Theorem archnq 10983

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	⊢ (𝐴 ∈ Q → ∃𝑥 ∈ N 𝐴 <Q ⟨𝑥, 1o⟩)

Distinct variable group:   𝑥,𝐴

Proof of Theorem archnq

Step	Hyp	Ref	Expression
1		elpqn 10928	. . . 4 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
2		xp1st 8027	. . . 4 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
3	1, 2	syl 18	. . 3 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) ∈ N)
4		1pi 10886	. . 3 ⊢ 1o ∈ N
5		addclpi 10895	. . 3 ⊢ (((1st ‘𝐴) ∈ N ∧ 1o ∈ N) → ((1st ‘𝐴) +N 1o) ∈ N)
6	3, 4, 5	sylancl 598	. 2 ⊢ (𝐴 ∈ Q → ((1st ‘𝐴) +N 1o) ∈ N)
7		xp2nd 8028	. . . . . 6 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
8	1, 7	syl 18	. . . . 5 ⊢ (𝐴 ∈ Q → (2nd ‘𝐴) ∈ N)
9		mulclpi 10896	. . . . 5 ⊢ ((((1st ‘𝐴) +N 1o) ∈ N ∧ (2nd ‘𝐴) ∈ N) → (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) ∈ N)
10	6, 8, 9	syl2anc 596	. . . 4 ⊢ (𝐴 ∈ Q → (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) ∈ N)
11		eqid 2766	. . . . . . 7 ⊢ ((1st ‘𝐴) +N 1o) = ((1st ‘𝐴) +N 1o)
12		oveq2 7431	. . . . . . . . 9 ⊢ (𝑥 = 1o → ((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1o))
13	12	eqeq1d 2768	. . . . . . . 8 ⊢ (𝑥 = 1o → (((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1o) ↔ ((1st ‘𝐴) +N 1o) = ((1st ‘𝐴) +N 1o)))
14	13	rspcev 3584	. . . . . . 7 ⊢ ((1o ∈ N ∧ ((1st ‘𝐴) +N 1o) = ((1st ‘𝐴) +N 1o)) → ∃𝑥 ∈ N ((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1o))
15	4, 11, 14	mp2an 705	. . . . . 6 ⊢ ∃𝑥 ∈ N ((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1o)
16		ltexpi 10905	. . . . . 6 ⊢ (((1st ‘𝐴) ∈ N ∧ ((1st ‘𝐴) +N 1o) ∈ N) → ((1st ‘𝐴) <N ((1st ‘𝐴) +N 1o) ↔ ∃𝑥 ∈ N ((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1o)))
17	15, 16	mpbiri 261	. . . . 5 ⊢ (((1st ‘𝐴) ∈ N ∧ ((1st ‘𝐴) +N 1o) ∈ N) → (1st ‘𝐴) <N ((1st ‘𝐴) +N 1o))
18	3, 6, 17	syl2anc 596	. . . 4 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) <N ((1st ‘𝐴) +N 1o))
19		nlt1pi 10909	. . . . 5 ⊢ ¬ (2nd ‘𝐴) <N 1o
20		ltmpi 10907	. . . . . . 7 ⊢ (((1st ‘𝐴) +N 1o) ∈ N → ((2nd ‘𝐴) <N 1o ↔ (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) <N (((1st ‘𝐴) +N 1o) ·N 1o)))
21	6, 20	syl 18	. . . . . 6 ⊢ (𝐴 ∈ Q → ((2nd ‘𝐴) <N 1o ↔ (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) <N (((1st ‘𝐴) +N 1o) ·N 1o)))
22		mulidpi 10889	. . . . . . . 8 ⊢ (((1st ‘𝐴) +N 1o) ∈ N → (((1st ‘𝐴) +N 1o) ·N 1o) = ((1st ‘𝐴) +N 1o))
23	6, 22	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ Q → (((1st ‘𝐴) +N 1o) ·N 1o) = ((1st ‘𝐴) +N 1o))
24	23	breq2d 5126	. . . . . 6 ⊢ (𝐴 ∈ Q → ((((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) <N (((1st ‘𝐴) +N 1o) ·N 1o) ↔ (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) <N ((1st ‘𝐴) +N 1o)))
25	21, 24	bitrd 282	. . . . 5 ⊢ (𝐴 ∈ Q → ((2nd ‘𝐴) <N 1o ↔ (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) <N ((1st ‘𝐴) +N 1o)))
26	19, 25	mtbii 329	. . . 4 ⊢ (𝐴 ∈ Q → ¬ (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) <N ((1st ‘𝐴) +N 1o))
27		ltsopi 10891	. . . . 5 ⊢ <N Or N
28		ltrelpi 10892	. . . . 5 ⊢ <N ⊆ (N × N)
