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Theorem ltsonq 9670
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 9626 . . . . . . 7 (𝑥Q𝑥 ∈ (N × N))
21adantr 480 . . . . . 6 ((𝑥Q𝑦Q) → 𝑥 ∈ (N × N))
3 xp1st 7089 . . . . . 6 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
42, 3syl 17 . . . . 5 ((𝑥Q𝑦Q) → (1st𝑥) ∈ N)
5 elpqn 9626 . . . . . . 7 (𝑦Q𝑦 ∈ (N × N))
65adantl 481 . . . . . 6 ((𝑥Q𝑦Q) → 𝑦 ∈ (N × N))
7 xp2nd 7090 . . . . . 6 (𝑦 ∈ (N × N) → (2nd𝑦) ∈ N)
86, 7syl 17 . . . . 5 ((𝑥Q𝑦Q) → (2nd𝑦) ∈ N)
9 mulclpi 9594 . . . . 5 (((1st𝑥) ∈ N ∧ (2nd𝑦) ∈ N) → ((1st𝑥) ·N (2nd𝑦)) ∈ N)
104, 8, 9syl2anc 691 . . . 4 ((𝑥Q𝑦Q) → ((1st𝑥) ·N (2nd𝑦)) ∈ N)
11 xp1st 7089 . . . . . 6 (𝑦 ∈ (N × N) → (1st𝑦) ∈ N)
126, 11syl 17 . . . . 5 ((𝑥Q𝑦Q) → (1st𝑦) ∈ N)
13 xp2nd 7090 . . . . . 6 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
142, 13syl 17 . . . . 5 ((𝑥Q𝑦Q) → (2nd𝑥) ∈ N)
15 mulclpi 9594 . . . . 5 (((1st𝑦) ∈ N ∧ (2nd𝑥) ∈ N) → ((1st𝑦) ·N (2nd𝑥)) ∈ N)
1612, 14, 15syl2anc 691 . . . 4 ((𝑥Q𝑦Q) → ((1st𝑦) ·N (2nd𝑥)) ∈ N)
17 ltsopi 9589 . . . . 5 <N Or N
18 sotric 4985 . . . . 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 702 . . . 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 691 . . 3 ((𝑥Q𝑦Q) → (((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)) ↔ ¬ (((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)) ∨ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦)))))
21 ordpinq 9644 . . 3 ((𝑥Q𝑦Q) → (𝑥 <Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥))))
22 fveq2 6103 . . . . . . 7 (𝑥 = 𝑦 → (1st𝑥) = (1st𝑦))
23 fveq2 6103 . . . . . . . 8 (𝑥 = 𝑦 → (2nd𝑥) = (2nd𝑦))
2423eqcomd 2616 . . . . . . 7 (𝑥 = 𝑦 → (2nd𝑦) = (2nd𝑥))
2522, 24oveq12d 6567 . . . . . 6 (𝑥 = 𝑦 → ((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)))
26 enqbreq2 9621 . . . . . . . 8 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (𝑥 ~Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥))))
271, 5, 26syl2an 493 . . . . . . 7 ((𝑥Q𝑦Q) → (𝑥 ~Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥))))
28 enqeq 9635 . . . . . . . 8 ((𝑥Q𝑦Q𝑥 ~Q 𝑦) → 𝑥 = 𝑦)
29283expia 1259 . . . . . . 7 ((𝑥Q𝑦Q) → (𝑥 ~Q 𝑦𝑥 = 𝑦))
3027, 29sylbird 249 . . . . . 6 ((𝑥Q𝑦Q) → (((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)) → 𝑥 = 𝑦))
3125, 30impbid2 215 . . . . 5 ((𝑥Q𝑦Q) → (𝑥 = 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥))))
32 ordpinq 9644 . . . . . 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 742 . . . 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 299 . 2 ((𝑥Q𝑦Q) → (𝑥 <Q 𝑦 ↔ ¬ (𝑥 = 𝑦𝑦 <Q 𝑥)))
37213adant3 1074 . . . . . 6 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥))))
38 elpqn 9626 . . . . . . . 8 (𝑧Q𝑧 ∈ (N × N))
39383ad2ant3 1077 . . . . . . 7 ((𝑥Q𝑦Q𝑧Q) → 𝑧 ∈ (N × N))
40 xp2nd 7090 . . . . . . 7 (𝑧 ∈ (N × N) → (2nd𝑧) ∈ N)
41 ltmpi 9605 . . . . . . 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 267 . . . . 5 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ ((2nd𝑧) ·N ((1st𝑥) ·N (2nd𝑦))) <N ((2nd𝑧) ·N ((1st𝑦) ·N (2nd𝑥)))))
