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Theorem ltsonq 10044
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 10000 . . . . . . 7 (𝑥Q𝑥 ∈ (N × N))
21adantr 472 . . . . . 6 ((𝑥Q𝑦Q) → 𝑥 ∈ (N × N))
3 xp1st 7398 . . . . . 6 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
42, 3syl 17 . . . . 5 ((𝑥Q𝑦Q) → (1st𝑥) ∈ N)
5 elpqn 10000 . . . . . . 7 (𝑦Q𝑦 ∈ (N × N))
65adantl 473 . . . . . 6 ((𝑥Q𝑦Q) → 𝑦 ∈ (N × N))
7 xp2nd 7399 . . . . . 6 (𝑦 ∈ (N × N) → (2nd𝑦) ∈ N)
86, 7syl 17 . . . . 5 ((𝑥Q𝑦Q) → (2nd𝑦) ∈ N)
9 mulclpi 9968 . . . . 5 (((1st𝑥) ∈ N ∧ (2nd𝑦) ∈ N) → ((1st𝑥) ·N (2nd𝑦)) ∈ N)
104, 8, 9syl2anc 579 . . . 4 ((𝑥Q𝑦Q) → ((1st𝑥) ·N (2nd𝑦)) ∈ N)
11 xp1st 7398 . . . . . 6 (𝑦 ∈ (N × N) → (1st𝑦) ∈ N)
126, 11syl 17 . . . . 5 ((𝑥Q𝑦Q) → (1st𝑦) ∈ N)
13 xp2nd 7399 . . . . . 6 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
142, 13syl 17 . . . . 5 ((𝑥Q𝑦Q) → (2nd𝑥) ∈ N)
15 mulclpi 9968 . . . . 5 (((1st𝑦) ∈ N ∧ (2nd𝑥) ∈ N) → ((1st𝑦) ·N (2nd𝑥)) ∈ N)
1612, 14, 15syl2anc 579 . . . 4 ((𝑥Q𝑦Q) → ((1st𝑦) ·N (2nd𝑥)) ∈ N)
17 ltsopi 9963 . . . . 5 <N Or N
18 sotric 5224 . . . . 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 681 . . . 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 579 . . 3 ((𝑥Q𝑦Q) → (((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)) ↔ ¬ (((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)) ∨ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦)))))
21 ordpinq 10018 . . 3 ((𝑥Q𝑦Q) → (𝑥 <Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥))))
22 fveq2 6375 . . . . . . 7 (𝑥 = 𝑦 → (1st𝑥) = (1st𝑦))
23 fveq2 6375 . . . . . . . 8 (𝑥 = 𝑦 → (2nd𝑥) = (2nd𝑦))
2423eqcomd 2771 . . . . . . 7 (𝑥 = 𝑦 → (2nd𝑦) = (2nd𝑥))
2522, 24oveq12d 6860 . . . . . 6 (𝑥 = 𝑦 → ((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)))
26 enqbreq2 9995 . . . . . . . 8 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (𝑥 ~Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥))))
271, 5, 26syl2an 589 . . . . . . 7 ((𝑥Q𝑦Q) → (𝑥 ~Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥))))
28 enqeq 10009 . . . . . . . 8 ((𝑥Q𝑦Q𝑥 ~Q 𝑦) → 𝑥 = 𝑦)
29283expia 1150 . . . . . . 7 ((𝑥Q𝑦Q) → (𝑥 ~Q 𝑦𝑥 = 𝑦))
3027, 29sylbird 251 . . . . . 6 ((𝑥Q𝑦Q) → (((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)) → 𝑥 = 𝑦))
3125, 30impbid2 217 . . . . 5 ((𝑥Q𝑦Q) → (𝑥 = 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥))))
32 ordpinq 10018 . . . . . 6 ((𝑦Q𝑥Q) → (𝑦 <Q 𝑥 ↔ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦))))
3332ancoms 450 . . . . 5 ((𝑥Q𝑦Q) → (𝑦 <Q 𝑥 ↔ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦))))
3431, 33orbi12d 942 . . . 4 ((𝑥Q𝑦Q) → ((𝑥 = 𝑦𝑦 <Q 𝑥) ↔ (((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)) ∨ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦)))))
3534notbid 309 . . 3 ((𝑥Q𝑦Q) → (¬ (𝑥 = 𝑦𝑦 <Q 𝑥) ↔ ¬ (((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)) ∨ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦)))))
3620, 21, 353bitr4d 302 . 2 ((𝑥Q𝑦Q) → (𝑥 <Q 𝑦 ↔ ¬ (𝑥 = 𝑦𝑦 <Q 𝑥)))
37213adant3 1162 . . . . . 6 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥))))
38 elpqn 10000 . . . . . . . 8 (𝑧Q𝑧 ∈ (N × N))
39383ad2ant3 1165 . . . . . . 7 ((𝑥Q𝑦Q𝑧Q) → 𝑧 ∈ (N × N))
40 xp2nd 7399 . . . . . . 7 (𝑧 ∈ (N × N) → (2nd𝑧) ∈ N)
