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Theorem ltsonq 9981
Description: 'Less than' is a strict ordering on positive fractions. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 4-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltsonq <Q Or Q

Proof of Theorem ltsonq
Dummy variables 𝑠 𝑟 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpqn 9937 . . . . . . 7 (𝑥Q𝑥 ∈ (N × N))
21adantr 472 . . . . . 6 ((𝑥Q𝑦Q) → 𝑥 ∈ (N × N))
3 xp1st 7363 . . . . . 6 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
42, 3syl 17 . . . . 5 ((𝑥Q𝑦Q) → (1st𝑥) ∈ N)
5 elpqn 9937 . . . . . . 7 (𝑦Q𝑦 ∈ (N × N))
65adantl 473 . . . . . 6 ((𝑥Q𝑦Q) → 𝑦 ∈ (N × N))
7 xp2nd 7364 . . . . . 6 (𝑦 ∈ (N × N) → (2nd𝑦) ∈ N)
86, 7syl 17 . . . . 5 ((𝑥Q𝑦Q) → (2nd𝑦) ∈ N)
9 mulclpi 9905 . . . . 5 (((1st𝑥) ∈ N ∧ (2nd𝑦) ∈ N) → ((1st𝑥) ·N (2nd𝑦)) ∈ N)
104, 8, 9syl2anc 696 . . . 4 ((𝑥Q𝑦Q) → ((1st𝑥) ·N (2nd𝑦)) ∈ N)
11 xp1st 7363 . . . . . 6 (𝑦 ∈ (N × N) → (1st𝑦) ∈ N)
126, 11syl 17 . . . . 5 ((𝑥Q𝑦Q) → (1st𝑦) ∈ N)
13 xp2nd 7364 . . . . . 6 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
142, 13syl 17 . . . . 5 ((𝑥Q𝑦Q) → (2nd𝑥) ∈ N)
15 mulclpi 9905 . . . . 5 (((1st𝑦) ∈ N ∧ (2nd𝑥) ∈ N) → ((1st𝑦) ·N (2nd𝑥)) ∈ N)
1612, 14, 15syl2anc 696 . . . 4 ((𝑥Q𝑦Q) → ((1st𝑦) ·N (2nd𝑥)) ∈ N)
17 ltsopi 9900 . . . . 5 <N Or N
18 sotric 5211 . . . . 5 (( <N Or N ∧ (((1st𝑥) ·N (2nd𝑦)) ∈ N ∧ ((1st𝑦) ·N (2nd𝑥)) ∈ N)) → (((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)) ↔ ¬ (((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)) ∨ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦)))))
1917, 18mpan 708 . . . 4 ((((1st𝑥) ·N (2nd𝑦)) ∈ N ∧ ((1st𝑦) ·N (2nd𝑥)) ∈ N) → (((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)) ↔ ¬ (((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)) ∨ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦)))))
2010, 16, 19syl2anc 696 . . 3 ((𝑥Q𝑦Q) → (((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)) ↔ ¬ (((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)) ∨ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦)))))
21 ordpinq 9955 . . 3 ((𝑥Q𝑦Q) → (𝑥 <Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥))))
22 fveq2 6350 . . . . . . 7 (𝑥 = 𝑦 → (1st𝑥) = (1st𝑦))
23 fveq2 6350 . . . . . . . 8 (𝑥 = 𝑦 → (2nd𝑥) = (2nd𝑦))
2423eqcomd 2764 . . . . . . 7 (𝑥 = 𝑦 → (2nd𝑦) = (2nd𝑥))
2522, 24oveq12d 6829 . . . . . 6 (𝑥 = 𝑦 → ((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)))
26 enqbreq2 9932 . . . . . . . 8 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (𝑥 ~Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥))))
271, 5, 26syl2an 495 . . . . . . 7 ((𝑥Q𝑦Q) → (𝑥 ~Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥))))
28 enqeq 9946 . . . . . . . 8 ((𝑥Q𝑦Q𝑥 ~Q 𝑦) → 𝑥 = 𝑦)
29283expia 1115 . . . . . . 7 ((𝑥Q𝑦Q) → (𝑥 ~Q 𝑦𝑥 = 𝑦))
3027, 29sylbird 250 . . . . . 6 ((𝑥Q𝑦Q) → (((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)) → 𝑥 = 𝑦))
3125, 30impbid2 216 . . . . 5 ((𝑥Q𝑦Q) → (𝑥 = 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥))))
