Theorem archnq 9681

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 9626	. . . 4 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
2		xp1st 7089	. . . 4 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) ∈ N)
4		1pi 9584	. . 3 ⊢ 1𝑜 ∈ N
5		addclpi 9593	. . 3 ⊢ (((1st ‘𝐴) ∈ N ∧ 1𝑜 ∈ N) → ((1st ‘𝐴) +N 1𝑜) ∈ N)
6	3, 4, 5	sylancl 693	. 2 ⊢ (𝐴 ∈ Q → ((1st ‘𝐴) +N 1𝑜) ∈ N)
7		xp2nd 7090	. . . . . 6 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
8	1, 7	syl 17	. . . . 5 ⊢ (𝐴 ∈ Q → (2nd ‘𝐴) ∈ N)
9		mulclpi 9594	. . . . 5 ⊢ ((((1st ‘𝐴) +N 1𝑜) ∈ N ∧ (2nd ‘𝐴) ∈ N) → (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴)) ∈ N)
10	6, 8, 9	syl2anc 691	. . . 4 ⊢ (𝐴 ∈ Q → (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴)) ∈ N)
11		eqid 2610	. . . . . . 7 ⊢ ((1st ‘𝐴) +N 1𝑜) = ((1st ‘𝐴) +N 1𝑜)
12		oveq2 6557	. . . . . . . . 9 ⊢ (𝑥 = 1𝑜 → ((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1𝑜))
13	12	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑥 = 1𝑜 → (((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1𝑜) ↔ ((1st ‘𝐴) +N 1𝑜) = ((1st ‘𝐴) +N 1𝑜)))
14	13	rspcev 3282	. . . . . . 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 9603	. . . . . 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 247	. . . . 5 ⊢ (((1st ‘𝐴) ∈ N ∧ ((1st ‘𝐴) +N 1𝑜) ∈ N) → (1st ‘𝐴) <N ((1st ‘𝐴) +N 1𝑜))
18	3, 6, 17	syl2anc 691	. . . 4 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) <N ((1st ‘𝐴) +N 1𝑜))
19		nlt1pi 9607	. . . . 5 ⊢ ¬ (2nd ‘𝐴) <N 1𝑜
20		ltmpi 9605	. . . . . . 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 9587	. . . . . . . 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 4595	. . . . . 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 267	. . . . 5 ⊢ (𝐴 ∈ Q → ((2nd ‘𝐴) <N 1𝑜 ↔ (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴)) <N ((1st ‘𝐴) +N 1𝑜)))
26	19, 25	mtbii 315	. . . 4 ⊢ (𝐴 ∈ Q → ¬ (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴)) <N ((1st ‘𝐴) +N 1𝑜))
27		ltsopi 9589	. . . . 5 ⊢ <N Or N
28		ltrelpi 9590	. . . . 5 ⊢ <N ⊆ (N × N)
29	27, 28	sotri3 5445	. . . 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 1318	. . 3 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) <N (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴)))
31		pinq 9628	. . . . . 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 9644	. . . . 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 6577	. . . . . . . 8 ⊢ ((1st ‘𝐴) +N 1𝑜) ∈ V
36	4	elexi 3186	. . . . . . . 8 ⊢ 1𝑜 ∈ V
37	35, 36	op2nd 7068	. . . . . . 7 ⊢ (2nd ‘⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩) = 1𝑜
38	37	oveq2i 6560	. . . . . 6 ⊢ ((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩)) = ((1st ‘𝐴) ·N 1𝑜)
39		mulidpi 9587	. . . . . . 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	syl5eq 2656	. . . . 5 ⊢ (𝐴 ∈ Q → ((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩)) = (1st ‘𝐴))
42	35, 36	op1st 7067	. . . . . . 7 ⊢ (1st ‘⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩) = ((1st ‘𝐴) +N 1𝑜)
43	42	oveq1i 6559	. . . . . 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 4596	. . . 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 267	. . 3 ⊢ (𝐴 ∈ Q → (𝐴 <Q ⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩ ↔ (1st ‘𝐴) <N (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴))))
47	30, 46	mpbird 246	. 2 ⊢ (𝐴 ∈ Q → 𝐴 <Q ⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩)
48		opeq1 4340	. . . 4 ⊢ (𝑥 = ((1st ‘𝐴) +N 1𝑜) → ⟨𝑥, 1𝑜⟩ = ⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩)
49	48	breq2d 4595	. . 3 ⊢ (𝑥 = ((1st ‘𝐴) +N 1𝑜) → (𝐴 <Q ⟨𝑥, 1𝑜⟩ ↔ 𝐴 <Q ⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩))
50	49	rspcev 3282	. 2 ⊢ ((((1st ‘𝐴) +N 1𝑜) ∈ N ∧ 𝐴 <Q ⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩) → ∃𝑥 ∈ N 𝐴 <Q ⟨𝑥, 1𝑜⟩)
51	6, 47, 50	syl2anc 691	1 ⊢ (𝐴 ∈ Q → ∃𝑥 ∈ N 𝐴 <Q ⟨𝑥, 1𝑜⟩)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  ⟨cop 4131   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  1_𝑜c1o 7440  Ncnpi 9545   +_N cpli 9546   ·_N cmi 9547   <_N clti 9548  Qcnq 9553   <_Q cltq 9559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-ni 9573  df-pli 9574  df-mi 9575  df-lti 9576  df-ltpq 9611  df-nq 9613  df-ltnq 9619

This theorem is referenced by:  prlem934  9734