Theorem archnq 9253

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 9198	. . . 4 |- ( A e. Q. -> A e. ( N. X. N. ) )
2		xp1st 6709	. . . 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 9156	. . 3 |- 1o e. N.
5		addclpi 9165	. . 3 |- ( ( ( 1st ` A ) e. N. /\ 1o e. N. ) -> ( ( 1st ` A ) +N 1o ) e. N. )
6	3, 4, 5	sylancl 662	. 2 |- ( A e. Q. -> ( ( 1st ` A ) +N 1o ) e. N. )
7		xp2nd 6710	. . . . . 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 9166	. . . . 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 661	. . . 4 |- ( A e. Q. -> ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) e. N. )
11		eqid 2451	. . . . . . 7 |- ( ( 1st ` A ) +N 1o ) = ( ( 1st ` A ) +N 1o )
12		oveq2 6201	. . . . . . . . 9 |- ( x = 1o -> ( ( 1st ` A ) +N x ) = ( ( 1st ` A ) +N 1o ) )
13	12	eqeq1d 2453	. . . . . . . 8 |- ( x = 1o -> ( ( ( 1st ` A ) +N x ) = ( ( 1st ` A ) +N 1o ) <-> ( ( 1st ` A ) +N 1o ) = ( ( 1st ` A ) +N 1o ) ) )
14	13	rspcev 3172	. . . . . . 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 672	. . . . . 6 |- E. x e. N. ( ( 1st ` A ) +N x ) = ( ( 1st ` A ) +N 1o )
16		ltexpi 9175	. . . . . 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 661	. . . 4 |- ( A e. Q. -> ( 1st ` A ) <N ( ( 1st ` A ) +N 1o ) )
19		nlt1pi 9179	. . . . 5 |- -. ( 2nd ` A ) <N 1o
20		ltmpi 9177	. . . . . . 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 9159	. . . . . . . 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 4405	. . . . . 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 302	. . . 4 |- ( A e. Q. -> -. ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) <N ( ( 1st ` A ) +N 1o ) )
27		ltsopi 9161	. . . . 5 |- <N Or N.
28		ltrelpi 9162	. . . . 5 |- <N C_ ( N. X. N. )
29	27, 28	sotri3 5329	. . . 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 1219	. . 3 |- ( A e. Q. -> ( 1st ` A ) <N ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) )
31		pinq 9200	. . . . . 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 9216	. . . . 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 668	. . . 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 6218	. . . . . . . 8 |- ( ( 1st ` A ) +N 1o ) e. _V
36	4	elexi 3081	. . . . . . . 8 |- 1o e. _V
37	35, 36	op2nd 6689	. . . . . . 7 |- ( 2nd ` <. ( ( 1st ` A ) +N 1o ) , 1o >. ) = 1o
38	37	oveq2i 6204	. . . . . 6 |- ( ( 1st ` A ) .N ( 2nd ` <. ( ( 1st ` A ) +N 1o ) , 1o >. ) ) = ( ( 1st ` A ) .N 1o )
39		mulidpi 9159	. . . . . . 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 2504	. . . . 5 |- ( A e. Q. -> ( ( 1st ` A ) .N ( 2nd ` <. ( ( 1st ` A ) +N 1o ) , 1o >. ) ) = ( 1st ` A ) )
42	35, 36	op1st 6688	. . . . . . 7 |- ( 1st ` <. ( ( 1st ` A ) +N 1o ) , 1o >. ) = ( ( 1st ` A ) +N 1o )
43	42	oveq1i 6203	. . . . . 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 4406	. . . 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 4160	. . . 4 |- ( x = ( ( 1st ` A ) +N 1o ) -> <. x , 1o >. = <. ( ( 1st ` A ) +N 1o ) , 1o >. )
49	48	breq2d 4405	. . 3 |- ( x = ( ( 1st ` A ) +N 1o ) -> ( A <Q <. x , 1o >. <-> A <Q <. ( ( 1st ` A ) +N 1o ) , 1o >. ) )
50	49	rspcev 3172	. 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 661	1 |- ( A e. Q. -> E. x e. N. A <Q <. x , 1o >. )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   E.wrex 2796   <.cop 3984   class class class wbr 4393   X. cxp 4939   ` cfv 5519  (class class class)co 6193   1stc1st 6678   2ndc2nd 6679   1oc1o 7016   N.cnpi 9115   +N cpli 9116   .N cmi 9117   <N clti 9118   Q.cnq 9123   <Q cltq 9129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-omul 7028  df-ni 9145  df-pli 9146  df-mi 9147  df-lti 9148  df-ltpq 9183  df-nq 9185  df-ltnq 9191

This theorem is referenced by:  prlem934  9306