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Theorem archnq 10841
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 10786 . . . 4 (𝐴Q𝐴 ∈ (N × N))
2 xp1st 7935 . . . 4 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
31, 2syl 17 . . 3 (𝐴Q → (1st𝐴) ∈ N)
4 1pi 10744 . . 3 1oN
5 addclpi 10753 . . 3 (((1st𝐴) ∈ N ∧ 1oN) → ((1st𝐴) +N 1o) ∈ N)
63, 4, 5sylancl 587 . 2 (𝐴Q → ((1st𝐴) +N 1o) ∈ N)
7 xp2nd 7936 . . . . . 6 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
81, 7syl 17 . . . . 5 (𝐴Q → (2nd𝐴) ∈ N)
9 mulclpi 10754 . . . . 5 ((((1st𝐴) +N 1o) ∈ N ∧ (2nd𝐴) ∈ N) → (((1st𝐴) +N 1o) ·N (2nd𝐴)) ∈ N)
106, 8, 9syl2anc 585 . . . 4 (𝐴Q → (((1st𝐴) +N 1o) ·N (2nd𝐴)) ∈ N)
11 eqid 2737 . . . . . . 7 ((1st𝐴) +N 1o) = ((1st𝐴) +N 1o)
12 oveq2 7349 . . . . . . . . 9 (𝑥 = 1o → ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1o))
1312eqeq1d 2739 . . . . . . . 8 (𝑥 = 1o → (((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1o) ↔ ((1st𝐴) +N 1o) = ((1st𝐴) +N 1o)))
1413rspcev 3573 . . . . . . 7 ((1oN ∧ ((1st𝐴) +N 1o) = ((1st𝐴) +N 1o)) → ∃𝑥N ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1o))
154, 11, 14mp2an 690 . . . . . 6 𝑥N ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1o)
16 ltexpi 10763 . . . . . 6 (((1st𝐴) ∈ N ∧ ((1st𝐴) +N 1o) ∈ N) → ((1st𝐴) <N ((1st𝐴) +N 1o) ↔ ∃𝑥N ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1o)))
1715, 16mpbiri 258 . . . . 5 (((1st𝐴) ∈ N ∧ ((1st𝐴) +N 1o) ∈ N) → (1st𝐴) <N ((1st𝐴) +N 1o))
183, 6, 17syl2anc 585 . . . 4 (𝐴Q → (1st𝐴) <N ((1st𝐴) +N 1o))
19 nlt1pi 10767 . . . . 5 ¬ (2nd𝐴) <N 1o
20 ltmpi 10765 . . . . . . 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 10747 . . . . . . . 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 5108 . . . . . 6 (𝐴Q → ((((1st𝐴) +N 1o) ·N (2nd𝐴)) <N (((1st𝐴) +N 1o) ·N 1o) ↔ (((1st𝐴) +N 1o) ·N (2nd𝐴)) <N ((1st𝐴) +N 1o)))
2521, 24bitrd 279 . . . . 5 (𝐴Q → ((2nd𝐴) <N 1o ↔ (((1st𝐴) +N 1o) ·N (2nd𝐴)) <N ((1st𝐴) +N 1o)))
2619, 25mtbii 326 . . . 4 (𝐴Q → ¬ (((1st𝐴) +N 1o) ·N (2nd𝐴)) <N ((1st𝐴) +N 1o))
27 ltsopi 10749 . . . . 5 <N Or N
28 ltrelpi 10750 . . . . 5 <N ⊆ (N × N)
2927, 28sotri3 6074 . . . 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 1371 . . 3 (𝐴Q → (1st𝐴) <N (((1st𝐴) +N 1o) ·N (2nd𝐴)))
31 pinq 10788 . . . . . 6 (((1st𝐴) +N 1o) ∈ N → ⟨((1st𝐴) +N 1o), 1o⟩ ∈ Q)
326, 31syl 17 . . . . 5 (𝐴Q → ⟨((1st𝐴) +N 1o), 1o⟩ ∈ Q)
33 ordpinq 10804 . . . . 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 685 . . . 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 7374 . . . . . . . 8 ((1st𝐴) +N 1o) ∈ V
36 1oex 8381 . . . . . . . 8 1o ∈ V
3735, 36op2nd 7912 . . . . . . 7 (2nd ‘⟨((1st𝐴) +N 1o), 1o⟩) = 1o
3837oveq2i 7352 . . . . . 6 ((1st𝐴) ·N (2nd ‘⟨((1st𝐴) +N 1o), 1o⟩)) = ((1st𝐴) ·N 1o)
39 mulidpi 10747 . . . . . . 7 ((1st𝐴) ∈ N → ((1st𝐴) ·N 1o) = (1st𝐴))
403, 39syl 17 . . . . . 6 (𝐴Q → ((1st𝐴) ·N 1o) = (1st𝐴))
4138, 40eqtrid 2789 . . . . 5 (𝐴Q → ((1st𝐴) ·N (2nd ‘⟨((1st𝐴) +N 1o), 1o⟩)) = (1st𝐴))
4235, 36op1st 7911 . . . . . . 7 (1st ‘⟨((1st𝐴) +N 1o), 1o⟩) = ((1st𝐴) +N 1o)
4342oveq1i 7351 . . . . . 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 5109 . . . 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 279 . . 3 (𝐴Q → (𝐴 <Q ⟨((1st𝐴) +N 1o), 1o⟩ ↔ (1st𝐴) <N (((1st𝐴) +N 1o) ·N (2nd𝐴))))
4730, 46mpbird 257 . 2 (𝐴Q𝐴 <Q ⟨((1st𝐴) +N 1o), 1o⟩)
48 opeq1 4821 . . . 4 (𝑥 = ((1st𝐴) +N 1o) → ⟨𝑥, 1o⟩ = ⟨((1st𝐴) +N 1o), 1o⟩)
4948breq2d 5108 . . 3 (𝑥 = ((1st𝐴) +N 1o) → (𝐴 <Q𝑥, 1o⟩ ↔ 𝐴 <Q ⟨((1st𝐴) +N 1o), 1o⟩))
5049rspcev 3573 . 2 ((((1st𝐴) +N 1o) ∈ N𝐴 <Q ⟨((1st𝐴) +N 1o), 1o⟩) → ∃𝑥N 𝐴 <Q𝑥, 1o⟩)
516, 47, 50syl2anc 585 1 (𝐴Q → ∃𝑥N 𝐴 <Q𝑥, 1o⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1541  wcel 2106  wrex 3071  cop 4583   class class class wbr 5096   × cxp 5622  cfv 6483  (class class class)co 7341  1st c1st 7901  2nd c2nd 7902  1oc1o 8364  Ncnpi 10705   +N cpli 10706   ·N cmi 10707   <N clti 10708  Qcnq 10713   <Q cltq 10719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5247  ax-nul 5254  ax-pr 5376  ax-un 7654
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3920  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-int 4899  df-iun 4947  df-br 5097  df-opab 5159  df-mpt 5180  df-tr 5214  df-id 5522  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5579  df-we 5581  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6242  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7785  df-1st 7903  df-2nd 7904  df-frecs 8171  df-wrecs 8202  df-recs 8276  df-rdg 8315  df-1o 8371  df-oadd 8375  df-omul 8376  df-ni 10733  df-pli 10734  df-mi 10735  df-lti 10736  df-ltpq 10771  df-nq 10773  df-ltnq 10779
This theorem is referenced by:  prlem934  10894
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