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Theorem archnq 9840
 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 9785 . . . 4 (𝐴Q𝐴 ∈ (N × N))
2 xp1st 7242 . . . 4 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
31, 2syl 17 . . 3 (𝐴Q → (1st𝐴) ∈ N)
4 1pi 9743 . . 3 1𝑜N
5 addclpi 9752 . . 3 (((1st𝐴) ∈ N ∧ 1𝑜N) → ((1st𝐴) +N 1𝑜) ∈ N)
63, 4, 5sylancl 695 . 2 (𝐴Q → ((1st𝐴) +N 1𝑜) ∈ N)
7 xp2nd 7243 . . . . . 6 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
81, 7syl 17 . . . . 5 (𝐴Q → (2nd𝐴) ∈ N)
9 mulclpi 9753 . . . . 5 ((((1st𝐴) +N 1𝑜) ∈ N ∧ (2nd𝐴) ∈ N) → (((1st𝐴) +N 1𝑜) ·N (2nd𝐴)) ∈ N)
106, 8, 9syl2anc 694 . . . 4 (𝐴Q → (((1st𝐴) +N 1𝑜) ·N (2nd𝐴)) ∈ N)
11 eqid 2651 . . . . . . 7 ((1st𝐴) +N 1𝑜) = ((1st𝐴) +N 1𝑜)
12 oveq2 6698 . . . . . . . . 9 (𝑥 = 1𝑜 → ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1𝑜))
1312eqeq1d 2653 . . . . . . . 8 (𝑥 = 1𝑜 → (((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1𝑜) ↔ ((1st𝐴) +N 1𝑜) = ((1st𝐴) +N 1𝑜)))
1413rspcev 3340 . . . . . . 7 ((1𝑜N ∧ ((1st𝐴) +N 1𝑜) = ((1st𝐴) +N 1𝑜)) → ∃𝑥N ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1𝑜))
154, 11, 14mp2an 708 . . . . . 6 𝑥N ((1st𝐴) +N 𝑥) = ((1st𝐴) +N 1𝑜)
16 ltexpi 9762 . . . . . 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 694 . . . 4 (𝐴Q → (1st𝐴) <N ((1st𝐴) +N 1𝑜))
19 nlt1pi 9766 . . . . 5 ¬ (2nd𝐴) <N 1𝑜
20 ltmpi 9764 . . . . . . 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 9746 . . . . . . . 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 4697 . . . . . 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 9748 . . . . 5 <N Or N
28 ltrelpi 9749 . . . . 5 <N ⊆ (N × N)
2927, 28sotri3 5561 . . . 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 1366 . . 3 (𝐴Q → (1st𝐴) <N (((1st𝐴) +N 1𝑜) ·N (2nd𝐴)))
31 pinq 9787 . . . . . 6 (((1st𝐴) +N 1𝑜) ∈ N → ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩ ∈ Q)
326, 31syl 17 . . . . 5 (𝐴Q → ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩ ∈ Q)
33 ordpinq 9803 . . . . 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 703 . . . 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 6718 . . . . . . . 8 ((1st𝐴) +N 1𝑜) ∈ V
364elexi 3244 . . . . . . . 8 1𝑜 ∈ V
3735, 36op2nd 7219 . . . . . . 7 (2nd ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩) = 1𝑜
3837oveq2i 6701 . . . . . 6 ((1st𝐴) ·N (2nd ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩)) = ((1st𝐴) ·N 1𝑜)
39 mulidpi 9746 . . . . . . 7 ((1st𝐴) ∈ N → ((1st𝐴) ·N 1𝑜) = (1st𝐴))
403, 39syl 17 . . . . . 6 (𝐴Q → ((1st𝐴) ·N 1𝑜) = (1st𝐴))
4138, 40syl5eq 2697 . . . . 5 (𝐴Q → ((1st𝐴) ·N (2nd ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩)) = (1st𝐴))
4235, 36op1st 7218 . . . . . . 7 (1st ‘⟨((1st𝐴) +N 1𝑜), 1𝑜⟩) = ((1st𝐴) +N 1𝑜)
4342oveq1i 6700 . . . . . 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 4698 . . . 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 4433 . . . 4 (𝑥 = ((1st𝐴) +N 1𝑜) → ⟨𝑥, 1𝑜⟩ = ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩)
4948breq2d 4697 . . 3 (𝑥 = ((1st𝐴) +N 1𝑜) → (𝐴 <Q𝑥, 1𝑜⟩ ↔ 𝐴 <Q ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩))
5049rspcev 3340 . 2 ((((1st𝐴) +N 1𝑜) ∈ N𝐴 <Q ⟨((1st𝐴) +N 1𝑜), 1𝑜⟩) → ∃𝑥N 𝐴 <Q𝑥, 1𝑜⟩)
516, 47, 50syl2anc 694 1 (𝐴Q → ∃𝑥N 𝐴 <Q𝑥, 1𝑜⟩)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∃wrex 2942  ⟨cop 4216   class class class wbr 4685   × cxp 5141  ‘cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  1𝑜c1o 7598  Ncnpi 9704   +N cpli 9705   ·N cmi 9706
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