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Theorem archnq 9840
Description: For any fraction, there is an integer that is greater than it. This is also known as the "archimedean property". (Contributed by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
archnq (𝐴Q → ∃𝑥N 𝐴 <Q𝑥, 1𝑜⟩)
Distinct variable group:   𝑥,𝐴

Proof of Theorem archnq
StepHypRef Expression
1 elpqn 9785 . . . 4 (𝐴Q𝐴 ∈ (N × N))
2 xp1st 7242 . . . 4 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
31, 2syl 17 . . 3 (𝐴Q → (1st𝐴) ∈ N)
4 1pi 9743 . . 3 1𝑜N
5 addclpi 9752 . . 3 (((1st𝐴) ∈ N ∧ 1𝑜N) → ((1st𝐴) +N 1𝑜) ∈ N)
63, 4, 5sylancl 695 . 2 (𝐴Q → ((1st𝐴) +N 1𝑜) ∈ N)
7 xp2nd 7243 . . . . . 6 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
81, 7syl 17 . . . . 5 (𝐴Q → (2nd𝐴) ∈ N)
9 mulclpi 9753 . . . . 5 ((((1st𝐴) +N 1𝑜) ∈ N ∧ (2nd𝐴) ∈ N) → (((1st𝐴) +N 1𝑜) ·N (2nd𝐴)) ∈ N)
106, 8, 9syl2anc 694 . . . 4 (𝐴Q → (((1st𝐴) +N 1𝑜) ·N (2nd𝐴)) ∈ N)
11 eqid 2651 . . . . . . 7 ((1st𝐴) +N 1𝑜) = ((1st𝐴) +N 1𝑜)
12 oveq2 6698 . . . . . . . . 9 (𝑥 = 1𝑜 → ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1𝑜))
1312eqeq1d 2653 . . . . . . . 8 (𝑥 = 1𝑜 → (((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1𝑜) ↔ ((1st𝐴) +N 1𝑜) = ((1st𝐴) +N 1𝑜)))
1413rspcev 3340 . . . . . . 7 ((1𝑜N ∧ ((1st𝐴) +N 1𝑜) = ((1st𝐴) +N 1𝑜)) → ∃𝑥N ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1𝑜))
154, 11, 14mp2an 708 . . . . . 6 𝑥N ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1𝑜)
16 ltexpi 9762 . . . . . 6 (((1st𝐴) ∈ N ∧ ((1st𝐴) +N 1𝑜) ∈ N) → ((1st𝐴) <N ((1st𝐴) +N 1𝑜) ↔ ∃𝑥N ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1𝑜)))
1715, 16mpbiri 248 . . . . 5 (((1st𝐴) ∈ N ∧ ((1st𝐴) +N 1𝑜) ∈ N) → (1st𝐴) <N ((1st𝐴) +N 1𝑜))
183, 6, 17syl2anc 694 . . . 4 (𝐴Q → (1st𝐴) <N ((1st𝐴) +N 1𝑜))
19 nlt1pi 9766 . . . . 5 ¬ (2nd𝐴) <N 1𝑜
20 ltmpi 9764 . . . . . . 7 (((1st𝐴) +N 1𝑜) ∈ N → ((2nd𝐴) <N 1𝑜 ↔ (((1st𝐴) +N 1𝑜) ·N (2nd𝐴)) <N (((1st𝐴) +N 1𝑜) ·N 1𝑜)))
216, 20syl 17 . . . . . 6 (𝐴Q → ((2nd𝐴) <N 1𝑜 ↔ (((1st𝐴) +N 1𝑜) ·N (2nd𝐴)) <N (((1st𝐴) +N 1𝑜) ·N 1𝑜)))
22 mulidpi 9746 . . . . . . . 8 (((1st𝐴) +N 1𝑜) ∈ N → (((1st𝐴) +N 1𝑜) ·N 1𝑜) = ((1st𝐴) +N 1𝑜))
236, 22syl 17 . . . . . . 7 (𝐴Q → (((1st𝐴) +N 1𝑜) ·N 1𝑜) = ((1st𝐴) +N 1𝑜))
2423breq2d 4697 . . . . . 6 (𝐴Q → ((((1st𝐴) +N 1𝑜) ·N (2nd𝐴)) <N (((1st𝐴) +N 1𝑜) ·N 1𝑜) ↔ (((1st𝐴) +N 1𝑜) ·N (2nd𝐴)) <N ((1st𝐴) +N 1𝑜)))
2521, 24bitrd 268 . . . . 5 (𝐴Q → ((2nd𝐴) <N 1𝑜 ↔ (((1st𝐴) +N 1𝑜) ·N (2nd𝐴)) <N ((1st𝐴) +N 1𝑜)))
2619, 25mtbii 315 . . . 4 (𝐴Q → ¬ (((1st𝐴) +N 1𝑜) ·N (2nd𝐴)) <N ((1st𝐴) +N 1𝑜))
27 ltsopi 9748 . . . . 5 <N Or N
28 ltrelpi 9749 . . . . 5 <N ⊆ (N × N)
2927, 28sotri3 5561 . . . 4 (((((1st𝐴) +N 1𝑜) ·N (2nd𝐴)) ∈ N ∧ (1st𝐴) <N ((1st𝐴) +N 1𝑜) ∧ ¬ (((1st𝐴) +N 1𝑜) ·N (2nd𝐴)) <N ((1st𝐴) +N 1𝑜)) → (1st𝐴) <N (((1st𝐴) +N 1𝑜) ·N (2nd𝐴)))
3010, 18, 26, 29syl3anc 1366 . . 3 (𝐴Q → (1st𝐴) <N (((1st𝐴) +N 1𝑜) ·N (2nd𝐴)))
