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Theorem ltmnq 10908
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 10885 . . 3 ·Q :(Q × Q)⟶Q
21fdmi 6680 . 2 dom ·Q = (Q × Q)
3 ltrelnq 10862 . 2 <Q ⊆ (Q × Q)
4 0nnq 10860 . 2 ¬ ∅ ∈ Q
5 elpqn 10861 . . . . . . . . . 10 (𝐶Q𝐶 ∈ (N × N))
653ad2ant3 1135 . . . . . . . . 9 ((𝐴Q𝐵Q𝐶Q) → 𝐶 ∈ (N × N))
7 xp1st 7953 . . . . . . . . 9 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
86, 7syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (1st𝐶) ∈ N)
9 xp2nd 7954 . . . . . . . . 9 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
106, 9syl 17 . . . . . . . 8 ((𝐴Q𝐵Q𝐶Q) → (2nd𝐶) ∈ N)
11 mulclpi 10829 . . . . . . . 8 (((1st𝐶) ∈ N ∧ (2nd𝐶) ∈ N) → ((1st𝐶) ·N (2nd𝐶)) ∈ N)
128, 10, 11syl2anc 584 . . . . . . 7 ((𝐴Q𝐵Q𝐶Q) → ((1st𝐶) ·N (2nd𝐶)) ∈ N)
13 ltmpi 10840 . . . . . . 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 6855 . . . . . . . 8 (1st𝐶) ∈ V
16 fvex 6855 . . . . . . . 8 (2nd𝐶) ∈ V
17 fvex 6855 . . . . . . . 8 (1st𝐴) ∈ V
18 mulcompi 10832 . . . . . . . 8 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
19 mulasspi 10833 . . . . . . . 8 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
20 fvex 6855 . . . . . . . 8 (2nd𝐵) ∈ V
2115, 16, 17, 18, 19, 20caov4 7585 . . . . . . 7 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐶) ·N (1st𝐴)) ·N ((2nd𝐶) ·N (2nd𝐵)))
22 fvex 6855 . . . . . . . 8 (1st𝐵) ∈ V
23 fvex 6855 . . . . . . . 8 (2nd𝐴) ∈ V
2415, 16, 22, 18, 19, 23caov4 7585 . . . . . . 7 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((1st𝐶) ·N (1st𝐵)) ·N ((2nd𝐶) ·N (2nd𝐴)))
2521, 24breq12i 5114 . . . . . 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 10878 . . . . 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 10861 . . . . . . 7 (𝐴Q𝐴 ∈ (N × N))
30293ad2ant1 1133 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → 𝐴 ∈ (N × N))
31 mulpipq2 10875 . . . . . 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 10861 . . . . . . 7 (𝐵Q𝐵 ∈ (N × N))
34333ad2ant2 1134 . . . . . 6 ((𝐴Q𝐵Q𝐶Q) → 𝐵 ∈ (N × N))
35 mulpipq2 10875 . . . . . 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 5118 . . . 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 10879 . . . 4 ((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
40393adant3 1132 . . 3 ((𝐴Q𝐵Q𝐶Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
41 mulpqnq 10877 . . . . . . 7 ((𝐶Q𝐴Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
4241ancoms 459 . . . . . 6 ((𝐴Q𝐶Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
43423adant2 1131 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 ·Q 𝐴) = ([Q]‘(𝐶 ·pQ 𝐴)))
44 mulpqnq 10877 . . . . . . 7 ((𝐶Q𝐵Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
4544ancoms 459 . . . . . 6 ((𝐵Q𝐶Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
46453adant1 1130 . . . . 5 ((𝐴Q𝐵Q𝐶Q) → (𝐶 ·Q 𝐵) = ([Q]‘(𝐶 ·pQ 𝐵)))
4743, 46breq12d 5118 . . . 4 ((𝐴Q𝐵Q𝐶Q) → ((𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵) ↔ ([Q]‘(𝐶 ·pQ 𝐴)) <Q ([Q]‘(𝐶 ·pQ 𝐵))))
48 lterpq 10906 . . . 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 7544 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 1087   = wceq 1541  wcel 2106  cop 4592   class class class wbr 5105   × cxp 5631  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  Ncnpi 10780   ·N cmi 10782   <N clti 10783   ·pQ cmpq 10785   <pQ cltpq 10786  Qcnq 10788  [Q]cerq 10790   ·Q cmq 10792   <Q cltq 10794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417  df-er 8648  df-ni 10808  df-mi 10810  df-lti 10811  df-mpq 10845  df-ltpq 10846  df-enq 10847  df-nq 10848  df-erq 10849  df-mq 10851  df-1nq 10852  df-ltnq 10854
This theorem is referenced by:  ltaddnq  10910  ltrnq  10915  addclprlem1  10952  mulclprlem  10955  mulclpr  10956  distrlem4pr  10962  1idpr  10965  prlem934  10969  prlem936  10983  reclem3pr  10985  reclem4pr  10986
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