Theorem archpr 7758

Description: For any positive real, there is an integer that is greater than it. This is also known as the "archimedean property". The integer x is embedded into the reals as described at nnprlu 7668. (Contributed by Jim Kingdon, 22-Apr-2020.)

Assertion

Ref	Expression
archpr	|- ( A e. P. -> E. x e. N. A <P <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. )

Distinct variable group:   A, l, u, x

Proof of Theorem archpr

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7590	. . 3 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prmu 7593	. . 3 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. z e. Q. z e. ( 2nd ` A ) )
3	1, 2	syl 14	. 2 |- ( A e. P. -> E. z e. Q. z e. ( 2nd ` A ) )
4		archnqq 7532	. . . 4 |- ( z e. Q. -> E. x e. N. z <Q [ <. x , 1o >. ] ~Q )
5	4	ad2antrl 490	. . 3 |- ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) -> E. x e. N. z <Q [ <. x , 1o >. ] ~Q )
6		simprl 529	. . . . . . . 8 |- ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) -> z e. Q. )
7	6	ad2antrr 488	. . . . . . 7 |- ( ( ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) /\ x e. N. ) /\ z <Q [ <. x , 1o >. ] ~Q ) -> z e. Q. )
8		simprr 531	. . . . . . . 8 |- ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) -> z e. ( 2nd ` A ) )
9	8	ad2antrr 488	. . . . . . 7 |- ( ( ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) /\ x e. N. ) /\ z <Q [ <. x , 1o >. ] ~Q ) -> z e. ( 2nd ` A ) )
10		simpr 110	. . . . . . . 8 |- ( ( ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) /\ x e. N. ) /\ z <Q [ <. x , 1o >. ] ~Q ) -> z <Q [ <. x , 1o >. ] ~Q )
11		vex 2775	. . . . . . . . 9 |- z e. _V
12		breq1 4048	. . . . . . . . 9 |- ( l = z -> ( l <Q [ <. x , 1o >. ] ~Q <-> z <Q [ <. x , 1o >. ] ~Q ) )
13		ltnqex 7664	. . . . . . . . . 10 |- { l | l <Q [ <. x , 1o >. ] ~Q } e. _V
14		gtnqex 7665	. . . . . . . . . 10 |- { u | [ <. x , 1o >. ] ~Q <Q u } e. _V
15	13, 14	op1st 6234	. . . . . . . . 9 |- ( 1st ` <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. ) = { l | l <Q [ <. x , 1o >. ] ~Q }
16	11, 12, 15	elab2 2921	. . . . . . . 8 |- ( z e. ( 1st ` <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. ) <-> z <Q [ <. x , 1o >. ] ~Q )
17	10, 16	sylibr 134	. . . . . . 7 |- ( ( ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) /\ x e. N. ) /\ z <Q [ <. x , 1o >. ] ~Q ) -> z e. ( 1st ` <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. ) )
18		eleq1 2268	. . . . . . . . 9 |- ( w = z -> ( w e. ( 2nd ` A ) <-> z e. ( 2nd ` A ) ) )
19		eleq1 2268	. . . . . . . . 9 |- ( w = z -> ( w e. ( 1st ` <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. ) <-> z e. ( 1st ` <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. ) ) )
20	18, 19	anbi12d 473	. . . . . . . 8 |- ( w = z -> ( ( w e. ( 2nd ` A ) /\ w e. ( 1st ` <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. ) ) <-> ( z e. ( 2nd ` A ) /\ z e. ( 1st ` <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. ) ) ) )
21	20	rspcev 2877	. . . . . . 7 |- ( ( z e. Q. /\ ( z e. ( 2nd ` A ) /\ z e. ( 1st ` <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. ) ) ) -> E. w e. Q. ( w e. ( 2nd ` A ) /\ w e. ( 1st ` <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. ) ) )
22	7, 9, 17, 21	syl12anc 1248	. . . . . 6 |- ( ( ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) /\ x e. N. ) /\ z <Q [ <. x , 1o >. ] ~Q ) -> E. w e. Q. ( w e. ( 2nd ` A ) /\ w e. ( 1st ` <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. ) ) )
23		simplll 533	. . . . . . 7 |- ( ( ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) /\ x e. N. ) /\ z <Q [ <. x , 1o >. ] ~Q ) -> A e. P. )
24		nnprlu 7668	. . . . . . . 8 |- ( x e. N. -> <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. e. P. )
25	24	ad2antlr 489	. . . . . . 7 |- ( ( ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) /\ x e. N. ) /\ z <Q [ <. x , 1o >. ] ~Q ) -> <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. e. P. )
26		ltdfpr 7621	. . . . . . 7 |- ( ( A e. P. /\ <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. e. P. ) -> ( A <P <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. <-> E. w e. Q. ( w e. ( 2nd ` A ) /\ w e. ( 1st ` <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. ) ) ) )
27	23, 25, 26	syl2anc 411	. . . . . 6 |- ( ( ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) /\ x e. N. ) /\ z <Q [ <. x , 1o >. ] ~Q ) -> ( A <P <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. <-> E. w e. Q. ( w e. ( 2nd ` A ) /\ w e. ( 1st ` <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. ) ) ) )
28	22, 27	mpbird 167	. . . . 5 |- ( ( ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) /\ x e. N. ) /\ z <Q [ <. x , 1o >. ] ~Q ) -> A <P <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. )
29	28	ex 115	. . . 4 |- ( ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) /\ x e. N. ) -> ( z <Q [ <. x , 1o >. ] ~Q -> A <P <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. ) )
30	29	reximdva 2608	. . 3 |- ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) -> ( E. x e. N. z <Q [ <. x , 1o >. ] ~Q -> E. x e. N. A <P <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. ) )
31	5, 30	mpd 13	. 2 |- ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) -> E. x e. N. A <P <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. )
32	3, 31	rexlimddv 2628	1 |- ( A e. P. -> E. x e. N. A <P <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2176   {cab 2191   E.wrex 2485   <.cop 3636   class class class wbr 4045   ` cfv 5272   1stc1st 6226   2ndc2nd 6227   1oc1o 6497   [cec 6620   N.cnpi 7387   ~Q ceq 7394   Q.cnq 7395   <Q cltq 7400   P.cnp 7406   <P cltp 7410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-eprel 4337  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-1o 6504  df-oadd 6508  df-omul 6509  df-er 6622  df-ec 6624  df-qs 6628  df-ni 7419  df-pli 7420  df-mi 7421  df-lti 7422  df-plpq 7459  df-mpq 7460  df-enq 7462  df-nqqs 7463  df-plqqs 7464  df-mqqs 7465  df-1nqqs 7466  df-rq 7467  df-ltnqqs 7468  df-inp 7581  df-iltp 7585

This theorem is referenced by:  archsr  7897