Theorem archnq 10891

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 10836	. . . 4 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
2		xp1st 7965	. . . 4 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) ∈ N)
4		1pi 10794	. . 3 ⊢ 1o ∈ N
5		addclpi 10803	. . 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 7966	. . . . . 6 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
8	1, 7	syl 17	. . . . 5 ⊢ (𝐴 ∈ Q → (2nd ‘𝐴) ∈ N)
9		mulclpi 10804	. . . . 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 2736	. . . . . . 7 ⊢ ((1st ‘𝐴) +N 1o) = ((1st ‘𝐴) +N 1o)
12		oveq2 7366	. . . . . . . . 9 ⊢ (𝑥 = 1o → ((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1o))
13	12	eqeq1d 2738	. . . . . . . 8 ⊢ (𝑥 = 1o → (((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1o) ↔ ((1st ‘𝐴) +N 1o) = ((1st ‘𝐴) +N 1o)))
14	13	rspcev 3576	. . . . . . 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 692	. . . . . 6 ⊢ ∃𝑥 ∈ N ((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1o)
16		ltexpi 10813	. . . . . 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 258	. . . . 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 10817	. . . . 5 ⊢ ¬ (2nd ‘𝐴) <N 1o
20		ltmpi 10815	. . . . . . 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 10797	. . . . . . . 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 5110	. . . . . 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 279	. . . . 5 ⊢ (𝐴 ∈ Q → ((2nd ‘𝐴) <N 1o ↔ (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) <N ((1st ‘𝐴) +N 1o)))
26	19, 25	mtbii 326	. . . 4 ⊢ (𝐴 ∈ Q → ¬ (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)) <N ((1st ‘𝐴) +N 1o))
27		ltsopi 10799	. . . . 5 ⊢ <N Or N
28		ltrelpi 10800	. . . . 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 1373	. . 3 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) <N (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴)))
31		pinq 10838	. . . . . 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 10854	. . . . 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 687	. . . 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 7391	. . . . . . . 8 ⊢ ((1st ‘𝐴) +N 1o) ∈ V
36		1oex 8407	. . . . . . . 8 ⊢ 1o ∈ V
37	35, 36	op2nd 7942	. . . . . . 7 ⊢ (2nd ‘⟨((1st ‘𝐴) +N 1o), 1o⟩) = 1o
38	37	oveq2i 7369	. . . . . 6 ⊢ ((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1o), 1o⟩)) = ((1st ‘𝐴) ·N 1o)
39		mulidpi 10797	. . . . . . 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 7941	. . . . . . 7 ⊢ (1st ‘⟨((1st ‘𝐴) +N 1o), 1o⟩) = ((1st ‘𝐴) +N 1o)
43	42	oveq1i 7368	. . . . . 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 5111	. . . 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 279	. . 3 ⊢ (𝐴 ∈ Q → (𝐴 <Q ⟨((1st ‘𝐴) +N 1o), 1o⟩ ↔ (1st ‘𝐴) <N (((1st ‘𝐴) +N 1o) ·N (2nd ‘𝐴))))
47	30, 46	mpbird 257	. 2 ⊢ (𝐴 ∈ Q → 𝐴 <Q ⟨((1st ‘𝐴) +N 1o), 1o⟩)
48		opeq1 4829	. . . 4 ⊢ (𝑥 = ((1st ‘𝐴) +N 1o) → ⟨𝑥, 1o⟩ = ⟨((1st ‘𝐴) +N 1o), 1o⟩)
49	48	breq2d 5110	. . 3 ⊢ (𝑥 = ((1st ‘𝐴) +N 1o) → (𝐴 <Q ⟨𝑥, 1o⟩ ↔ 𝐴 <Q ⟨((1st ‘𝐴) +N 1o), 1o⟩))
50	49	rspcev 3576	. 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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3060  ⟨cop 4586   class class class wbr 5098   × cxp 5622  ‘cfv 6492  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  1_oc1o 8390  Ncnpi 10755   +_N cpli 10756   ·_N cmi 10757   <_N clti 10758  Qcnq 10763   <_Q cltq 10769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-omul 8402  df-ni 10783  df-pli 10784  df-mi 10785  df-lti 10786  df-ltpq 10821  df-nq 10823  df-ltnq 10829

This theorem is referenced by:  prlem934  10944