41 ltmpi 9979 . . . . . . 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 270 . . . . 5 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ ((2nd𝑧) ·N ((1st𝑥) ·N (2nd𝑦))) <N ((2nd𝑧) ·N ((1st𝑦) ·N (2nd𝑥)))))
44 ordpinq 10018 . . . . . . 7 ((𝑦Q𝑧Q) → (𝑦 <Q 𝑧 ↔ ((1st𝑦) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑦))))
45443adant1 1160 . . . . . 6 ((𝑥Q𝑦Q𝑧Q) → (𝑦 <Q 𝑧 ↔ ((1st𝑦) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑦))))
4613ad2ant1 1163 . . . . . . 7 ((𝑥Q𝑦Q𝑧Q) → 𝑥 ∈ (N × N))
47 ltmpi 9979 . . . . . . 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 270 . . . . 5 ((𝑥Q𝑦Q𝑧Q) → (𝑦 <Q 𝑧 ↔ ((2nd𝑥) ·N ((1st𝑦) ·N (2nd𝑧))) <N ((2nd𝑥) ·N ((1st𝑧) ·N (2nd𝑦)))))
5043, 49anbi12d 624 . . . 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 6388 . . . . . . 7 (2nd𝑥) ∈ V
52 fvex 6388 . . . . . . 7 (1st𝑦) ∈ V
53 fvex 6388 . . . . . . 7 (2nd𝑧) ∈ V
54 mulcompi 9971 . . . . . . 7 (𝑟 ·N 𝑠) = (𝑠 ·N 𝑟)
55 mulasspi 9972 . . . . . . 7 ((𝑟 ·N 𝑠) ·N 𝑡) = (𝑟 ·N (𝑠 ·N 𝑡))
5651, 52, 53, 54, 55caov13 7062 . . . . . 6 ((2nd𝑥) ·N ((1st𝑦) ·N (2nd𝑧))) = ((2nd𝑧) ·N ((1st𝑦) ·N (2nd𝑥)))
57 fvex 6388 . . . . . . 7 (1st𝑧) ∈ V
58 fvex 6388 . . . . . . 7 (2nd𝑦) ∈ V
5951, 57, 58, 54, 55caov13 7062 . . . . . 6 ((2nd𝑥) ·N ((1st𝑧) ·N (2nd𝑦))) = ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥)))
6056, 59breq12i 4818 . . . . 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 6388 . . . . . . 7 (1st𝑥) ∈ V
6253, 61, 58, 54, 55caov13 7062 . . . . . 6 ((2nd𝑧) ·N ((1st𝑥) ·N (2nd𝑦))) = ((2nd𝑦) ·N ((1st𝑥) ·N (2nd𝑧)))
63 ltrelpi 9964 . . . . . . 7 <N ⊆ (N × N)
6417, 63sotri 5706 . . . . . 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 4845 . . . . 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 587 . . . 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 244 . . 3 ((𝑥Q𝑦Q𝑧Q) → ((𝑥 <Q 𝑦𝑦 <Q 𝑧) → ((2nd𝑦) ·N ((1st𝑥) ·N (2nd𝑧))) <N ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥)))))
68 ordpinq 10018 . . . . 5 ((𝑥Q𝑧Q) → (𝑥 <Q 𝑧 ↔ ((1st𝑥) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑥))))
69683adant2 1161 . . . 4 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑧 ↔ ((1st𝑥) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑥))))
7053ad2ant2 1164 . . . . 5 ((𝑥Q𝑦Q𝑧Q) → 𝑦 ∈ (N × N))
71 ltmpi 9979 . . . . 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 270 . . 3 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑧 ↔ ((2nd𝑦) ·N ((1st𝑥) ·N (2nd𝑧))) <N ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥)))))
7467, 73sylibrd 250 . 2 ((𝑥Q𝑦Q𝑧Q) → ((𝑥 <Q 𝑦𝑦 <Q 𝑧) → 𝑥 <Q 𝑧))
7536, 74isso2i 5230 1 <Q Or Q
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155   class class class wbr 4809   Or wor 5197   × cxp 5275  cfv 6068  (class class class)co 6842  1st c1st 7364  2nd c2nd 7365  Ncnpi 9919   ·N cmi 9921   <N clti 9922   ~Q ceq 9926  Qcnq 9927   <Q cltq 9933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-oadd 7768  df-omul 7769  df-er 7947  df-ni 9947  df-mi 9949  df-lti 9950  df-ltpq 9985  df-enq 9986  df-nq 9987  df-ltnq 9993
This theorem is referenced by:  ltbtwnnq  10053  prub  10069  npomex  10071  genpnnp  10080  nqpr  10089  distrlem4pr  10101  prlem934  10108  ltexprlem4  10114  reclem2pr  10123  reclem4pr  10125
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