44 ordpinq 9644 . . . . . . 7 ((𝑦Q𝑧Q) → (𝑦 <Q 𝑧 ↔ ((1st𝑦) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑦))))
45443adant1 1072 . . . . . 6 ((𝑥Q𝑦Q𝑧Q) → (𝑦 <Q 𝑧 ↔ ((1st𝑦) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑦))))
4613ad2ant1 1075 . . . . . . 7 ((𝑥Q𝑦Q𝑧Q) → 𝑥 ∈ (N × N))
47 ltmpi 9605 . . . . . . 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 267 . . . . 5 ((𝑥Q𝑦Q𝑧Q) → (𝑦 <Q 𝑧 ↔ ((2nd𝑥) ·N ((1st𝑦) ·N (2nd𝑧))) <N ((2nd𝑥) ·N ((1st𝑧) ·N (2nd𝑦)))))
5043, 49anbi12d 743 . . . 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 6113 . . . . . . 7 (2nd𝑥) ∈ V
52 fvex 6113 . . . . . . 7 (1st𝑦) ∈ V
53 fvex 6113 . . . . . . 7 (2nd𝑧) ∈ V
54 mulcompi 9597 . . . . . . 7 (𝑟 ·N 𝑠) = (𝑠 ·N 𝑟)
55 mulasspi 9598 . . . . . . 7 ((𝑟 ·N 𝑠) ·N 𝑡) = (𝑟 ·N (𝑠 ·N 𝑡))
5651, 52, 53, 54, 55caov13 6762 . . . . . 6 ((2nd𝑥) ·N ((1st𝑦) ·N (2nd𝑧))) = ((2nd𝑧) ·N ((1st𝑦) ·N (2nd𝑥)))
57 fvex 6113 . . . . . . 7 (1st𝑧) ∈ V
58 fvex 6113 . . . . . . 7 (2nd𝑦) ∈ V
5951, 57, 58, 54, 55caov13 6762 . . . . . 6 ((2nd𝑥) ·N ((1st𝑧) ·N (2nd𝑦))) = ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥)))
6056, 59breq12i 4592 . . . . 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 6113 . . . . . . 7 (1st𝑥) ∈ V
6253, 61, 58, 54, 55caov13 6762 . . . . . 6 ((2nd𝑧) ·N ((1st𝑥) ·N (2nd𝑦))) = ((2nd𝑦) ·N ((1st𝑥) ·N (2nd𝑧)))
63 ltrelpi 9590 . . . . . . 7 <N ⊆ (N × N)
6417, 63sotri 5442 . . . . . 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 4619 . . . . 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 491 . . . 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 242 . . 3 ((𝑥Q𝑦Q𝑧Q) → ((𝑥 <Q 𝑦𝑦 <Q 𝑧) → ((2nd𝑦) ·N ((1st𝑥) ·N (2nd𝑧))) <N ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥)))))
68 ordpinq 9644 . . . . 5 ((𝑥Q𝑧Q) → (𝑥 <Q 𝑧 ↔ ((1st𝑥) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑥))))
69683adant2 1073 . . . 4 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑧 ↔ ((1st𝑥) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑥))))
7053ad2ant2 1076 . . . . 5 ((𝑥Q𝑦Q𝑧Q) → 𝑦 ∈ (N × N))
71 ltmpi 9605 . . . . 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 267 . . 3 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑧 ↔ ((2nd𝑦) ·N ((1st𝑥) ·N (2nd𝑧))) <N ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥)))))
7467, 73sylibrd 248 . 2 ((𝑥Q𝑦Q𝑧Q) → ((𝑥 <Q 𝑦𝑦 <Q 𝑧) → 𝑥 <Q 𝑧))
7536, 74isso2i 4991 1 <Q Or Q
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 195  wo 382  wa 383  w3a 1031   = wceq 1475  wcel 1977   class class class wbr 4583   Or wor 4958   × cxp 5036  cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Ncnpi 9545   ·N cmi 9547   <N clti 9548   ~Q ceq 9552  Qcnq 9553   <Q cltq 9559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-omul 7452  df-er 7629  df-ni 9573  df-mi 9575  df-lti 9576  df-ltpq 9611  df-enq 9612  df-nq 9613  df-ltnq 9619
This theorem is referenced by:  ltbtwnnq  9679  prub  9695  npomex  9697  genpnnp  9706  nqpr  9715  distrlem4pr  9727  prlem934  9734  ltexprlem4  9740  reclem2pr  9749  reclem4pr  9751
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