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Theorem ltsonq 9993
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 9949 . . . . . . 7 (𝑥Q𝑥 ∈ (N × N))
21adantr 466 . . . . . 6 ((𝑥Q𝑦Q) → 𝑥 ∈ (N × N))
3 xp1st 7347 . . . . . 6 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
42, 3syl 17 . . . . 5 ((𝑥Q𝑦Q) → (1st𝑥) ∈ N)
5 elpqn 9949 . . . . . . 7 (𝑦Q𝑦 ∈ (N × N))
65adantl 467 . . . . . 6 ((𝑥Q𝑦Q) → 𝑦 ∈ (N × N))
7 xp2nd 7348 . . . . . 6 (𝑦 ∈ (N × N) → (2nd𝑦) ∈ N)
86, 7syl 17 . . . . 5 ((𝑥Q𝑦Q) → (2nd𝑦) ∈ N)
9 mulclpi 9917 . . . . 5 (((1st𝑥) ∈ N ∧ (2nd𝑦) ∈ N) → ((1st𝑥) ·N (2nd𝑦)) ∈ N)
104, 8, 9syl2anc 573 . . . 4 ((𝑥Q𝑦Q) → ((1st𝑥) ·N (2nd𝑦)) ∈ N)
11 xp1st 7347 . . . . . 6 (𝑦 ∈ (N × N) → (1st𝑦) ∈ N)
126, 11syl 17 . . . . 5 ((𝑥Q𝑦Q) → (1st𝑦) ∈ N)
13 xp2nd 7348 . . . . . 6 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
142, 13syl 17 . . . . 5 ((𝑥Q𝑦Q) → (2nd𝑥) ∈ N)
15 mulclpi 9917 . . . . 5 (((1st𝑦) ∈ N ∧ (2nd𝑥) ∈ N) → ((1st𝑦) ·N (2nd𝑥)) ∈ N)
1612, 14, 15syl2anc 573 . . . 4 ((𝑥Q𝑦Q) → ((1st𝑦) ·N (2nd𝑥)) ∈ N)
17 ltsopi 9912 . . . . 5 <N Or N
18 sotric 5196 . . . . 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 670 . . . 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 573 . . 3 ((𝑥Q𝑦Q) → (((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)) ↔ ¬ (((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)) ∨ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦)))))
21 ordpinq 9967 . . 3 ((𝑥Q𝑦Q) → (𝑥 <Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥))))
22 fveq2 6332 . . . . . . 7 (𝑥 = 𝑦 → (1st𝑥) = (1st𝑦))
23 fveq2 6332 . . . . . . . 8 (𝑥 = 𝑦 → (2nd𝑥) = (2nd𝑦))
2423eqcomd 2777 . . . . . . 7 (𝑥 = 𝑦 → (2nd𝑦) = (2nd𝑥))
2522, 24oveq12d 6811 . . . . . 6 (𝑥 = 𝑦 → ((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)))
26 enqbreq2 9944 . . . . . . . 8 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (𝑥 ~Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥))))
271, 5, 26syl2an 583 . . . . . . 7 ((𝑥Q𝑦Q) → (𝑥 ~Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥))))
28 enqeq 9958 . . . . . . . 8 ((𝑥Q𝑦Q𝑥 ~Q 𝑦) → 𝑥 = 𝑦)
29283expia 1114 . . . . . . 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 9967 . . . . . 6 ((𝑦Q𝑥Q) → (𝑦 <Q 𝑥 ↔ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦))))
3332ancoms 455 . . . . 5 ((𝑥Q𝑦Q) → (𝑦 <Q 𝑥 ↔ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦))))
3431, 33orbi12d 904 . . . 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 1126 . . . . . 6 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥))))
38 elpqn 9949 . . . . . . . 8 (𝑧Q𝑧 ∈ (N × N))
39383ad2ant3 1129 . . . . . . 7 ((𝑥Q𝑦Q𝑧Q) → 𝑧 ∈ (N × N))
40 xp2nd 7348 . . . . . . 7 (𝑧 ∈ (N × N) → (2nd𝑧) ∈ N)
41 ltmpi 9928 . . . . . . 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 9967 . . . . . . 7 ((𝑦Q𝑧Q) → (𝑦 <Q 𝑧 ↔ ((1st𝑦) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑦))))
45443adant1 1124 . . . . . 6 ((𝑥Q𝑦Q𝑧Q) → (𝑦 <Q 𝑧 ↔ ((1st𝑦) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑦))))
4613ad2ant1 1127 . . . . . . 7 ((𝑥Q𝑦Q𝑧Q) → 𝑥 ∈ (N × N))
47 ltmpi 9928 . . . . . . 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 616 . . . 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 6342 . . . . . . 7 (2nd𝑥) ∈ V
52 fvex 6342 . . . . . . 7 (1st𝑦) ∈ V
53 fvex 6342 . . . . . . 7 (2nd𝑧) ∈ V
54 mulcompi 9920 . . . . . . 7 (𝑟 ·N 𝑠) = (𝑠 ·N 𝑟)
55 mulasspi 9921 . . . . . . 7 ((𝑟 ·N 𝑠) ·N 𝑡) = (𝑟 ·N (𝑠 ·N 𝑡))
5651, 52, 53, 54, 55caov13 7011 . . . . . 6 ((2nd𝑥) ·N ((1st𝑦) ·N (2nd𝑧))) = ((2nd𝑧) ·N ((1st𝑦) ·N (2nd𝑥)))
57 fvex 6342 . . . . . . 7 (1st𝑧) ∈ V
58 fvex 6342 . . . . . . 7 (2nd𝑦) ∈ V
5951, 57, 58, 54, 55caov13 7011 . . . . . 6 ((2nd𝑥) ·N ((1st𝑧) ·N (2nd𝑦))) = ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥)))
6056, 59breq12i 4795 . . . . 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 6342 . . . . . . 7 (1st𝑥) ∈ V
6253, 61, 58, 54, 55caov13 7011 . . . . . 6 ((2nd𝑧) ·N ((1st𝑥) ·N (2nd𝑦))) = ((2nd𝑦) ·N ((1st𝑥) ·N (2nd𝑧)))
63 ltrelpi 9913 . . . . . . 7 <N ⊆ (N × N)
6417, 63sotri 5664 . . . . . 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 4822 . . . . 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 581 . . . 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 9967 . . . . 5 ((𝑥Q𝑧Q) → (𝑥 <Q 𝑧 ↔ ((1st𝑥) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑥))))
69683adant2 1125 . . . 4 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑧 ↔ ((1st𝑥) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑥))))
7053ad2ant2 1128 . . . . 5 ((𝑥Q𝑦Q𝑧Q) → 𝑦 ∈ (N × N))
71 ltmpi 9928 . . . . 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 5202 1 <Q Or Q
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382  wo 836  w3a 1071   = wceq 1631  wcel 2145   class class class wbr 4786   Or wor 5169   × cxp 5247  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  Ncnpi 9868   ·N cmi 9870   <N clti 9871   ~Q ceq 9875  Qcnq 9876   <Q cltq 9882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-oadd 7717  df-omul 7718  df-er 7896  df-ni 9896  df-mi 9898  df-lti 9899  df-ltpq 9934  df-enq 9935  df-nq 9936  df-ltnq 9942
This theorem is referenced by:  ltbtwnnq  10002  prub  10018  npomex  10020  genpnnp  10029  nqpr  10038  distrlem4pr  10050  prlem934  10057  ltexprlem4  10063  reclem2pr  10072  reclem4pr  10074
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