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Theorem ltsonq 9735
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 9691 . . . . . . 7 (𝑥Q𝑥 ∈ (N × N))
21adantr 481 . . . . . 6 ((𝑥Q𝑦Q) → 𝑥 ∈ (N × N))
3 xp1st 7143 . . . . . 6 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
42, 3syl 17 . . . . 5 ((𝑥Q𝑦Q) → (1st𝑥) ∈ N)
5 elpqn 9691 . . . . . . 7 (𝑦Q𝑦 ∈ (N × N))
65adantl 482 . . . . . 6 ((𝑥Q𝑦Q) → 𝑦 ∈ (N × N))
7 xp2nd 7144 . . . . . 6 (𝑦 ∈ (N × N) → (2nd𝑦) ∈ N)
86, 7syl 17 . . . . 5 ((𝑥Q𝑦Q) → (2nd𝑦) ∈ N)
9 mulclpi 9659 . . . . 5 (((1st𝑥) ∈ N ∧ (2nd𝑦) ∈ N) → ((1st𝑥) ·N (2nd𝑦)) ∈ N)
104, 8, 9syl2anc 692 . . . 4 ((𝑥Q𝑦Q) → ((1st𝑥) ·N (2nd𝑦)) ∈ N)
11 xp1st 7143 . . . . . 6 (𝑦 ∈ (N × N) → (1st𝑦) ∈ N)
126, 11syl 17 . . . . 5 ((𝑥Q𝑦Q) → (1st𝑦) ∈ N)
13 xp2nd 7144 . . . . . 6 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
142, 13syl 17 . . . . 5 ((𝑥Q𝑦Q) → (2nd𝑥) ∈ N)
15 mulclpi 9659 . . . . 5 (((1st𝑦) ∈ N ∧ (2nd𝑥) ∈ N) → ((1st𝑦) ·N (2nd𝑥)) ∈ N)
1612, 14, 15syl2anc 692 . . . 4 ((𝑥Q𝑦Q) → ((1st𝑦) ·N (2nd𝑥)) ∈ N)
17 ltsopi 9654 . . . . 5 <N Or N
18 sotric 5021 . . . . 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 705 . . . 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 692 . . 3 ((𝑥Q𝑦Q) → (((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)) ↔ ¬ (((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)) ∨ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦)))))
21 ordpinq 9709 . . 3 ((𝑥Q𝑦Q) → (𝑥 <Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥))))
22 fveq2 6148 . . . . . . 7 (𝑥 = 𝑦 → (1st𝑥) = (1st𝑦))
23 fveq2 6148 . . . . . . . 8 (𝑥 = 𝑦 → (2nd𝑥) = (2nd𝑦))
2423eqcomd 2627 . . . . . . 7 (𝑥 = 𝑦 → (2nd𝑦) = (2nd𝑥))
2522, 24oveq12d 6622 . . . . . 6 (𝑥 = 𝑦 → ((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)))
26 enqbreq2 9686 . . . . . . . 8 ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (𝑥 ~Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥))))
271, 5, 26syl2an 494 . . . . . . 7 ((𝑥Q𝑦Q) → (𝑥 ~Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥))))
28 enqeq 9700 . . . . . . . 8 ((𝑥Q𝑦Q𝑥 ~Q 𝑦) → 𝑥 = 𝑦)
29283expia 1264 . . . . . . 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 9709 . . . . . 6 ((𝑦Q𝑥Q) → (𝑦 <Q 𝑥 ↔ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦))))
3332ancoms 469 . . . . 5 ((𝑥Q𝑦Q) → (𝑦 <Q 𝑥 ↔ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦))))
3431, 33orbi12d 745 . . . 4 ((𝑥Q𝑦Q) → ((𝑥 = 𝑦𝑦 <Q 𝑥) ↔ (((1st𝑥) ·N (2nd𝑦)) = ((1st𝑦) ·N (2nd𝑥)) ∨ ((1st𝑦) ·N (2nd𝑥)) <N ((1st𝑥) ·N (2nd𝑦)))))
3534notbid 308 . . 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 1079 . . . . . 6 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥))))
