Theorem archnq 10901

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 10846	. . . 4 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
2		xp1st 7970	. . . 4 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) ∈ N)
4		1pi 10804	. . 3 ⊢ 1o ∈ N
5		addclpi 10813	. . 3 ⊢ (((1st ‘𝐴) ∈ N ∧ 1o ∈ N) → ((1st ‘𝐴) +N 1o) ∈ N)
6	3, 4, 5	sylancl 592	. 2 ⊢ (𝐴 ∈ Q → ((1st ‘𝐴) +N 1o) ∈ N)
7		xp2nd 7971	. . . . . 6 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
8	1, 7	syl 17	. . . . 5 ⊢ (𝐴 ∈ Q → (2nd ‘𝐴) ∈ N)
9		mulclpi 10814	. . . . 5 ⊢ ((((1st ‘𝐴) +N 1o) ∈ N ∧ (2nd ‘𝐴) ∈ N) → (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) ∈ N)
10	6, 8, 9	syl2anc 590	. . . 4 ⊢ (𝐴 ∈ Q → (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) ∈ N)
11		eqid 2740	. . . . . . 7 ⊢ ((1st ‘𝐴) +N 1o) = ((1st ‘𝐴) +N 1o)
12		oveq2 7371	. . . . . . . . 9 ⊢ (𝑥 = 1o → ((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1o))
13	12	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑥 = 1o → (((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1o) ↔ ((1st ‘𝐴) +N 1o) = ((1st ‘𝐴) +N 1o)))
14	13	rspcev 3567	. . . . . . 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 698	. . . . . 6 ⊢ ∃𝑥 ∈ N ((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1o)
16		ltexpi 10823	. . . . . 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 259	. . . . 5 ⊢ (((1st ‘𝐴) ∈ N ∧ ((1st ‘𝐴) +N 1o) ∈ N) → (1st ‘𝐴) <N ((1st ‘𝐴) +N 1o))
18	3, 6, 17	syl2anc 590	. . . 4 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) <N ((1st ‘𝐴) +N 1o))
19		nlt1pi 10827	. . . . 5 ⊢ ¬ (2nd ‘𝐴) <N 1o
20		ltmpi 10825	. . . . . . 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 17	. . . . . 6 ⊢ (𝐴 ∈ Q → ((2nd ‘𝐴) <N 1o ↔ (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) <N (((1st ‘𝐴) +N 1o) ·N 1o)))
22		mulidpi 10807	. . . . . . . 8 ⊢ (((1st ‘𝐴) +N 1o) ∈ N → (((1st ‘𝐴) +N 1o) ·N 1o) = ((1st ‘𝐴) +N 1o))
23	6, 22	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ Q → (((1st ‘𝐴) +N 1o) ·N 1o) = ((1st ‘𝐴) +N 1o))
24	23	breq2d 5091	. . . . . 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 280	. . . . 5 ⊢ (𝐴 ∈ Q → ((2nd ‘𝐴) <N 1o ↔ (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) <N ((1st ‘𝐴) +N 1o)))
26	19, 25	mtbii 327	. . . 4 ⊢ (𝐴 ∈ Q → ¬ (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) <N ((1st ‘𝐴) +N 1o))
27		ltsopi 10809	. . . . 5 ⊢ <N Or N
28		ltrelpi 10810	. . . . 5 ⊢ <N ⊆ (N × N)
29	27, 28	sotri3 6087	. . . 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 1379	. . 3 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) <N (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)))
31		pinq 10848	. . . . . 6 ⊢ (((1st ‘𝐴) +N 1o) ∈ N → ⟨((1st ‘𝐴) +N 1o), 1o⟩ ∈ Q)
32	6, 31	syl 17	. . . . 5 ⊢ (𝐴 ∈ Q → ⟨((1st ‘𝐴) +N 1o), 1o⟩ ∈ Q)
33		ordpinq 10864	. . . . 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 693	. . . 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 7396	. . . . . . . 8 ⊢ ((1st ‘𝐴) +N 1o) ∈ V
36		1oex 8412	. . . . . . . 8 ⊢ 1o ∈ V
37	35, 36	op2nd 7947	. . . . . . 7 ⊢ (2nd ‘⟨((1st ‘𝐴) +N 1o), 1o⟩) = 1o
38	37	oveq2i 7374	. . . . . 6 ⊢ ((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1o), 1o⟩)) = ((1st ‘𝐴) ·N 1o)
39		mulidpi 10807	. . . . . . 7 ⊢ ((1st ‘𝐴) ∈ N → ((1st ‘𝐴) ·N 1o) = (1st ‘𝐴))
40	3, 39	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Q → ((1st ‘𝐴) ·N 1o) = (1st ‘𝐴))
41	38, 40	eqtrid 2787	. . . . 5 ⊢ (𝐴 ∈ Q → ((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1o), 1o⟩)) = (1st ‘𝐴))
42	35, 36	op1st 7946	. . . . . . 7 ⊢ (1st ‘⟨((1st ‘𝐴) +N 1o), 1o⟩) = ((1st ‘𝐴) +N 1o)
43	42	oveq1i 7373	. . . . . 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 5092	. . . 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 280	. . 3 ⊢ (𝐴 ∈ Q → (𝐴 <Q ⟨((1st ‘𝐴) +N 1o), 1o⟩ ↔ (1st ‘𝐴) <N (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴))))
47	30, 46	mpbird 258	. 2 ⊢ (𝐴 ∈ Q → 𝐴 <Q ⟨((1st ‘𝐴) +N 1o), 1o⟩)
48		opeq1 4811	. . . 4 ⊢ (𝑥 = ((1st ‘𝐴) +N 1o) → ⟨𝑥, 1o⟩ = ⟨((1st ‘𝐴) +N 1o), 1o⟩)
49	48	breq2d 5091	. . 3 ⊢ (𝑥 = ((1st ‘𝐴) +N 1o) → (𝐴 <Q ⟨𝑥, 1o⟩ ↔ 𝐴 <Q ⟨((1st ‘𝐴) +N 1o), 1o⟩))
50	49	rspcev 3567	. 2 ⊢ ((((1st ‘𝐴) +N 1o) ∈ N ∧ 𝐴 <Q ⟨((1st ‘𝐴) +N 1o), 1o⟩) → ∃𝑥 ∈ N 𝐴 <Q ⟨𝑥, 1o⟩)
51	6, 47, 50	syl2anc 590	1 ⊢ (𝐴 ∈ Q → ∃𝑥 ∈ N 𝐴 <Q ⟨𝑥, 1o⟩)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃wrex 3064  ⟨cop 4568   class class class wbr 5079   × cxp 5623  ‘cfv 6492  (class class class)co 7363  1^st c1st 7936  2^nd c2nd 7937  1_oc1o 8395  Ncnpi 10765   +_N cpli 10766   ·_N cmi 10767   <_N clti 10768  Qcnq 10773   <_Q cltq 10779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-oadd 8406  df-omul 8407  df-ni 10793  df-pli 10794  df-mi 10795  df-lti 10796  df-ltpq 10831  df-nq 10833  df-ltnq 10839

This theorem is referenced by:  prlem934  10954