Theorem archnq 9356

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 9301	. . . 4 |- ( A e. Q. -> A e. ( N. X. N. ) )
2		xp1st 6781	. . . 4 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
3	1, 2	syl 17	. . 3 |- ( A e. Q. -> ( 1st ` A ) e. N. )
4		1pi 9259	. . 3 |- 1o e. N.
5		addclpi 9268	. . 3 |- ( ( ( 1st ` A ) e. N. /\ 1o e. N. ) -> ( ( 1st ` A ) +N 1o ) e. N. )
6	3, 4, 5	sylancl 666	. 2 |- ( A e. Q. -> ( ( 1st ` A ) +N 1o ) e. N. )
7		xp2nd 6782	. . . . . 6 |- ( A e. ( N. X. N. ) -> ( 2nd ` A ) e. N. )
8	1, 7	syl 17	. . . . 5 |- ( A e. Q. -> ( 2nd ` A ) e. N. )
9		mulclpi 9269	. . . . 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 665	. . . 4 |- ( A e. Q. -> ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) e. N. )
11		eqid 2428	. . . . . . 7 |- ( ( 1st ` A ) +N 1o ) = ( ( 1st ` A ) +N 1o )
12		oveq2 6257	. . . . . . . . 9 |- ( x = 1o -> ( ( 1st ` A ) +N x ) = ( ( 1st ` A ) +N 1o ) )
13	12	eqeq1d 2430	. . . . . . . 8 |- ( x = 1o -> ( ( ( 1st ` A ) +N x ) = ( ( 1st ` A ) +N 1o ) <-> ( ( 1st ` A ) +N 1o ) = ( ( 1st ` A ) +N 1o ) ) )
14	13	rspcev 3125	. . . . . . 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 676	. . . . . 6 |- E. x e. N. ( ( 1st ` A ) +N x ) = ( ( 1st ` A ) +N 1o )
16		ltexpi 9278	. . . . . 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 236	. . . . 5 |- ( ( ( 1st ` A ) e. N. /\ ( ( 1st ` A ) +N 1o ) e. N. ) -> ( 1st ` A ) <N ( ( 1st ` A ) +N 1o ) )
18	3, 6, 17	syl2anc 665	. . . 4 |- ( A e. Q. -> ( 1st ` A ) <N ( ( 1st ` A ) +N 1o ) )
19		nlt1pi 9282	. . . . 5 |- -. ( 2nd ` A ) <N 1o
20		ltmpi 9280	. . . . . . 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 17	. . . . . 6 |- ( A e. Q. -> ( ( 2nd ` A ) <N 1o <-> ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) <N ( ( ( 1st ` A ) +N 1o ) .N 1o ) ) )
22		mulidpi 9262	. . . . . . . 8 |- ( ( ( 1st ` A ) +N 1o ) e. N. -> ( ( ( 1st ` A ) +N 1o ) .N 1o ) = ( ( 1st ` A ) +N 1o ) )
23	6, 22	syl 17	. . . . . . 7 |- ( A e. Q. -> ( ( ( 1st ` A ) +N 1o ) .N 1o ) = ( ( 1st ` A ) +N 1o ) )
24	23	breq2d 4378	. . . . . 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 256	. . . . 5 |- ( A e. Q. -> ( ( 2nd ` A ) <N 1o <-> ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) <N ( ( 1st ` A ) +N 1o ) ) )
26	19, 25	mtbii 303	. . . 4 |- ( A e. Q. -> -. ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) <N ( ( 1st ` A ) +N 1o ) )
27		ltsopi 9264	. . . . 5 |- <N Or N.
28		ltrelpi 9265	. . . . 5 |- <N C_ ( N. X. N. )
29	27, 28	sotri3 5192	. . . 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 1264	. . 3 |- ( A e. Q. -> ( 1st ` A ) <N ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) )
31		pinq 9303	. . . . . 6 |- ( ( ( 1st ` A ) +N 1o ) e. N. -> <. ( ( 1st ` A ) +N 1o ) , 1o >. e. Q. )
32	6, 31	syl 17	. . . . 5 |- ( A e. Q. -> <. ( ( 1st ` A ) +N 1o ) , 1o >. e. Q. )
33		ordpinq 9319	. . . . 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 672	. . . 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 6277	. . . . . . . 8 |- ( ( 1st ` A ) +N 1o ) e. _V
36	4	elexi 3032	. . . . . . . 8 |- 1o e. _V
37	35, 36	op2nd 6760	. . . . . . 7 |- ( 2nd ` <. ( ( 1st ` A ) +N 1o ) , 1o >. ) = 1o
38	37	oveq2i 6260	. . . . . 6 |- ( ( 1st ` A ) .N ( 2nd ` <. ( ( 1st ` A ) +N 1o ) , 1o >. ) ) = ( ( 1st ` A ) .N 1o )
39		mulidpi 9262	. . . . . . 7 |- ( ( 1st ` A ) e. N. -> ( ( 1st ` A ) .N 1o ) = ( 1st ` A ) )
40	3, 39	syl 17	. . . . . 6 |- ( A e. Q. -> ( ( 1st ` A ) .N 1o ) = ( 1st ` A ) )
41	38, 40	syl5eq 2474	. . . . 5 |- ( A e. Q. -> ( ( 1st ` A ) .N ( 2nd ` <. ( ( 1st ` A ) +N 1o ) , 1o >. ) ) = ( 1st ` A ) )
42	35, 36	op1st 6759	. . . . . . 7 |- ( 1st ` <. ( ( 1st ` A ) +N 1o ) , 1o >. ) = ( ( 1st ` A ) +N 1o )
43	42	oveq1i 6259	. . . . . 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 4379	. . . 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 256	. . 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 235	. 2 |- ( A e. Q. -> A <Q <. ( ( 1st ` A ) +N 1o ) , 1o >. )
48		opeq1 4130	. . . 4 |- ( x = ( ( 1st ` A ) +N 1o ) -> <. x , 1o >. = <. ( ( 1st ` A ) +N 1o ) , 1o >. )
49	48	breq2d 4378	. . 3 |- ( x = ( ( 1st ` A ) +N 1o ) -> ( A <Q <. x , 1o >. <-> A <Q <. ( ( 1st ` A ) +N 1o ) , 1o >. ) )
50	49	rspcev 3125	. 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 665	1 |- ( A e. Q. -> E. x e. N. A <Q <. x , 1o >. )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1872   E.wrex 2715   <.cop 3947   class class class wbr 4366   X. cxp 4794   ` cfv 5544  (class class class)co 6249   1stc1st 6749   2ndc2nd 6750   1oc1o 7130   N.cnpi 9220   +N cpli 9221   .N cmi 9222   <N clti 9223   Q.cnq 9228   <Q cltq 9234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-oadd 7141  df-omul 7142  df-ni 9248  df-pli 9249  df-mi 9250  df-lti 9251  df-ltpq 9286  df-nq 9288  df-ltnq 9294

This theorem is referenced by:  prlem934  9409