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Theorem archnq 11023
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 10968 . . . 4 (𝐴Q𝐴 ∈ (N × N))
2 xp1st 8035 . . . 4 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
31, 2syl 17 . . 3 (𝐴Q → (1st𝐴) ∈ N)
4 1pi 10926 . . 3 1oN
5 addclpi 10935 . . 3 (((1st𝐴) ∈ N ∧ 1oN) → ((1st𝐴) +N 1o) ∈ N)
63, 4, 5sylancl 584 . 2 (𝐴Q → ((1st𝐴) +N 1o) ∈ N)
7 xp2nd 8036 . . . . . 6 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
81, 7syl 17 . . . . 5 (𝐴Q → (2nd𝐴) ∈ N)
9 mulclpi 10936 . . . . 5 ((((1st𝐴) +N 1o) ∈ N ∧ (2nd𝐴) ∈ N) → (((1st𝐴) +N 1o) ·N (2nd𝐴)) ∈ N)
106, 8, 9syl2anc 582 . . . 4 (𝐴Q → (((1st𝐴) +N 1o) ·N (2nd𝐴)) ∈ N)
11 eqid 2726 . . . . . . 7 ((1st𝐴) +N 1o) = ((1st𝐴) +N 1o)
12 oveq2 7432 . . . . . . . . 9 (𝑥 = 1o → ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1o))
1312eqeq1d 2728 . . . . . . . 8 (𝑥 = 1o → (((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1o) ↔ ((1st𝐴) +N 1o) = ((1st𝐴) +N 1o)))
1413rspcev 3608 . . . . . . 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 10945 . . . . . 6 (((1st𝐴) ∈ N ∧ ((1st𝐴) +N 1o) ∈ N) → ((1st𝐴) <N ((1st𝐴) +N 1o) ↔ ∃𝑥N ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1o)))
1715, 16mpbiri 257 . . . . 5 (((1st𝐴) ∈ N ∧ ((1st𝐴) +N 1o) ∈ N) → (1st𝐴) <N ((1st𝐴) +N 1o))
183, 6, 17syl2anc 582 . . . 4 (𝐴Q → (1st𝐴) <N ((1st𝐴) +N 1o))
19 nlt1pi 10949 . . . . 5 ¬ (2nd𝐴) <N 1o
20 ltmpi 10947 . . . . . . 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 10929 . . . . . . . 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 5165 . . . . . 6 (𝐴Q → ((((1st𝐴) +N 1o) ·N (2nd𝐴)) <N (((1st𝐴) +N 1o) ·N 1o) ↔ (((1st𝐴) +N 1o) ·N (2nd𝐴)) <N ((1st𝐴) +N 1o)))
2521, 24bitrd 278 . . . . 5 (𝐴Q → ((2nd𝐴) <N 1o ↔ (((1st𝐴) +N 1o) ·N (2nd𝐴)) <N ((1st𝐴) +N 1o)))
2619, 25mtbii 325 . . . 4 (𝐴Q → ¬ (((1st𝐴) +N 1o) ·N (2nd𝐴)) <N ((1st𝐴) +N 1o))
27 ltsopi 10931 . . . . 5 <N Or N
28 ltrelpi 10932 . . . . 5 <N ⊆ (N × N)
2927, 28sotri3 6142 . . . 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 10970 . . . . . 6 (((1st𝐴) +N 1o) ∈ N → ⟨((1st𝐴) +N 1o), 1o⟩ ∈ Q)
326, 31syl 17 . . . . 5 (𝐴Q → ⟨((1st𝐴) +N 1o), 1o⟩ ∈ Q)
33 ordpinq 10986 . . . . 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 7457 . . . . . . . 8 ((1st𝐴) +N 1o) ∈ V
36 1oex 8506 . . . . . . . 8 1o ∈ V
3735, 36op2nd 8012 . . . . . . 7 (2nd ‘⟨((1st𝐴) +N 1o), 1o⟩) = 1o
3837oveq2i 7435 . . . . . 6 ((1st𝐴) ·N (2nd ‘⟨((1st𝐴) +N 1o), 1o⟩)) = ((1st𝐴) ·N 1o)
39 mulidpi 10929 . . . . . . 7 ((1st𝐴) ∈ N → ((1st𝐴) ·N 1o) = (1st𝐴))
403, 39syl 17 . . . . . 6 (𝐴Q → ((1st𝐴) ·N 1o) = (1st𝐴))
4138, 40eqtrid 2778 . . . . 5 (𝐴Q → ((1st𝐴) ·N (2nd ‘⟨((1st𝐴) +N 1o), 1o⟩)) = (1st𝐴))
4235, 36op1st 8011 . . . . . . 7 (1st ‘⟨((1st𝐴) +N 1o), 1o⟩) = ((1st𝐴) +N 1o)
4342oveq1i 7434 . . . . . 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 5166 . . . 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 278 . . 3 (𝐴Q → (𝐴 <Q ⟨((1st𝐴) +N 1o), 1o⟩ ↔ (1st𝐴) <N (((1st𝐴) +N 1o) ·N (2nd𝐴))))
4730, 46mpbird 256 . 2 (𝐴Q𝐴 <Q ⟨((1st𝐴) +N 1o), 1o⟩)
48 opeq1 4879 . . . 4 (𝑥 = ((1st𝐴) +N 1o) → ⟨𝑥, 1o⟩ = ⟨((1st𝐴) +N 1o), 1o⟩)
4948breq2d 5165 . . 3 (𝑥 = ((1st𝐴) +N 1o) → (𝐴 <Q𝑥, 1o⟩ ↔ 𝐴 <Q ⟨((1st𝐴) +N 1o), 1o⟩))
5049rspcev 3608 . 2 ((((1st𝐴) +N 1o) ∈ N𝐴 <Q ⟨((1st𝐴) +N 1o), 1o⟩) → ∃𝑥N 𝐴 <Q𝑥, 1o⟩)
516, 47, 50syl2anc 582 1 (𝐴Q → ∃𝑥N 𝐴 <Q𝑥, 1o⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wrex 3060  cop 4639   class class class wbr 5153   × cxp 5680  cfv 6554  (class class class)co 7424  1st c1st 8001  2nd c2nd 8002  1oc1o 8489  Ncnpi 10887   +N cpli 10888   ·N cmi 10889   <N clti 10890  Qcnq 10895   <Q cltq 10901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-oadd 8500  df-omul 8501  df-ni 10915  df-pli 10916  df-mi 10917  df-lti 10918  df-ltpq 10953  df-nq 10955  df-ltnq 10961
This theorem is referenced by:  prlem934  11076
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