38 elpqn 9691 . . . . . . . 8 (𝑧Q𝑧 ∈ (N × N))
39383ad2ant3 1082 . . . . . . 7 ((𝑥Q𝑦Q𝑧Q) → 𝑧 ∈ (N × N))
40 xp2nd 7144 . . . . . . 7 (𝑧 ∈ (N × N) → (2nd𝑧) ∈ N)
41 ltmpi 9670 . . . . . . 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 9709 . . . . . . 7 ((𝑦Q𝑧Q) → (𝑦 <Q 𝑧 ↔ ((1st𝑦) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑦))))
45443adant1 1077 . . . . . 6 ((𝑥Q𝑦Q𝑧Q) → (𝑦 <Q 𝑧 ↔ ((1st𝑦) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑦))))
4613ad2ant1 1080 . . . . . . 7 ((𝑥Q𝑦Q𝑧Q) → 𝑥 ∈ (N × N))
47 ltmpi 9670 . . . . . . 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 746 . . . 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 6158 . . . . . . 7 (2nd𝑥) ∈ V
52 fvex 6158 . . . . . . 7 (1st𝑦) ∈ V
53 fvex 6158 . . . . . . 7 (2nd𝑧) ∈ V
54 mulcompi 9662 . . . . . . 7 (𝑟 ·N 𝑠) = (𝑠 ·N 𝑟)
55 mulasspi 9663 . . . . . . 7 ((𝑟 ·N 𝑠) ·N 𝑡) = (𝑟 ·N (𝑠 ·N 𝑡))
5651, 52, 53, 54, 55caov13 6817 . . . . . 6 ((2nd𝑥) ·N ((1st𝑦) ·N (2nd𝑧))) = ((2nd𝑧) ·N ((1st𝑦) ·N (2nd𝑥)))
57 fvex 6158 . . . . . . 7 (1st𝑧) ∈ V
58 fvex 6158 . . . . . . 7 (2nd𝑦) ∈ V
5951, 57, 58, 54, 55caov13 6817 . . . . . 6 ((2nd𝑥) ·N ((1st𝑧) ·N (2nd𝑦))) = ((2nd𝑦) ·N ((1st𝑧) ·N (2nd𝑥)))
6056, 59breq12i 4622 . . . . 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 6158 . . . . . . 7 (1st𝑥) ∈ V
6253, 61, 58, 54, 55caov13 6817 . . . . . 6 ((2nd𝑧) ·N ((1st𝑥) ·N (2nd𝑦))) = ((2nd𝑦) ·N ((1st𝑥) ·N (2nd𝑧)))
63 ltrelpi 9655 . . . . . . 7 <N ⊆ (N × N)
6417, 63sotri 5482 . . . . . 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 4649 . . . . 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 492 . . . 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 9709 . . . . 5 ((𝑥Q𝑧Q) → (𝑥 <Q 𝑧 ↔ ((1st𝑥) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑥))))
69683adant2 1078 . . . 4 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑧 ↔ ((1st𝑥) ·N (2nd𝑧)) <N ((1st𝑧) ·N (2nd𝑥))))
7053ad2ant2 1081 . . . . 5 ((𝑥Q𝑦Q𝑧Q) → 𝑦 ∈ (N × N))
71 ltmpi 9670 . . . . 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 5027 1 <Q Or Q
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987   class class class wbr 4613   Or wor 4994   × cxp 5072  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  Ncnpi 9610   ·N cmi 9612   <N clti 9613   ~Q ceq 9617  Qcnq 9618   <Q cltq 9624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-omul 7510  df-er 7687  df-ni 9638  df-mi 9640  df-lti 9641  df-ltpq 9676  df-enq 9677  df-nq 9678  df-ltnq 9684
This theorem is referenced by:  ltbtwnnq  9744  prub  9760  npomex  9762  genpnnp  9771  nqpr  9780  distrlem4pr  9792  prlem934  9799  ltexprlem4  9805  reclem2pr  9814  reclem4pr  9816
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