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Theorem archnq 9994
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 9939 . . . 4 (𝐴Q𝐴 ∈ (N × N))
2 xp1st 7365 . . . 4 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
31, 2syl 17 . . 3 (𝐴Q → (1st𝐴) ∈ N)
4 1pi 9897 . . 3 1𝑜N
5 addclpi 9906 . . 3 (((1st𝐴) ∈ N ∧ 1𝑜N) → ((1st𝐴) +N 1𝑜) ∈ N)
63, 4, 5sylancl 697 . 2 (𝐴Q → ((1st𝐴) +N 1𝑜) ∈ N)
7 xp2nd 7366 . . . . . 6 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
81, 7syl 17 . . . . 5 (𝐴Q → (2nd𝐴) ∈ N)
9 mulclpi 9907 . . . . 5 ((((1st𝐴) +N 1𝑜) ∈ N ∧ (2nd𝐴) ∈ N) → (((1st𝐴) +N 1𝑜) ·N (2nd𝐴)) ∈ N)
106, 8, 9syl2anc 696 . . . 4 (𝐴Q → (((1st𝐴) +N 1𝑜) ·N (2nd𝐴)) ∈ N)
11 eqid 2760 . . . . . . 7 ((1st𝐴) +N 1𝑜) = ((1st𝐴) +N 1𝑜)
12 oveq2 6821 . . . . . . . . 9 (𝑥 = 1𝑜 → ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1𝑜))
1312eqeq1d 2762 . . . . . . . 8 (𝑥 = 1𝑜 → (((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1𝑜) ↔ ((1st𝐴) +N 1𝑜) = ((1st𝐴) +N 1𝑜)))
1413rspcev 3449 . . . . . . 7 ((1𝑜N ∧ ((1st𝐴) +N 1𝑜) = ((1st𝐴) +N 1𝑜)) → ∃𝑥N ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1𝑜))
154, 11, 14mp2an 710 . . . . . 6 𝑥N ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1𝑜)
16 ltexpi 9916 . . . . . 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 696 . . . 4 (𝐴Q → (1st𝐴) <N ((1st𝐴) +N 1𝑜))
19 nlt1pi 9920 . . . . 5 ¬ (2nd𝐴) <N 1𝑜
20 ltmpi 9918 . . . . . . 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 9900 . . . . . . . 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 4816 . . . . . 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 9902 . . . . 5 <N Or N
28 ltrelpi 9903 . . . . 5 <N ⊆ (N × N)
2927, 28sotri3 5684 . . . 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 1477 . . 3 (𝐴Q → (1st𝐴) <N (((1st𝐴) +N 1𝑜) ·N (2nd𝐴)))
31 pinq 9941 . . . . . 6 (((1st𝐴) +N 1𝑜) ∈ N → ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩ ∈ Q)
326, 31syl 17 . . . . 5 (𝐴Q → ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩ ∈ Q)
33 ordpinq 9957 . . . . 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 705 . . . 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 6841 . . . . . . . 8 ((1st𝐴) +N 1𝑜) ∈ V
364elexi 3353 . . . . . . . 8 1𝑜 ∈ V
3735, 36op2nd 7342 . . . . . . 7 (2nd ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩) = 1𝑜
3837oveq2i 6824 . . . . . 6 ((1st𝐴) ·N (2nd ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩)) = ((1st𝐴) ·N 1𝑜)
39 mulidpi 9900 . . . . . . 7 ((1st𝐴) ∈ N → ((1st𝐴) ·N 1𝑜) = (1st𝐴))
403, 39syl 17 . . . . . 6 (𝐴Q → ((1st𝐴) ·N 1𝑜) = (1st𝐴))
4138, 40syl5eq 2806 . . . . 5 (𝐴Q → ((1st𝐴) ·N (2nd ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩)) = (1st𝐴))
4235, 36op1st 7341 . . . . . . 7 (1st ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩) = ((1st𝐴) +N 1𝑜)
4342oveq1i 6823 . . . . . 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 4817 . . . 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 4553 . . . 4 (𝑥 = ((1st𝐴) +N 1𝑜) → ⟨𝑥, 1𝑜⟩ = ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩)
4948breq2d 4816 . . 3 (𝑥 = ((1st𝐴) +N 1𝑜) → (𝐴 <Q𝑥, 1𝑜⟩ ↔ 𝐴 <Q ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩))
5049rspcev 3449 . 2 ((((1st𝐴) +N 1𝑜) ∈ N𝐴 <Q ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩) → ∃𝑥N 𝐴 <Q𝑥, 1𝑜⟩)
516, 47, 50syl2anc 696 1 (𝐴Q → ∃𝑥N 𝐴 <Q𝑥, 1𝑜⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wrex 3051  cop 4327   class class class wbr 4804   × cxp 5264  cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332  1𝑜c1o 7722  Ncnpi 9858   +N cpli 9859   ·N cmi 9860   <N clti 9861  Qcnq 9866   <Q cltq 9872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-omul 7734  df-ni 9886  df-pli 9887  df-mi 9888  df-lti 9889  df-ltpq 9924  df-nq 9926  df-ltnq 9932
This theorem is referenced by:  prlem934  10047
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