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Theorem ltmnq 10997
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 10974 . . 3 ·Q :(Q × Q)⟶Q
21fdmi 6734 . 2 dom ·Q = (Q × Q)
3 ltrelnq 10951 . 2 <Q ⊆ (Q × Q)
4 0nnq 10949 . 2 ¬ ∅ ∈ Q
5 elpqn 10950 . . . . . . . . . 10 (𝐶Q𝐶 ∈ (N × N))
653ad2ant3 1132 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐶 ∈ (N × N))
7 xp1st 8026 . . . . . . . . 9 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
86, 7syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (1st𝐶) ∈ N)
9 xp2nd 8027 . . . . . . . . 9 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
106, 9syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐶) ∈ N)
11 mulclpi 10918 . . . . . . . 8 (((1st𝐶) ∈ N ∧ (2nd𝐶) ∈ N) → ((1st𝐶) ·N (2nd𝐶)) ∈ N)
128, 10, 11syl2anc 582 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐶) ·N (2nd𝐶)) ∈ N)
13 ltmpi 10929 . . . . . . 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 6909 . . . . . . . 8 (1st𝐶) ∈ V
16 fvex 6909 . . . . . . . 8 (2nd𝐶) ∈ V
17 fvex 6909 . . . . . . . 8 (1st𝐴) ∈ V
18 mulcompi 10921 . . . . . . . 8 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
19 mulasspi 10922 . . . . . . . 8 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
20 fvex 6909 . . . . . . . 8 (2nd𝐵) ∈ V
2115, 16, 17, 18, 19, 20caov4 7652 . . . . . . 7 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐶) ·N (1st𝐴)) ·N ((2nd𝐶) ·N (2nd𝐵)))
22 fvex 6909 . . . . . . . 8 (1st𝐵) ∈ V
23 fvex 6909 . . . . . . . 8 (2nd𝐴) ∈ V
2415, 16, 22, 18, 19, 23caov4 7652 . . . . . . 7 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((1st𝐶) ·N (1st𝐵)) ·N ((2nd𝐶) ·N (2nd𝐴)))
2521, 24breq12i 5158 . . . . . 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 286 . . . . 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 10967 . . . . 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 288 . . . 4 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ ⟨((1st𝐶) ·N (1st𝐴)), ((2nd𝐶) ·N (2nd𝐴))⟩ <pQ ⟨((1st𝐶) ·N (1st𝐵)), ((2nd𝐶) ·N (2nd𝐵))⟩))
29 elpqn 10950 . . . . . . 7 (𝐴Q𝐴 ∈ (N × N))
30293ad2ant1 1130 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → 𝐴 ∈ (N × N))
31 mulpipq2 10964 . . . . . 6 ((𝐶 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐶 ·pQ 𝐴) = ⟨((1st𝐶) ·N (1st𝐴)), ((2nd𝐶) ·N (2nd𝐴))⟩)
326, 30, 31syl2anc 582 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 ·pQ 𝐴) = ⟨((1st𝐶) ·N (1st𝐴)), ((2nd𝐶) ·N (2nd𝐴))⟩)
33 elpqn 10950 . . . . . . 7 (𝐵Q𝐵 ∈ (N × N))
34333ad2ant2 1131 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → 𝐵 ∈ (N × N))
35 mulpipq2 10964 . . . . . 6 ((𝐶 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐶 ·pQ 𝐵) = ⟨((1st𝐶) ·N (1st𝐵)), ((2nd𝐶) ·N (2nd𝐵))⟩)
366, 34, 35syl2anc 582 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 ·pQ 𝐵) = ⟨((1st𝐶) ·N (1st𝐵)), ((2nd𝐶) ·N (2nd𝐵))⟩)
3732, 36breq12d 5162 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ((𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵) ↔ ⟨((1st𝐶) ·N (1st𝐴)), ((2nd𝐶) ·N (2nd𝐴))⟩ <pQ ⟨((1st𝐶) ·N (1st𝐵)), ((2nd𝐶) ·N (2nd𝐵))⟩))
3828, 37bitr4d 281 . . 3 ((𝐴Q𝐵Q𝐶Q) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵)))
39 ordpinq 10968 . . . 4 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
40393adant3 1129 . . 3 ((𝐴Q𝐵Q𝐶Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
41 mulpqnq 10966 . . . . . . 7 ((𝐶Q𝐴Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
4241ancoms 457 . . . . . 6 ((𝐴Q𝐶Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
43423adant2 1128 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
44 mulpqnq 10966 . . . . . . 7 ((𝐶Q𝐵Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
4544ancoms 457 . . . . . 6 ((𝐵Q𝐶Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
46453adant1 1127 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
4743, 46breq12d 5162 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ((𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵) ↔ ([Q]‘(𝐶 ·pQ 𝐴)) <Q ([Q]‘(𝐶 ·pQ 𝐵))))
48 lterpq 10995 . . . 4 ((𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵) ↔ ([Q]‘(𝐶 ·pQ 𝐴)) <Q ([Q]‘(𝐶 ·pQ 𝐵)))
4947, 48bitr4di 288 . . 3 ((𝐴Q𝐵Q𝐶Q) → ((𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵) ↔ (𝐶 ·pQ 𝐴) <pQ (𝐶 ·pQ 𝐵)))
5038, 40, 493bitr4d 310 . 2 ((𝐴Q𝐵Q𝐶Q) → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))
512, 3, 4, 50ndmovord 7611 1 (𝐶Q → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084   = wceq 1533  wcel 2098  cop 4636   class class class wbr 5149   × cxp 5676  cfv 6549  (class class class)co 7419  1st c1st 7992  2nd c2nd 7993  Ncnpi 10869   ·N cmi 10871   <N clti 10872   ·pQ cmpq 10874   <pQ cltpq 10875  Qcnq 10877  [Q]cerq 10879   ·Q cmq 10881   <Q cltq 10883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-omul 8492  df-er 8725  df-ni 10897  df-mi 10899  df-lti 10900  df-mpq 10934  df-ltpq 10935  df-enq 10936  df-nq 10937  df-erq 10938  df-mq 10940  df-1nq 10941  df-ltnq 10943
This theorem is referenced by:  ltaddnq  10999  ltrnq  11004  addclprlem1  11041  mulclprlem  11044  mulclpr  11045  distrlem4pr  11051  1idpr  11054  prlem934  11058  prlem936  11072  reclem3pr  11074  reclem4pr  11075
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