32 ordpinq 9955 . . . . . 6 ((𝑦Q𝑥Q) → (𝑦 <Q 𝑥 ↔ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦))))
3332ancoms 468 . . . . 5 ((𝑥Q𝑦Q) → (𝑦 <Q 𝑥 ↔ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦))))
3431, 33orbi12d 748 . . . 4 ((𝑥Q𝑦Q) → ((𝑥 = 𝑦𝑦 <Q 𝑥) ↔ (((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)) ∨ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦)))))
3534notbid 307 . . 3 ((𝑥Q𝑦Q) → (¬ (𝑥 = 𝑦𝑦 <Q 𝑥) ↔ ¬ (((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)) ∨ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦)))))
3620, 21, 353bitr4d 300 . 2 ((𝑥Q𝑦Q) → (𝑥 <Q 𝑦 ↔ ¬ (𝑥 = 𝑦𝑦 <Q 𝑥)))
37213adant3 1127 . . . . . 6 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥))))
38 elpqn 9937 . . . . . . . 8 (𝑧Q𝑧 ∈ (N × N))
39383ad2ant3 1130 . . . . . . 7 ((𝑥Q𝑦Q𝑧Q) → 𝑧 ∈ (N × N))
40 xp2nd 7364 . . . . . . 7 (𝑧 ∈ (N × N) → (2nd𝑧) ∈ N)
41 ltmpi 9916 . . . . . . 7 ((2nd𝑧) ∈ N → (((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)) ↔ ((2nd𝑧) ·N ((1st𝑥) ·N (2nd𝑦))) <N ((2nd𝑧) ·N ((1st𝑦) ·N (2nd𝑥)))))
4239, 40, 413syl 18 . . . . . 6 ((𝑥Q𝑦Q𝑧Q) → (((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)) ↔ ((2nd𝑧) ·N ((1st𝑥) ·N (2nd𝑦))) <N ((2nd𝑧) ·N ((1st𝑦) ·N (2nd𝑥)))))
4337, 42bitrd 268 . . . . 5 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ ((2nd𝑧) ·N ((1st𝑥) ·N (2nd𝑦))) <N ((2nd𝑧) ·N ((1st𝑦) ·N (2nd𝑥)))))
44 ordpinq 9955 . . . . . . 7 ((𝑦Q𝑧Q) → (𝑦 <Q 𝑧 ↔ ((1st𝑦) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑦))))
45443adant1 1125 . . . . . 6 ((𝑥Q𝑦Q𝑧Q) → (𝑦 <Q 𝑧 ↔ ((1st𝑦) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑦))))
4613ad2ant1 1128 . . . . . . 7 ((𝑥Q𝑦Q𝑧Q) → 𝑥 ∈ (N × N))
47 ltmpi 9916 . . . . . . 7 ((2nd𝑥) ∈ N → (((1st𝑦) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑦)) ↔ ((2nd𝑥) ·N ((1st𝑦) ·N (2nd𝑧))) <N ((2nd𝑥) ·N ((1st𝑧) ·N (2nd𝑦)))))
4846, 13, 473syl 18 . . . . . 6 ((𝑥Q𝑦Q𝑧Q) → (((1st𝑦) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑦)) ↔ ((2nd𝑥) ·N ((1st𝑦) ·N (2nd𝑧))) <N ((2nd𝑥) ·N ((1st𝑧) ·N (2nd𝑦)))))
4945, 48bitrd 268 . . . . 5 ((𝑥Q𝑦Q𝑧Q) → (𝑦 <Q 𝑧 ↔ ((2nd𝑥) ·N ((1st𝑦) ·N (2nd𝑧))) <N ((2nd𝑥) ·N ((1st𝑧) ·N (2nd𝑦)))))
5043, 49anbi12d 749 . . . 4 ((𝑥Q𝑦Q𝑧Q) → ((𝑥 <Q 𝑦𝑦 <Q 𝑧) ↔ (((2nd𝑧) ·N ((1st𝑥) ·N (2nd𝑦))) <N ((2nd𝑧) ·N ((1st𝑦) ·N (2nd𝑥))) ∧ ((2nd𝑥) ·N ((1st𝑦) ·N (2nd𝑧))) <N ((2nd𝑥) ·N ((1st𝑧) ·N (2nd𝑦))))))
51 fvex 6360 . . . . . . 7 (2nd𝑥) ∈ V
52 fvex 6360 . . . . . . 7 (1st𝑦) ∈ V
53 fvex 6360 . . . . . . 7 (2nd𝑧) ∈ V
54 mulcompi 9908 . . . . . . 7 (𝑟 ·N 𝑠) = (𝑠 ·N 𝑟)
55 mulasspi 9909 . . . . . . 7 ((𝑟 ·N 𝑠) ·N 𝑡) = (𝑟 ·N (𝑠 ·N 𝑡))
5651, 52, 53, 54, 55caov13 7027 . . . . . 6 ((2nd𝑥) ·N ((1st𝑦) ·N (2nd𝑧))) = ((2nd𝑧) ·N ((1st𝑦) ·N (2nd𝑥)))
57 fvex 6360 . . . . . . 7 (1st𝑧) ∈ V
58 fvex 6360 . . . . . . 7 (2nd𝑦) ∈ V
5951, 57, 58, 54, 55caov13 7027 . . . . . 6 ((2nd𝑥) ·N ((1st𝑧) ·N (2nd𝑦))) = ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥)))
6056, 59breq12i 4811 . . . . 5 (((2nd𝑥) ·N ((1st𝑦) ·N (2nd𝑧))) <N ((2nd𝑥) ·N ((1st𝑧) ·N (2nd𝑦))) ↔ ((2nd𝑧) ·N ((1st𝑦) ·N (2nd𝑥))) <N ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥))))
