Theorem archnq 9347

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 e. Q. -> E. x e. N. A <Q <. x , 1o >. )

Distinct variable group:   x, A

Proof of Theorem archnq

Step	Hyp	Ref	Expression
1		elpqn 9292	. . . 4 |- ( A e. Q. -> A e. ( N. X. N. ) )
2		xp1st 6803	. . . 4 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
3	1, 2	syl 16	. . 3 |- ( A e. Q. -> ( 1st ` A ) e. N. )
4		1pi 9250	. . 3 |- 1o e. N.
5		addclpi 9259	. . 3 |- ( ( ( 1st ` A ) e. N. /\ 1o e. N. ) -> ( ( 1st ` A ) +N 1o ) e. N. )
6	3, 4, 5	sylancl 660	. 2 |- ( A e. Q. -> ( ( 1st ` A ) +N 1o ) e. N. )
7		xp2nd 6804	. . . . . 6 |- ( A e. ( N. X. N. ) -> ( 2nd ` A ) e. N. )
8	1, 7	syl 16	. . . . 5 |- ( A e. Q. -> ( 2nd ` A ) e. N. )
9		mulclpi 9260	. . . . 5 |- ( ( ( ( 1st ` A ) +N 1o ) e. N. /\ ( 2nd ` A ) e. N. ) -> ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) e. N. )
10	6, 8, 9	syl2anc 659	. . . 4 |- ( A e. Q. -> ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) e. N. )
11		eqid 2454	. . . . . . 7 |- ( ( 1st ` A ) +N 1o ) = ( ( 1st ` A ) +N 1o )
12		oveq2 6278	. . . . . . . . 9 |- ( x = 1o -> ( ( 1st ` A ) +N x ) = ( ( 1st ` A ) +N 1o ) )
13	12	eqeq1d 2456	. . . . . . . 8 |- ( x = 1o -> ( ( ( 1st ` A ) +N x ) = ( ( 1st ` A ) +N 1o ) <-> ( ( 1st ` A ) +N 1o ) = ( ( 1st ` A ) +N 1o ) ) )
14	13	rspcev 3207	. . . . . . 7 |- ( ( 1o e. N. /\ ( ( 1st ` A ) +N 1o ) = ( ( 1st ` A ) +N 1o ) ) -> E. x e. N. ( ( 1st ` A ) +N x ) = ( ( 1st ` A ) +N 1o ) )
15	4, 11, 14	mp2an 670	. . . . . 6 |- E. x e. N. ( ( 1st ` A ) +N x ) = ( ( 1st ` A ) +N 1o )
16		ltexpi 9269	. . . . . 6 |- ( ( ( 1st ` A ) e. N. /\ ( ( 1st ` A ) +N 1o ) e. N. ) -> ( ( 1st ` A ) <N ( ( 1st ` A ) +N 1o ) <-> E. x e. N. ( ( 1st ` A ) +N x ) = ( ( 1st ` A ) +N 1o ) ) )
17	15, 16	mpbiri 233	. . . . 5 |- ( ( ( 1st ` A ) e. N. /\ ( ( 1st ` A ) +N 1o ) e. N. ) -> ( 1st ` A ) <N ( ( 1st ` A ) +N 1o ) )
18	3, 6, 17	syl2anc 659	. . . 4 |- ( A e. Q. -> ( 1st ` A ) <N ( ( 1st ` A ) +N 1o ) )
19		nlt1pi 9273	. . . . 5 |- -. ( 2nd ` A ) <N 1o
20		ltmpi 9271	. . . . . . 7 |- ( ( ( 1st ` A ) +N 1o ) e. N. -> ( ( 2nd ` A ) <N 1o <-> ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) <N ( ( ( 1st ` A ) +N 1o ) .N 1o ) ) )
21	6, 20	syl 16	. . . . . 6 |- ( A e. Q. -> ( ( 2nd ` A ) <N 1o <-> ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) <N ( ( ( 1st ` A ) +N 1o ) .N 1o ) ) )
22		mulidpi 9253	. . . . . . . 8 |- ( ( ( 1st ` A ) +N 1o ) e. N. -> ( ( ( 1st ` A ) +N 1o ) .N 1o ) = ( ( 1st ` A ) +N 1o ) )
23	6, 22	syl 16	. . . . . . 7 |- ( A e. Q. -> ( ( ( 1st ` A ) +N 1o ) .N 1o ) = ( ( 1st ` A ) +N 1o ) )
24	23	breq2d 4451	. . . . . 6 |- ( A e. 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 253	. . . . 5 |- ( A e. Q. -> ( ( 2nd ` A ) <N 1o <-> ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) <N ( ( 1st ` A ) +N 1o ) ) )
26	19, 25	mtbii 300	. . . 4 |- ( A e. Q. -> -. ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) <N ( ( 1st ` A ) +N 1o ) )
27		ltsopi 9255	. . . . 5 |- <N Or N.
