Theorem archpr 7605

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 7515. (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 7437	. . 3 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prmu 7440	. . 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 7379	. . . 4 |- ( z e. Q. -> E. x e. N. z <Q [ <. x , 1o >. ] ~Q )
5	4	ad2antrl 487	. . 3 |- ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) -> E. x e. N. z <Q [ <. x , 1o >. ] ~Q )
6		simprl 526	. . . . . . . 8 |- ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) -> z e. Q. )
7	6	ad2antrr 485	. . . . . . 7 |- ( ( ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) /\ x e. N. ) /\ z <Q [ <. x , 1o >. ] ~Q ) -> z e. Q. )
8		simprr 527	. . . . . . . 8 |- ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) -> z e. ( 2nd ` A ) )
9	8	ad2antrr 485	. . . . . . 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 2733	. . . . . . . . 9 |- z e. _V
12		breq1 3992	. . . . . . . . 9 |- ( l = z -> ( l <Q [ <. x , 1o >. ] ~Q <-> z <Q [ <. x , 1o >. ] ~Q ) )
13		ltnqex 7511	. . . . . . . . . 10 |- { l | l <Q [ <. x , 1o >. ] ~Q } e. _V
14		gtnqex 7512	. . . . . . . . . 10 |- { u | [ <. x , 1o >. ] ~Q <Q u } e. _V
15	13, 14	op1st 6125	. . . . . . . . 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 2878	. . . . . . . 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 2233	. . . . . . . . 9 |- ( w = z -> ( w e. ( 2nd ` A ) <-> z e. ( 2nd ` A ) ) )
19		eleq1 2233	. . . . . . . . 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 470	. . . . . . . 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 2834	. . . . . . 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 1231	. . . . . 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 528	. . . . . . 7 |- ( ( ( ( A e. P. /\ ( z e. Q. /\ z e. ( 2nd ` A ) ) ) /\ x e. N. ) /\ z <Q [ <. x , 1o >. ] ~Q ) -> A e. P. )
24		nnprlu 7515	. . . . . . . 8 |- ( x e. N. -> <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. e. P. )
25	24	ad2antlr 486	. . . . . . 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 7468	. . . . . . 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 2572	. . 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 2592	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 2141   {cab 2156   E.wrex 2449   <.cop 3586   class class class wbr 3989   ` cfv 5198   1stc1st 6117   2ndc2nd 6118   1oc1o 6388   [cec 6511   N.cnpi 7234   ~Q ceq 7241   Q.cnq 7242   <Q cltq 7247   P.cnp 7253   <P cltp 7257

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-inp 7428  df-iltp 7432

This theorem is referenced by:  archsr  7744