61 fvex 6360 . . . . . . 7 (1st𝑥) ∈ V
6253, 61, 58, 54, 55caov13 7027 . . . . . 6 ((2nd𝑧) ·N ((1st𝑥) ·N (2nd𝑦))) = ((2nd𝑦) ·N ((1st𝑥) ·N (2nd𝑧)))
63 ltrelpi 9901 . . . . . . 7 <N ⊆ (N × N)
6417, 63sotri 5679 . . . . . 6 ((((2nd𝑧) ·N ((1st𝑥) ·N (2nd𝑦))) <N ((2nd𝑧) ·N ((1st𝑦) ·N (2nd𝑥))) ∧ ((2nd𝑧) ·N ((1st𝑦) ·N (2nd𝑥))) <N ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥)))) → ((2nd𝑧) ·N ((1st𝑥) ·N (2nd𝑦))) <N ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥))))
6562, 64syl5eqbrr 4838 . . . . 5 ((((2nd𝑧) ·N ((1st𝑥) ·N (2nd𝑦))) <N ((2nd𝑧) ·N ((1st𝑦) ·N (2nd𝑥))) ∧ ((2nd𝑧) ·N ((1st𝑦) ·N (2nd𝑥))) <N ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥)))) → ((2nd𝑦) ·N ((1st𝑥) ·N (2nd𝑧))) <N ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥))))
6660, 65sylan2b 493 . . . 4 ((((2nd𝑧) ·N ((1st𝑥) ·N (2nd𝑦))) <N ((2nd𝑧) ·N ((1st𝑦) ·N (2nd𝑥))) ∧ ((2nd𝑥) ·N ((1st𝑦) ·N (2nd𝑧))) <N ((2nd𝑥) ·N ((1st𝑧) ·N (2nd𝑦)))) → ((2nd𝑦) ·N ((1st𝑥) ·N (2nd𝑧))) <N ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥))))
6750, 66syl6bi 243 . . 3 ((𝑥Q𝑦Q𝑧Q) → ((𝑥 <Q 𝑦𝑦 <Q 𝑧) → ((2nd𝑦) ·N ((1st𝑥) ·N (2nd𝑧))) <N ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥)))))
68 ordpinq 9955 . . . . 5 ((𝑥Q𝑧Q) → (𝑥 <Q 𝑧 ↔ ((1st𝑥) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑥))))
69683adant2 1126 . . . 4 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑧 ↔ ((1st𝑥) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑥))))
7053ad2ant2 1129 . . . . 5 ((𝑥Q𝑦Q𝑧Q) → 𝑦 ∈ (N × N))
71 ltmpi 9916 . . . . 5 ((2nd𝑦) ∈ N → (((1st𝑥) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑥)) ↔ ((2nd𝑦) ·N ((1st𝑥) ·N (2nd𝑧))) <N ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥)))))
7270, 7, 713syl 18 . . . 4 ((𝑥Q𝑦Q𝑧Q) → (((1st𝑥) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑥)) ↔ ((2nd𝑦) ·N ((1st𝑥) ·N (2nd𝑧))) <N ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥)))))
7369, 72bitrd 268 . . 3 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑧 ↔ ((2nd𝑦) ·N ((1st𝑥) ·N (2nd𝑧))) <N ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥)))))
7467, 73sylibrd 249 . 2 ((𝑥Q𝑦Q𝑧Q) → ((𝑥 <Q 𝑦𝑦 <Q 𝑧) → 𝑥 <Q 𝑧))
7536, 74isso2i 5217 1 <Q Or Q
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 382  wa 383  w3a 1072   = wceq 1630  wcel 2137   class class class wbr 4802   Or wor 5184   × cxp 5262  cfv 6047  (class class class)co 6811  1st c1st 7329  2nd c2nd 7330  Ncnpi 9856   ·N cmi 9858   <N clti 9859   ~Q ceq 9863  Qcnq 9864   <Q cltq 9870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-oadd 7731  df-omul 7732  df-er 7909  df-ni 9884  df-mi 9886  df-lti 9887  df-ltpq 9922  df-enq 9923  df-nq 9924  df-ltnq 9930
This theorem is referenced by:  ltbtwnnq  9990  prub  10006  npomex  10008  genpnnp  10017  nqpr  10026  distrlem4pr  10038  prlem934  10045  ltexprlem4  10051  reclem2pr  10060  reclem4pr  10062
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