Theorem archnq 9802

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 9747	. . . 4 |- ( A e. Q. -> A e. ( N. X. N. ) )
2		xp1st 7198	. . . 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 9705	. . 3 |- 1o e. N.
5		addclpi 9714	. . 3 |- ( ( ( 1st ` A ) e. N. /\ 1o e. N. ) -> ( ( 1st ` A ) +N 1o ) e. N. )
6	3, 4, 5	sylancl 694	. 2 |- ( A e. Q. -> ( ( 1st ` A ) +N 1o ) e. N. )
7		xp2nd 7199	. . . . . 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 9715	. . . . 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 693	. . . 4 |- ( A e. Q. -> ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) e. N. )
11		eqid 2622	. . . . . . 7 |- ( ( 1st ` A ) +N 1o ) = ( ( 1st ` A ) +N 1o )
12		oveq2 6658	. . . . . . . . 9 |- ( x = 1o -> ( ( 1st ` A ) +N x ) = ( ( 1st ` A ) +N 1o ) )
13	12	eqeq1d 2624	. . . . . . . 8 |- ( x = 1o -> ( ( ( 1st ` A ) +N x ) = ( ( 1st ` A ) +N 1o ) <-> ( ( 1st ` A ) +N 1o ) = ( ( 1st ` A ) +N 1o ) ) )
14	13	rspcev 3309	. . . . . . 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 708	. . . . . 6 |- E. x e. N. ( ( 1st ` A ) +N x ) = ( ( 1st ` A ) +N 1o )
16		ltexpi 9724	. . . . . 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 248	. . . . 5 |- ( ( ( 1st ` A ) e. N. /\ ( ( 1st ` A ) +N 1o ) e. N. ) -> ( 1st ` A ) <N ( ( 1st ` A ) +N 1o ) )
18	3, 6, 17	syl2anc 693	. . . 4 |- ( A e. Q. -> ( 1st ` A ) <N ( ( 1st ` A ) +N 1o ) )
19		nlt1pi 9728	. . . . 5 |- -. ( 2nd ` A ) <N 1o
20		ltmpi 9726	. . . . . . 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 9708	. . . . . . . 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 4665	. . . . . 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 268	. . . . 5 |- ( A e. Q. -> ( ( 2nd ` A ) <N 1o <-> ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) <N ( ( 1st ` A ) +N 1o ) ) )
26	19, 25	mtbii 316	. . . 4 |- ( A e. Q. -> -. ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) <N ( ( 1st ` A ) +N 1o ) )
27		ltsopi 9710	. . . . 5 |- <N Or N.
28		ltrelpi 9711	. . . . 5 |- <N C_ ( N. X. N. )
29	27, 28	sotri3 5526	. . . 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 1326	. . 3 |- ( A e. Q. -> ( 1st ` A ) <N ( ( ( 1st ` A ) +N 1o ) .N ( 2nd ` A ) ) )
31		pinq 9749	. . . . . 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 9765	. . . . 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 702	. . . 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 6678	. . . . . . . 8 |- ( ( 1st ` A ) +N 1o ) e. _V
36	4	elexi 3213	. . . . . . . 8 |- 1o e. _V
37	35, 36	op2nd 7177	. . . . . . 7 |- ( 2nd ` <. ( ( 1st ` A ) +N 1o ) , 1o >. ) = 1o
38	37	oveq2i 6661	. . . . . 6 |- ( ( 1st ` A ) .N ( 2nd ` <. ( ( 1st ` A ) +N 1o ) , 1o >. ) ) = ( ( 1st ` A ) .N 1o )
39		mulidpi 9708	. . . . . . 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 2668	. . . . 5 |- ( A e. Q. -> ( ( 1st ` A ) .N ( 2nd ` <. ( ( 1st ` A ) +N 1o ) , 1o >. ) ) = ( 1st ` A ) )
42	35, 36	op1st 7176	. . . . . . 7 |- ( 1st ` <. ( ( 1st ` A ) +N 1o ) , 1o >. ) = ( ( 1st ` A ) +N 1o )
43	42	oveq1i 6660	. . . . . 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 4666	. . . 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 268	. . 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 247	. 2 |- ( A e. Q. -> A <Q <. ( ( 1st ` A ) +N 1o ) , 1o >. )
48		opeq1 4402	. . . 4 |- ( x = ( ( 1st ` A ) +N 1o ) -> <. x , 1o >. = <. ( ( 1st ` A ) +N 1o ) , 1o >. )
49	48	breq2d 4665	. . 3 |- ( x = ( ( 1st ` A ) +N 1o ) -> ( A <Q <. x , 1o >. <-> A <Q <. ( ( 1st ` A ) +N 1o ) , 1o >. ) )
50	49	rspcev 3309	. 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 693	1 |- ( A e. Q. -> E. x e. N. A <Q <. x , 1o >. )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   <.cop 4183   class class class wbr 4653   X. cxp 5112   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   1oc1o 7553   N.cnpi 9666   +N cpli 9667   .N cmi 9668   <N clti 9669   Q.cnq 9674   <Q cltq 9680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-ltpq 9732  df-nq 9734  df-ltnq 9740

This theorem is referenced by:  prlem934  9855