31 pinq 9787 . . . . . 6 (((1st𝐴) +N 1𝑜) ∈ N → ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩ ∈ Q)
326, 31syl 17 . . . . 5 (𝐴Q → ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩ ∈ Q)
33 ordpinq 9803 . . . . 5 ((𝐴Q ∧ ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩ ∈ Q) → (𝐴 <Q ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩ ↔ ((1st𝐴) ·N (2nd ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩)) <N ((1st ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩) ·N (2nd𝐴))))
3432, 33mpdan 703 . . . 4 (𝐴Q → (𝐴 <Q ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩ ↔ ((1st𝐴) ·N (2nd ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩)) <N ((1st ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩) ·N (2nd𝐴))))
35 ovex 6718 . . . . . . . 8 ((1st𝐴) +N 1𝑜) ∈ V
364elexi 3244 . . . . . . . 8 1𝑜 ∈ V
3735, 36op2nd 7219 . . . . . . 7 (2nd ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩) = 1𝑜
3837oveq2i 6701 . . . . . 6 ((1st𝐴) ·N (2nd ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩)) = ((1st𝐴) ·N 1𝑜)
39 mulidpi 9746 . . . . . . 7 ((1st𝐴) ∈ N → ((1st𝐴) ·N 1𝑜) = (1st𝐴))
403, 39syl 17 . . . . . 6 (𝐴Q → ((1st𝐴) ·N 1𝑜) = (1st𝐴))
4138, 40syl5eq 2697 . . . . 5 (𝐴Q → ((1st𝐴) ·N (2nd ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩)) = (1st𝐴))
4235, 36op1st 7218 . . . . . . 7 (1st ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩) = ((1st𝐴) +N 1𝑜)
4342oveq1i 6700 . . . . . 6 ((1st ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩) ·N (2nd𝐴)) = (((1st𝐴) +N 1𝑜) ·N (2nd𝐴))
4443a1i 11 . . . . 5 (𝐴Q → ((1st ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩) ·N (2nd𝐴)) = (((1st𝐴) +N 1𝑜) ·N (2nd𝐴)))
4541, 44breq12d 4698 . . . 4 (𝐴Q → (((1st𝐴) ·N (2nd ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩)) <N ((1st ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩) ·N (2nd𝐴)) ↔ (1st𝐴) <N (((1st𝐴) +N 1𝑜) ·N (2nd𝐴))))
4634, 45bitrd 268 . . 3 (𝐴Q → (𝐴 <Q ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩ ↔ (1st𝐴) <N (((1st𝐴) +N 1𝑜) ·N (2nd𝐴))))
4730, 46mpbird 247 . 2 (𝐴Q𝐴 <Q ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩)
48 opeq1 4433 . . . 4 (𝑥 = ((1st𝐴) +N 1𝑜) → ⟨𝑥, 1𝑜⟩ = ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩)
4948breq2d 4697 . . 3 (𝑥 = ((1st𝐴) +N 1𝑜) → (𝐴 <Q𝑥, 1𝑜⟩ ↔ 𝐴 <Q ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩))
5049rspcev 3340 . 2 ((((1st𝐴) +N 1𝑜) ∈ N𝐴 <Q ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩) → ∃𝑥N 𝐴 <Q𝑥, 1𝑜⟩)
516, 47, 50syl2anc 694 1 (𝐴Q → ∃𝑥N 𝐴 <Q𝑥, 1𝑜⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  cop 4216   class class class wbr 4685   × cxp 5141  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  1𝑜c1o 7598  Ncnpi 9704   +N cpli 9705   ·N cmi 9706   <N clti 9707  Qcnq 9712   <Q cltq 9718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-ltpq 9770  df-nq 9772  df-ltnq 9778
This theorem is referenced by:  prlem934  9893
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