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Theorem archnq 11018
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 10963 . . . 4 (𝐴Q𝐴 ∈ (N × N))
2 xp1st 8045 . . . 4 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
31, 2syl 17 . . 3 (𝐴Q → (1st𝐴) ∈ N)
4 1pi 10921 . . 3 1oN
5 addclpi 10930 . . 3 (((1st𝐴) ∈ N ∧ 1oN) → ((1st𝐴) +N 1o) ∈ N)
63, 4, 5sylancl 586 . 2 (𝐴Q → ((1st𝐴) +N 1o) ∈ N)
7 xp2nd 8046 . . . . . 6 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
81, 7syl 17 . . . . 5 (𝐴Q → (2nd𝐴) ∈ N)
9 mulclpi 10931 . . . . 5 ((((1st𝐴) +N 1o) ∈ N ∧ (2nd𝐴) ∈ N) → (((1st𝐴) +N 1o) ·N (2nd𝐴)) ∈ N)
106, 8, 9syl2anc 584 . . . 4 (𝐴Q → (((1st𝐴) +N 1o) ·N (2nd𝐴)) ∈ N)
11 eqid 2735 . . . . . . 7 ((1st𝐴) +N 1o) = ((1st𝐴) +N 1o)
12 oveq2 7439 . . . . . . . . 9 (𝑥 = 1o → ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1o))
1312eqeq1d 2737 . . . . . . . 8 (𝑥 = 1o → (((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1o) ↔ ((1st𝐴) +N 1o) = ((1st𝐴) +N 1o)))
1413rspcev 3622 . . . . . . 7 ((1oN ∧ ((1st𝐴) +N 1o) = ((1st𝐴) +N 1o)) → ∃𝑥N ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1o))
154, 11, 14mp2an 692 . . . . . 6 𝑥N ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1o)
16 ltexpi 10940 . . . . . 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 584 . . . 4 (𝐴Q → (1st𝐴) <N ((1st𝐴) +N 1o))
19 nlt1pi 10944 . . . . 5 ¬ (2nd𝐴) <N 1o
20 ltmpi 10942 . . . . . . 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 10924 . . . . . . . 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 5160 . . . . . 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 10926 . . . . 5 <N Or N
28 ltrelpi 10927 . . . . 5 <N ⊆ (N × N)
2927, 28sotri3 6153 . . . 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 1370 . . 3 (𝐴Q → (1st𝐴) <N (((1st𝐴) +N 1o) ·N (2nd𝐴)))
31 pinq 10965 . . . . . 6 (((1st𝐴) +N 1o) ∈ N → ⟨((1st𝐴) +N 1o), 1o⟩ ∈ Q)
326, 31syl 17 . . . . 5 (𝐴Q → ⟨((1st𝐴) +N 1o), 1o⟩ ∈ Q)
33 ordpinq 10981 . . . . 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 687 . . . 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 7464 . . . . . . . 8 ((1st𝐴) +N 1o) ∈ V
36 1oex 8515 . . . . . . . 8 1o ∈ V
3735, 36op2nd 8022 . . . . . . 7 (2nd ‘⟨((1st𝐴) +N 1o), 1o⟩) = 1o
3837oveq2i 7442 . . . . . 6 ((1st𝐴) ·N (2nd ‘⟨((1st𝐴) +N 1o), 1o⟩)) = ((1st𝐴) ·N 1o)
39 mulidpi 10924 . . . . . . 7 ((1st𝐴) ∈ N → ((1st𝐴) ·N 1o) = (1st𝐴))
403, 39syl 17 . . . . . 6 (𝐴Q → ((1st𝐴) ·N 1o) = (1st𝐴))
4138, 40eqtrid 2787 . . . . 5 (𝐴Q → ((1st𝐴) ·N (2nd ‘⟨((1st𝐴) +N 1o), 1o⟩)) = (1st𝐴))
4235, 36op1st 8021 . . . . . . 7 (1st ‘⟨((1st𝐴) +N 1o), 1o⟩) = ((1st𝐴) +N 1o)
4342oveq1i 7441 . . . . . 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 5161 . . . 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 4878 . . . 4 (𝑥 = ((1st𝐴) +N 1o) → ⟨𝑥, 1o⟩ = ⟨((1st𝐴) +N 1o), 1o⟩)
4948breq2d 5160 . . 3 (𝑥 = ((1st𝐴) +N 1o) → (𝐴 <Q𝑥, 1o⟩ ↔ 𝐴 <Q ⟨((1st𝐴) +N 1o), 1o⟩))
5049rspcev 3622 . 2 ((((1st𝐴) +N 1o) ∈ N𝐴 <Q ⟨((1st𝐴) +N 1o), 1o⟩) → ∃𝑥N 𝐴 <Q𝑥, 1o⟩)
516, 47, 50syl2anc 584 1 (𝐴Q → ∃𝑥N 𝐴 <Q𝑥, 1o⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  cop 4637   class class class wbr 5148   × cxp 5687  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  1oc1o 8498  Ncnpi 10882   +N cpli 10883   ·N cmi 10884   <N clti 10885  Qcnq 10890   <Q cltq 10896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-ni 10910  df-pli 10911  df-mi 10912  df-lti 10913  df-ltpq 10948  df-nq 10950  df-ltnq 10956
This theorem is referenced by:  prlem934  11071
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