Theorem archpr 8000

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 7910. (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 7832	. . 3 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prmu 7835	. . 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 7774	. . . 4 |- ( z e. Q. -> E. x e. N. z <Q [ <. x , 1o >. ] ~Q )
5	4	ad2antrl 494	. . 3 |- ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) -> E. x e. N. z <Q [ <. x , 1o >. ] ~Q )
6		simprl 535	. . . . . . . 8 |- ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) -> z e. Q. )
7	6	ad2antrr 492	. . . . . . 7 |- ( ( ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) /\ x e. N. ) /\ z <Q [ <. x , 1o >. ] ~Q ) -> z e. Q. )
8		simprr 537	. . . . . . . 8 |- ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) -> z e. ( 2nd ` A ) )
9	8	ad2antrr 492	. . . . . . 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 2824	. . . . . . . . 9 |- z e. _V
12		breq1 4128	. . . . . . . . 9 |- ( l = z -> ( l <Q [ <. x , 1o >. ] ~Q <-> z <Q [ <. x , 1o >. ] ~Q ) )
13		ltnqex 7906	. . . . . . . . . 10 |- { l | l <Q [ <. x , 1o >. ] ~Q } e. _V
14		gtnqex 7907	. . . . . . . . . 10 |- { u | [ <. x , 1o >. ] ~Q <Q u } e. _V
15	13, 14	op1st 6370	. . . . . . . . 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 2974	. . . . . . . 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 2301	. . . . . . . . 9 |- ( w = z -> ( w e. ( 2nd ` A ) <-> z e. ( 2nd ` A ) ) )
19		eleq1 2301	. . . . . . . . 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 477	. . . . . . . 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 2929	. . . . . . 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 1276	. . . . . 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 539	. . . . . . 7 |- ( ( ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) /\ x e. N. ) /\ z <Q [ <. x , 1o >. ] ~Q ) -> A e. P. )
24		nnprlu 7910	. . . . . . . 8 |- ( x e. N. -> <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. e. P. )
25	24	ad2antlr 493	. . . . . . 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 7863	. . . . . . 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 415	. . . . . 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 2652	. . 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 2673	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 2209   {cab 2224   E.wrex 2529   <.cop 3708   class class class wbr 4125   ` cfv 5372   1stc1st 6362   2ndc2nd 6363   1oc1o 6670   [cec 6795   N.cnpi 7629   ~Q ceq 7636   Q.cnq 7637   <Q cltq 7642   P.cnp 7648   <P cltp 7652

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-inp 7823  df-iltp 7827

This theorem is referenced by:  archsr  8139