Theorem archpr 7475

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 7385. (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 7307	. . 3 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prmu 7310	. . 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 7249	. . . 4 |- ( z e. Q. -> E. x e. N. z <Q [ <. x , 1o >. ] ~Q )
5	4	ad2antrl 482	. . 3 |- ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) -> E. x e. N. z <Q [ <. x , 1o >. ] ~Q )
6		simprl 521	. . . . . . . 8 |- ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) -> z e. Q. )
7	6	ad2antrr 480	. . . . . . 7 |- ( ( ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) /\ x e. N. ) /\ z <Q [ <. x , 1o >. ] ~Q ) -> z e. Q. )
8		simprr 522	. . . . . . . 8 |- ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) -> z e. ( 2nd ` A ) )
9	8	ad2antrr 480	. . . . . . 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 109	. . . . . . . 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 2692	. . . . . . . . 9 |- z e. _V
12		breq1 3940	. . . . . . . . 9 |- ( l = z -> ( l <Q [ <. x , 1o >. ] ~Q <-> z <Q [ <. x , 1o >. ] ~Q ) )
13		ltnqex 7381	. . . . . . . . . 10 |- { l | l <Q [ <. x , 1o >. ] ~Q } e. _V
14		gtnqex 7382	. . . . . . . . . 10 |- { u | [ <. x , 1o >. ] ~Q <Q u } e. _V
15	13, 14	op1st 6052	. . . . . . . . 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 2836	. . . . . . . 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 133	. . . . . . 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 2203	. . . . . . . . 9 |- ( w = z -> ( w e. ( 2nd ` A ) <-> z e. ( 2nd ` A ) ) )
19		eleq1 2203	. . . . . . . . 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 465	. . . . . . . 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 2793	. . . . . . 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 1215	. . . . . 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 523	. . . . . . 7 |- ( ( ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) /\ x e. N. ) /\ z <Q [ <. x , 1o >. ] ~Q ) -> A e. P. )
24		nnprlu 7385	. . . . . . . 8 |- ( x e. N. -> <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. e. P. )
25	24	ad2antlr 481	. . . . . . 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 7338	. . . . . . 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 409	. . . . . 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 166	. . . . 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 114	. . . 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 2537	. . 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 2557	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 103   <-> wb 104   e. wcel 1481   {cab 2126   E.wrex 2418   <.cop 3535   class class class wbr 3937   ` cfv 5131   1stc1st 6044   2ndc2nd 6045   1oc1o 6314   [cec 6435   N.cnpi 7104   ~Q ceq 7111   Q.cnq 7112   <Q cltq 7117   P.cnp 7123   <P cltp 7127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-inp 7298  df-iltp 7302

This theorem is referenced by:  archsr  7614