28		ltrelpi 9256	. . . . 5 |- <N C_ ( N. X. N. )
29	27, 28	sotri3 5385	. . . 4 |- ( ( ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) e. 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 1226	. . 3 |- ( A e. Q. -> ( 1st ` A ) <N ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) )
31		pinq 9294	. . . . . 6 |- ( ( ( 1st ` A ) +N 1o ) e. N. -> <. ( ( 1st ` A ) +N 1o ) , 1o >. e. Q. )
32	6, 31	syl 16	. . . . 5 |- ( A e. Q. -> <. ( ( 1st ` A ) +N 1o ) , 1o >. e. Q. )
33		ordpinq 9310	. . . . 5 |- ( ( A e. Q. /\ <. ( ( 1st ` A ) +N 1o ) , 1o >. e. 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 666	. . . 4 |- ( A e. 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 6298	. . . . . . . 8 |- ( ( 1st ` A ) +N 1o ) e. _V
36	4	elexi 3116	. . . . . . . 8 |- 1o e. _V
37	35, 36	op2nd 6782	. . . . . . 7 |- ( 2nd ` <. ( ( 1st ` A ) +N 1o ) , 1o >. ) = 1o
38	37	oveq2i 6281	. . . . . 6 |- ( ( 1st ` A ) .N ( 2nd ` <. ( ( 1st ` A ) +N 1o ) , 1o >. ) ) = ( ( 1st ` A ) .N 1o )
39		mulidpi 9253	. . . . . . 7 |- ( ( 1st ` A ) e. N. -> ( ( 1st ` A ) .N 1o ) = ( 1st ` A ) )
40	3, 39	syl 16	. . . . . 6 |- ( A e. Q. -> ( ( 1st ` A ) .N 1o ) = ( 1st ` A ) )
41	38, 40	syl5eq 2507	. . . . 5 |- ( A e. Q. -> ( ( 1st ` A ) .N ( 2nd ` <. ( ( 1st ` A ) +N 1o ) , 1o >. ) ) = ( 1st ` A ) )
42	35, 36	op1st 6781	. . . . . . 7 |- ( 1st ` <. ( ( 1st ` A ) +N 1o ) , 1o >. ) = ( ( 1st ` A ) +N 1o )
43	42	oveq1i 6280	. . . . . 6 |- ( ( 1st ` <. ( ( 1st ` A ) +N 1o ) , 1o >. ) .N ( 2nd ` A ) ) = ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) )
44	43	a1i 11	. . . . 5 |- ( A e. Q. -> ( ( 1st ` <. ( ( 1st ` A ) +N 1o ) , 1o >. ) .N ( 2nd ` A ) ) = ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) )
45	41, 44	breq12d 4452	. . . 4 |- ( A e. 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 253	. . 3 |- ( A e. Q. -> ( A <Q <. ( ( 1st ` A ) +N 1o ) , 1o >. <-> ( 1st ` A ) <N ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) ) )
47	30, 46	mpbird 232	. 2 |- ( A e. Q. -> A <Q <. ( ( 1st ` A ) +N 1o ) , 1o >. )
48		opeq1 4203	. . . 4 |- ( x = ( ( 1st ` A ) +N 1o ) -> <. x , 1o >. = <. ( ( 1st ` A ) +N 1o ) , 1o >. )
49	48	breq2d 4451	. . 3 |- ( x = ( ( 1st ` A ) +N 1o ) -> ( A <Q <. x , 1o >. <-> A <Q <. ( ( 1st ` A ) +N 1o ) , 1o >. ) )
50	49	rspcev 3207	. 2 |- ( ( ( ( 1st ` A ) +N 1o ) e. N. /\ A <Q <. ( ( 1st ` A ) +N 1o ) , 1o >. ) -> E. x e. N. A <Q <. x , 1o >. )
51	6, 47, 50	syl2anc 659	1 |- ( A e. Q. -> E. x e. N. A <Q <. x , 1o >. )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   E.wrex 2805   <.cop 4022   class class class wbr 4439   X. cxp 4986   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   1oc1o 7115   N.cnpi 9211   +N cpli 9212   .N cmi 9213   <N clti 9214   Q.cnq 9219   <Q cltq 9225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-omul 7127  df-ni 9239  df-pli 9240  df-mi 9241  df-lti 9242  df-ltpq 9277  df-nq 9279  df-ltnq 9285

This theorem is referenced by:  prlem934  9400