ ([Q]‘(𝐶 ·pQ 𝐴)) <Q ([Q]‘(𝐶 ·pQ 𝐵))))
48 lterpq 11010 . . . 4 ((𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵) ↔ ([Q]‘(𝐶 ·pQ 𝐴)) <Q ([Q]‘(𝐶 ·pQ 𝐵)))
4947, 48bitr4di 289 . . 3 ((𝐴Q𝐵Q𝐶Q) → ((𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵) ↔ (𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵)))
5038, 40, 493bitr4d 311 . 2 ((𝐴Q𝐵Q𝐶Q) → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))
512, 3, 4, 50ndmovord 7623 1 (𝐶Q → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   = wceq 1540  wcel 2108  cop 4632   class class class wbr 5143   × cxp 5683  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  Ncnpi 10884   ·N cmi 10886   <N clti 10887   ·pQ cmpq 10889   <pQ cltpq 10890  Qcnq 10892  [Q]cerq 10894   ·Q cmq 10896   <Q cltq 10898
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511  df-er 8745  df-ni 10912  df-mi 10914  df-lti 10915  df-mpq 10949  df-ltpq 10950  df-enq 10951  df-nq 10952  df-erq 10953  df-mq 10955  df-1nq 10956  df-ltnq 10958
This theorem is referenced by:  ltaddnq  11014  ltrnq  11019  addclprlem1  11056  mulclprlem  11059  mulclpr  11060  distrlem4pr  11066  1idpr  11069  prlem934  11073  prlem936  11087  reclem3pr  11089  reclem4pr  11090
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