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Theorem archnq 10395
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 10340 . . . 4 (𝐴Q𝐴 ∈ (N × N))
2 xp1st 7707 . . . 4 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
31, 2syl 17 . . 3 (𝐴Q → (1st𝐴) ∈ N)
4 1pi 10298 . . 3 1oN
5 addclpi 10307 . . 3 (((1st𝐴) ∈ N ∧ 1oN) → ((1st𝐴) +N 1o) ∈ N)
63, 4, 5sylancl 589 . 2 (𝐴Q → ((1st𝐴) +N 1o) ∈ N)
7 xp2nd 7708 . . . . . 6 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
81, 7syl 17 . . . . 5 (𝐴Q → (2nd𝐴) ∈ N)
9 mulclpi 10308 . . . . 5 ((((1st𝐴) +N 1o) ∈ N ∧ (2nd𝐴) ∈ N) → (((1st𝐴) +N 1o) ·N (2nd𝐴)) ∈ N)
106, 8, 9syl2anc 587 . . . 4 (𝐴Q → (((1st𝐴) +N 1o) ·N (2nd𝐴)) ∈ N)
11 eqid 2801 . . . . . . 7 ((1st𝐴) +N 1o) = ((1st𝐴) +N 1o)
12 oveq2 7147 . . . . . . . . 9 (𝑥 = 1o → ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1o))
1312eqeq1d 2803 . . . . . . . 8 (𝑥 = 1o → (((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1o) ↔ ((1st𝐴) +N 1o) = ((1st𝐴) +N 1o)))
1413rspcev 3574 . . . . . . 7 ((1oN ∧ ((1st𝐴) +N 1o) = ((1st𝐴) +N 1o)) → ∃𝑥N ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1o))
154, 11, 14mp2an 691 . . . . . 6 𝑥N ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1o)
16 ltexpi 10317 . . . . . 6 (((1st𝐴) ∈ N ∧ ((1st𝐴) +N 1o) ∈ N) → ((1st𝐴) <N ((1st𝐴) +N 1o) ↔ ∃𝑥N ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1o)))
1715, 16mpbiri 261 . . . . 5 (((1st𝐴) ∈ N ∧ ((1st𝐴) +N 1o) ∈ N) → (1st𝐴) <N ((1st𝐴) +N 1o))
183, 6, 17syl2anc 587 . . . 4 (𝐴Q → (1st𝐴) <N ((1st𝐴) +N 1o))
19 nlt1pi 10321 . . . . 5 ¬ (2nd𝐴) <N 1o
20 ltmpi 10319 . . . . . . 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 10301 . . . . . . . 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 5045 . . . . . 6 (𝐴Q → ((((1st𝐴) +N 1o) ·N (2nd𝐴)) <N (((1st𝐴) +N 1o) ·N 1o) ↔ (((1st𝐴) +N 1o) ·N (2nd𝐴)) <N ((1st𝐴) +N 1o)))
2521, 24bitrd 282 . . . . 5 (𝐴Q → ((2nd𝐴) <N 1o ↔ (((1st𝐴) +N 1o) ·N (2nd𝐴)) <N ((1st𝐴) +N 1o)))
2619, 25mtbii 329 . . . 4 (𝐴Q → ¬ (((1st𝐴) +N 1o) ·N (2nd𝐴)) <N ((1st𝐴) +N 1o))
27 ltsopi 10303 . . . . 5 <N Or N
28 ltrelpi 10304 . . . . 5 <N ⊆ (N × N)
2927, 28sotri3 5961 . . . 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 1368 . . 3 (𝐴Q → (1st𝐴) <N (((1st𝐴) +N 1o) ·N (2nd𝐴)))
31 pinq 10342 . . . . . 6 (((1st𝐴) +N 1o) ∈ N → ⟨((1st𝐴) +N 1o), 1o⟩ ∈ Q)
326, 31syl 17 . . . . 5 (𝐴Q → ⟨((1st𝐴) +N 1o), 1o⟩ ∈ Q)
33 ordpinq 10358 . . . . 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 686 . . . 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 7172 . . . . . . . 8 ((1st𝐴) +N 1o) ∈ V
36 1oex 8097 . . . . . . . 8 1o ∈ V
3735, 36op2nd 7684 . . . . . . 7 (2nd ‘⟨((1st𝐴) +N 1o), 1o⟩) = 1o
3837oveq2i 7150 . . . . . 6 ((1st𝐴) ·N (2nd ‘⟨((1st𝐴) +N 1o), 1o⟩)) = ((1st𝐴) ·N 1o)
39 mulidpi 10301 . . . . . . 7 ((1st𝐴) ∈ N → ((1st𝐴) ·N 1o) = (1st𝐴))
403, 39syl 17 . . . . . 6 (𝐴Q → ((1st𝐴) ·N 1o) = (1st𝐴))
4138, 40syl5eq 2848 . . . . 5 (𝐴Q → ((1st𝐴) ·N (2nd ‘⟨((1st𝐴) +N 1o), 1o⟩)) = (1st𝐴))
4235, 36op1st 7683 . . . . . . 7 (1st ‘⟨((1st𝐴) +N 1o), 1o⟩) = ((1st𝐴) +N 1o)
4342oveq1i 7149 . . . . . 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 5046 . . . 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 282 . . 3 (𝐴Q → (𝐴 <Q ⟨((1st𝐴) +N 1o), 1o⟩ ↔ (1st𝐴) <N (((1st𝐴) +N 1o) ·N (2nd𝐴))))
4730, 46mpbird 260 . 2 (𝐴Q𝐴 <Q ⟨((1st𝐴) +N 1o), 1o⟩)
48 opeq1 4766 . . . 4 (𝑥 = ((1st𝐴) +N 1o) → ⟨𝑥, 1o⟩ = ⟨((1st𝐴) +N 1o), 1o⟩)
4948breq2d 5045 . . 3 (𝑥 = ((1st𝐴) +N 1o) → (𝐴 <Q𝑥, 1o⟩ ↔ 𝐴 <Q ⟨((1st𝐴) +N 1o), 1o⟩))
5049rspcev 3574 . 2 ((((1st𝐴) +N 1o) ∈ N𝐴 <Q ⟨((1st𝐴) +N 1o), 1o⟩) → ∃𝑥N 𝐴 <Q𝑥, 1o⟩)
516, 47, 50syl2anc 587 1 (𝐴Q → ∃𝑥N 𝐴 <Q𝑥, 1o⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wrex 3110  cop 4534   class class class wbr 5033   × cxp 5521  cfv 6328  (class class class)co 7139  1st c1st 7673  2nd c2nd 7674  1oc1o 8082  Ncnpi 10259   +N cpli 10260   ·N cmi 10261   <N clti 10262  Qcnq 10267   <Q cltq 10273
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-omul 8094  df-ni 10287  df-pli 10288  df-mi 10289  df-lti 10290  df-ltpq 10325  df-nq 10327  df-ltnq 10333
This theorem is referenced by:  prlem934  10448
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