Theorem archnq 10953

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 10898	. . . 4 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
2		xp1st 8006	. . . 4 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
3	1, 2	syl 18	. . 3 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) ∈ N)
4		1pi 10856	. . 3 ⊢ 1o ∈ N
5		addclpi 10865	. . 3 ⊢ (((1st ‘𝐴) ∈ N ∧ 1o ∈ N) → ((1st ‘𝐴) +N 1o) ∈ N)
6	3, 4, 5	sylancl 597	. 2 ⊢ (𝐴 ∈ Q → ((1st ‘𝐴) +N 1o) ∈ N)
7		xp2nd 8007	. . . . . 6 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
8	1, 7	syl 18	. . . . 5 ⊢ (𝐴 ∈ Q → (2nd ‘𝐴) ∈ N)
9		mulclpi 10866	. . . . 5 ⊢ ((((1st ‘𝐴) +N 1o) ∈ N ∧ (2nd ‘𝐴) ∈ N) → (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) ∈ N)
10	6, 8, 9	syl2anc 595	. . . 4 ⊢ (𝐴 ∈ Q → (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) ∈ N)
11		eqid 2765	. . . . . . 7 ⊢ ((1st ‘𝐴) +N 1o) = ((1st ‘𝐴) +N 1o)
12		oveq2 7408	. . . . . . . . 9 ⊢ (𝑥 = 1o → ((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1o))
13	12	eqeq1d 2767	. . . . . . . 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 704	. . . . . 6 ⊢ ∃𝑥 ∈ N ((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1o)
16		ltexpi 10875	. . . . . 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 595	. . . 4 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) <N ((1st ‘𝐴) +N 1o))
19		nlt1pi 10879	. . . . 5 ⊢ ¬ (2nd ‘𝐴) <N 1o
20		ltmpi 10877	. . . . . . 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 10859	. . . . . . . 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 5117	. . . . . 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 10861	. . . . 5 ⊢ <N Or N
28		ltrelpi 10862	. . . . 5 ⊢ <N ⊆ (N × N)
29	27, 28	sotri3 6121	. . . 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 1394	. . 3 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) <N (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)))
31		pinq 10900	. . . . . 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 10916	. . . . 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 699	. . . 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 7433	. . . . . . . 8 ⊢ ((1st ‘𝐴) +N 1o) ∈ V
36		1oex 8451	. . . . . . . 8 ⊢ 1o ∈ V
37	35, 36	op2nd 7983	. . . . . . 7 ⊢ (2nd ‘⟨((1st ‘𝐴) +N 1o), 1o⟩) = 1o
38	37	oveq2i 7411	. . . . . 6 ⊢ ((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1o), 1o⟩)) = ((1st ‘𝐴) ·N 1o)
39		mulidpi 10859	. . . . . . 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 2812	. . . . 5 ⊢ (𝐴 ∈ Q → ((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1o), 1o⟩)) = (1st ‘𝐴))
42	35, 36	op1st 7982	. . . . . . 7 ⊢ (1st ‘⟨((1st ‘𝐴) +N 1o), 1o⟩) = ((1st ‘𝐴) +N 1o)
43	42	oveq1i 7410	. . . . . 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 5118	. . . 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 4834	. . . 4 ⊢ (𝑥 = ((1st ‘𝐴) +N 1o) → ⟨𝑥, 1o⟩ = ⟨((1st ‘𝐴) +N 1o), 1o⟩)
49	48	breq2d 5117	. . 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 595	1 ⊢ (𝐴 ∈ Q → ∃𝑥 ∈ N 𝐴 <Q ⟨𝑥, 1o⟩)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∃wrex 3089  ⟨cop 4591   class class class wbr 5105   × cxp 5650  ‘cfv 6525  (class class class)co 7400  1^st c1st 7972  2^nd c2nd 7973  1_oc1o 8434  Ncnpi 10817   +_N cpli 10818   ·_N cmi 10819   <_N clti 10820  Qcnq 10825   <_Q cltq 10831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-oadd 8445  df-omul 8446  df-ni 10845  df-pli 10846  df-mi 10847  df-lti 10848  df-ltpq 10883  df-nq 10885  df-ltnq 10891

This theorem is referenced by:  prlem934  11006