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Theorem archnq 11020
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 10965 . . . 4 (𝐴Q𝐴 ∈ (N × N))
2 xp1st 8046 . . . 4 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
31, 2syl 17 . . 3 (𝐴Q → (1st𝐴) ∈ N)
4 1pi 10923 . . 3 1oN
5 addclpi 10932 . . 3 (((1st𝐴) ∈ N ∧ 1oN) → ((1st𝐴) +N 1o) ∈ N)
63, 4, 5sylancl 586 . 2 (𝐴Q → ((1st𝐴) +N 1o) ∈ N)
7 xp2nd 8047 . . . . . 6 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
81, 7syl 17 . . . . 5 (𝐴Q → (2nd𝐴) ∈ N)
9 mulclpi 10933 . . . . 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 2737 . . . . . . 7 ((1st𝐴) +N 1o) = ((1st𝐴) +N 1o)
12 oveq2 7439 . . . . . . . . 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 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 10942 . . . . . 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 10946 . . . . 5 ¬ (2nd𝐴) <N 1o
20 ltmpi 10944 . . . . . . 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 10926 . . . . . . . 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 5155 . . . . . 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 10928 . . . . 5 <N Or N
28 ltrelpi 10929 . . . . 5 <N ⊆ (N × N)
2927, 28sotri3 6150 . . . 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 1373 . . 3 (𝐴Q → (1st𝐴) <N (((1st𝐴) +N 1o) ·N (2nd𝐴)))
31 pinq 10967 . . . . . 6 (((1st𝐴) +N 1o) ∈ N → ⟨((1st𝐴) +N 1o), 1o⟩ ∈ Q)
326, 31syl 17 . . . . 5 (𝐴Q → ⟨((1st𝐴) +N 1o), 1o⟩ ∈ Q)
33 ordpinq 10983 . . . . 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 8516 . . . . . . . 8 1o ∈ V
3735, 36op2nd 8023 . . . . . . 7 (2nd ‘⟨((1st𝐴) +N 1o), 1o⟩) = 1o
3837oveq2i 7442 . . . . . 6 ((1st𝐴) ·N (2nd ‘⟨((1st𝐴) +N 1o), 1o⟩)) = ((1st𝐴) ·N 1o)
39 mulidpi 10926 . . . . . . 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 8022 . . . . . . 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 5156 . . . 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 4873 . . . 4 (𝑥 = ((1st𝐴) +N 1o) → ⟨𝑥, 1o⟩ = ⟨((1st𝐴) +N 1o), 1o⟩)
4948breq2d 5155 . . 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 1540  wcel 2108  wrex 3070  cop 4632   class class class wbr 5143   × cxp 5683  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  1oc1o 8499  Ncnpi 10884   +N cpli 10885   ·N cmi 10886   <N clti 10887  Qcnq 10892   <Q cltq 10898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511  df-ni 10912  df-pli 10913  df-mi 10914  df-lti 10915  df-ltpq 10950  df-nq 10952  df-ltnq 10958
This theorem is referenced by:  prlem934  11073
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