29	27, 28	sotri3 6135	. . . 4 ⊢ (((((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) ∈ N ∧ (1st ‘𝐴) <N ((1st ‘𝐴) +N 1o) ∧ ¬ (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) <N ((1st ‘𝐴) +N 1o)) → (1st ‘𝐴) <N (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)))
30	10, 18, 26, 29	syl3anc 1398	. . 3 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) <N (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)))
31		pinq 10930	. . . . . 6 ⊢ (((1st ‘𝐴) +N 1o) ∈ N → ⟨((1st ‘𝐴) +N 1o), 1o⟩ ∈ Q)
32	6, 31	syl 18	. . . . 5 ⊢ (𝐴 ∈ Q → ⟨((1st ‘𝐴) +N 1o), 1o⟩ ∈ Q)
33		ordpinq 10946	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ ⟨((1st ‘𝐴) +N 1o), 1o⟩ ∈ Q) → (𝐴 <Q ⟨((1st ‘𝐴) +N 1o), 1o⟩ ↔ ((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1o), 1o⟩)) <N ((1st ‘⟨((1st ‘𝐴) +N 1o), 1o⟩) ·N (2nd ‘𝐴))))
34	32, 33	mpdan 700	. . . 4 ⊢ (𝐴 ∈ Q → (𝐴 <Q ⟨((1st ‘𝐴) +N 1o), 1o⟩ ↔ ((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1o), 1o⟩)) <N ((1st ‘⟨((1st ‘𝐴) +N 1o), 1o⟩) ·N (2nd ‘𝐴))))
35		ovex 7456	. . . . . . . 8 ⊢ ((1st ‘𝐴) +N 1o) ∈ V
36		1oex 8472	. . . . . . . 8 ⊢ 1o ∈ V
37	35, 36	op2nd 8004	. . . . . . 7 ⊢ (2nd ‘⟨((1st ‘𝐴) +N 1o), 1o⟩) = 1o
38	37	oveq2i 7434	. . . . . 6 ⊢ ((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1o), 1o⟩)) = ((1st ‘𝐴) ·N 1o)
39		mulidpi 10889	. . . . . . 7 ⊢ ((1st ‘𝐴) ∈ N → ((1st ‘𝐴) ·N 1o) = (1st ‘𝐴))
40	3, 39	syl 18	. . . . . 6 ⊢ (𝐴 ∈ Q → ((1st ‘𝐴) ·N 1o) = (1st ‘𝐴))
41	38, 40	eqtrid 2813	. . . . 5 ⊢ (𝐴 ∈ Q → ((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1o), 1o⟩)) = (1st ‘𝐴))
42	35, 36	op1st 8003	. . . . . . 7 ⊢ (1st ‘⟨((1st ‘𝐴) +N 1o), 1o⟩) = ((1st ‘𝐴) +N 1o)
43	42	oveq1i 7433	. . . . . 6 ⊢ ((1st ‘⟨((1st ‘𝐴) +N 1o), 1o⟩) ·N (2nd ‘𝐴)) = (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴))
44	43	a1i 11	. . . . 5 ⊢ (𝐴 ∈ Q → ((1st ‘⟨((1st ‘𝐴) +N 1o), 1o⟩) ·N (2nd ‘𝐴)) = (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)))
45	41, 44	breq12d 5127	. . . 4 ⊢ (𝐴 ∈ Q → (((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1o), 1o⟩)) <N ((1st ‘⟨((1st ‘𝐴) +N 1o), 1o⟩) ·N (2nd ‘𝐴)) ↔ (1st ‘𝐴) <N (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴))))
46	34, 45	bitrd 282	. . 3 ⊢ (𝐴 ∈ Q → (𝐴 <Q ⟨((1st ‘𝐴) +N 1o), 1o⟩ ↔ (1st ‘𝐴) <N (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴))))
47	30, 46	mpbird 260	. 2 ⊢ (𝐴 ∈ Q → 𝐴 <Q ⟨((1st ‘𝐴) +N 1o), 1o⟩)
48		opeq1 4843	. . . 4 ⊢ (𝑥 = ((1st ‘𝐴) +N 1o) → ⟨𝑥, 1o⟩ = ⟨((1st ‘𝐴) +N 1o), 1o⟩)
49	48	breq2d 5126	. . 3 ⊢ (𝑥 = ((1st ‘𝐴) +N 1o) → (𝐴 <Q ⟨𝑥, 1o⟩ ↔ 𝐴 <Q ⟨((1st ‘𝐴) +N 1o), 1o⟩))
50	49	rspcev 3584	. 2 ⊢ ((((1st ‘𝐴) +N 1o) ∈ N ∧ 𝐴 <Q ⟨((1st ‘𝐴) +N 1o), 1o⟩) → ∃𝑥 ∈ N 𝐴 <Q ⟨𝑥, 1o⟩)
51	6, 47, 50	syl2anc 596	1 ⊢ (𝐴 ∈ Q → ∃𝑥 ∈ N 𝐴 <Q ⟨𝑥, 1o⟩)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3092  ⟨cop 4600   class class class wbr 5114   × cxp 5664  ‘cfv 6543  (class class class)co 7423  1^st c1st 7993  2^nd c2nd 7994  1_oc1o 8455  Ncnpi 10847   +_N cpli 10848   ·_N cmi 10849   <_N clti 10850  Qcnq 10855   <_Q cltq 10861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-omul 8467  df-ni 10875  df-pli 10876  df-mi 10877  df-lti 10878  df-ltpq 10913  df-nq 10915  df-ltnq 10921

This theorem is used by:  prlem934  11036