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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 (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 10108 . . 3 ·Q :(Q × Q)⟶Q
21fdmi 6303 . 2 dom ·Q = (Q × Q)
3 ltrelnq 10085 . 2 <Q ⊆ (Q × Q)
4 0nnq 10083 . 2 ¬ ∅ ∈ Q
5 elpqn 10084 . . . . . . . . . 10 (𝐶Q𝐶 ∈ (N × N))
653ad2ant3 1126 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐶 ∈ (N × N))
7 xp1st 7479 . . . . . . . . 9 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
86, 7syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (1st𝐶) ∈ N)
9 xp2nd 7480 . . . . . . . . 9 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
106, 9syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐶) ∈ N)
11 mulclpi 10052 . . . . . . . 8 (((1st𝐶) ∈ N ∧ (2nd𝐶) ∈ N) → ((1st𝐶) ·N (2nd𝐶)) ∈ N)
128, 10, 11syl2anc 579 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐶) ·N (2nd𝐶)) ∈ N)
13 ltmpi 10063 . . . . . . 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 6461 . . . . . . . 8 (1st𝐶) ∈ V
16 fvex 6461 . . . . . . . 8 (2nd𝐶) ∈ V
17 fvex 6461 . . . . . . . 8 (1st𝐴) ∈ 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 (2nd𝐵) ∈ V
2115, 16, 17, 18, 19, 20caov4 7144 . . . . . . 7 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐶) ·N (1st𝐴)) ·N ((2nd𝐶) ·N (2nd𝐵)))
22 fvex 6461 . . . . . . . 8 (1st𝐵) ∈ V
23 fvex 6461 . . . . . . . 8 (2nd𝐴) ∈ V
2415, 16, 22, 18, 19, 23caov4 7144 . . . . . . 7 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((1st𝐶) ·N (1st𝐵)) ·N ((2nd𝐶) ·N (2nd𝐴)))
2521, 24breq12i 4897 . . . . . 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, 25syl6bb 279 . . . . 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 10101 . . . . 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, 27syl6bbr 281 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ ⟨((1st𝐶) ·N (1st𝐴)), ((2nd𝐶) ·N (2nd𝐴))⟩ <pQ ⟨((1st𝐶) ·N (1st𝐵)), ((2nd𝐶) ·N (2nd𝐵))⟩))
29 elpqn 10084 . . . . . . 7 (𝐴Q𝐴 ∈ (N × N))
30293ad2ant1 1124 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → 𝐴 ∈ (N × N))
31 mulpipq2 10098 . . . . . 6 ((𝐶 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐶 ·pQ 𝐴) = ⟨((1st𝐶) ·N (1st𝐴)), ((2nd𝐶) ·N (2nd𝐴))⟩)
326, 30, 31syl2anc 579 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 ·pQ 𝐴) = ⟨((1st𝐶) ·N (1st𝐴)), ((2nd𝐶) ·N (2nd𝐴))⟩)
33 elpqn 10084 . . . . . . 7 (𝐵Q𝐵 ∈ (N × N))
34333ad2ant2 1125 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → 𝐵 ∈ (N × N))
35 mulpipq2 10098 . . . . . 6 ((𝐶 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐶 ·pQ 𝐵) = ⟨((1st𝐶) ·N (1st𝐵)), ((2nd𝐶) ·N (2nd𝐵))⟩)
366, 34, 35syl2anc 579 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 ·pQ 𝐵) = ⟨((1st𝐶) ·N (1st𝐵)), ((2nd𝐶) ·N (2nd𝐵))⟩)
3732, 36breq12d 4901 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ((𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵) ↔ ⟨((1st𝐶) ·N (1st𝐴)), ((2nd𝐶) ·N (2nd𝐴))⟩ <pQ ⟨((1st𝐶) ·N (1st𝐵)), ((2nd𝐶) ·N (2nd𝐵))⟩))
3828, 37bitr4d 274 . . 3 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵)))
39 ordpinq 10102 . . . 4 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
40393adant3 1123 . . 3 ((𝐴Q𝐵Q𝐶Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
41 mulpqnq 10100 . . . . . . 7 ((𝐶Q𝐴Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
4241ancoms 452 . . . . . 6 ((𝐴Q𝐶Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
43423adant2 1122 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
44 mulpqnq 10100 . . . . . . 7 ((𝐶Q𝐵Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
4544ancoms 452 . . . . . 6 ((𝐵Q𝐶Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
46453adant1 1121 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
4743, 46breq12d 4901 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ((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)))
4947, 48syl6bbr 281 . . 3 ((𝐴Q𝐵Q𝐶Q) → ((𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵) ↔ (𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵)))
5038, 40, 493bitr4d 303 . 2 ((𝐴Q𝐵Q𝐶Q) → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))
512, 3, 4, 50ndmovord 7103 1 (𝐶Q → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))
Colors of variables: wff setvar class
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  1st c1st 7445  2nd 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
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