Theorem archnq 10925

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 10870	. . . 4 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
2		xp1st 7958	. . . 4 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) ∈ N)
4		1pi 10828	. . 3 ⊢ 1o ∈ N
5		addclpi 10837	. . 3 ⊢ (((1st ‘𝐴) ∈ N ∧ 1o ∈ N) → ((1st ‘𝐴) +N 1o) ∈ N)
6	3, 4, 5	sylancl 586	. 2 ⊢ (𝐴 ∈ Q → ((1st ‘𝐴) +N 1o) ∈ N)
7		xp2nd 7959	. . . . . 6 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
8	1, 7	syl 17	. . . . 5 ⊢ (𝐴 ∈ Q → (2nd ‘𝐴) ∈ N)
9		mulclpi 10838	. . . . 5 ⊢ ((((1st ‘𝐴) +N 1o) ∈ N ∧ (2nd ‘𝐴) ∈ N) → (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) ∈ N)
10	6, 8, 9	syl2anc 584	. . . 4 ⊢ (𝐴 ∈ Q → (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) ∈ N)
11		eqid 2731	. . . . . . 7 ⊢ ((1st ‘𝐴) +N 1o) = ((1st ‘𝐴) +N 1o)
12		oveq2 7370	. . . . . . . . 9 ⊢ (𝑥 = 1o → ((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1o))
13	12	eqeq1d 2733	. . . . . . . 8 ⊢ (𝑥 = 1o → (((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1o) ↔ ((1st ‘𝐴) +N 1o) = ((1st ‘𝐴) +N 1o)))
14	13	rspcev 3582	. . . . . . 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 690	. . . . . 6 ⊢ ∃𝑥 ∈ N ((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1o)
16		ltexpi 10847	. . . . . 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 257	. . . . 5 ⊢ (((1st ‘𝐴) ∈ N ∧ ((1st ‘𝐴) +N 1o) ∈ N) → (1st ‘𝐴) <N ((1st ‘𝐴) +N 1o))
18	3, 6, 17	syl2anc 584	. . . 4 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) <N ((1st ‘𝐴) +N 1o))
19		nlt1pi 10851	. . . . 5 ⊢ ¬ (2nd ‘𝐴) <N 1o
20		ltmpi 10849	. . . . . . 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 10831	. . . . . . . 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 5122	. . . . . 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 278	. . . . 5 ⊢ (𝐴 ∈ Q → ((2nd ‘𝐴) <N 1o ↔ (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) <N ((1st ‘𝐴) +N 1o)))
26	19, 25	mtbii 325	. . . 4 ⊢ (𝐴 ∈ Q → ¬ (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) <N ((1st ‘𝐴) +N 1o))
27		ltsopi 10833	. . . . 5 ⊢ <N Or N
28		ltrelpi 10834	. . . . 5 ⊢ <N ⊆ (N × N)
29	27, 28	sotri3 6089	. . . 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 1371	. . 3 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) <N (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)))
31		pinq 10872	. . . . . 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 10888	. . . . 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 685	. . . 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 7395	. . . . . . . 8 ⊢ ((1st ‘𝐴) +N 1o) ∈ V
36		1oex 8427	. . . . . . . 8 ⊢ 1o ∈ V
37	35, 36	op2nd 7935	. . . . . . 7 ⊢ (2nd ‘⟨((1st ‘𝐴) +N 1o), 1o⟩) = 1o
38	37	oveq2i 7373	. . . . . 6 ⊢ ((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1o), 1o⟩)) = ((1st ‘𝐴) ·N 1o)
39		mulidpi 10831	. . . . . . 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 2783	. . . . 5 ⊢ (𝐴 ∈ Q → ((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1o), 1o⟩)) = (1st ‘𝐴))
42	35, 36	op1st 7934	. . . . . . 7 ⊢ (1st ‘⟨((1st ‘𝐴) +N 1o), 1o⟩) = ((1st ‘𝐴) +N 1o)
43	42	oveq1i 7372	. . . . . 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 5123	. . . 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 278	. . 3 ⊢ (𝐴 ∈ Q → (𝐴 <Q ⟨((1st ‘𝐴) +N 1o), 1o⟩ ↔ (1st ‘𝐴) <N (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴))))
47	30, 46	mpbird 256	. 2 ⊢ (𝐴 ∈ Q → 𝐴 <Q ⟨((1st ‘𝐴) +N 1o), 1o⟩)
48		opeq1 4835	. . . 4 ⊢ (𝑥 = ((1st ‘𝐴) +N 1o) → ⟨𝑥, 1o⟩ = ⟨((1st ‘𝐴) +N 1o), 1o⟩)
49	48	breq2d 5122	. . 3 ⊢ (𝑥 = ((1st ‘𝐴) +N 1o) → (𝐴 <Q ⟨𝑥, 1o⟩ ↔ 𝐴 <Q ⟨((1st ‘𝐴) +N 1o), 1o⟩))
50	49	rspcev 3582	. 2 ⊢ ((((1st ‘𝐴) +N 1o) ∈ N ∧ 𝐴 <Q ⟨((1st ‘𝐴) +N 1o), 1o⟩) → ∃𝑥 ∈ N 𝐴 <Q ⟨𝑥, 1o⟩)
51	6, 47, 50	syl2anc 584	1 ⊢ (𝐴 ∈ Q → ∃𝑥 ∈ N 𝐴 <Q ⟨𝑥, 1o⟩)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3069  ⟨cop 4597   class class class wbr 5110   × cxp 5636  ‘cfv 6501  (class class class)co 7362  1^st c1st 7924  2^nd c2nd 7925  1_oc1o 8410  Ncnpi 10789   +_N cpli 10790   ·_N cmi 10791   <_N clti 10792  Qcnq 10797   <_Q cltq 10803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-omul 8422  df-ni 10817  df-pli 10818  df-mi 10819  df-lti 10820  df-ltpq 10855  df-nq 10857  df-ltnq 10863

This theorem is referenced by:  prlem934  10978