Theorem archsr 7969

Description: For any signed real, there is an integer that is greater than it. This is also known as the "archimedean property". The expression [ <. ( <. { l | l <Q [ <. x , 1o >. ] ~Q }, { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R is the embedding of the positive integer x into the signed reals. (Contributed by Jim Kingdon, 23-Apr-2020.)

Assertion

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

Distinct variable group:   A, l, u, x

Proof of Theorem archsr

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

Step	Hyp	Ref	Expression
1		df-nr 7914	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		breq1 4086	. . 3 |- ( [ <. z , w >. ] ~R = A -> ( [ <. z , w >. ] ~R <R [ <. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R <-> A <R [ <. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
3	2	rexbidv 2531	. 2 |- ( [ <. z , w >. ] ~R = A -> ( E. x e. N. [ <. z , w >. ] ~R <R [ <. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R <-> E. x e. N. A <R [ <. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
4		1pr 7741	. . . . . . 7 |- 1P e. P.
5		addclpr 7724	. . . . . . 7 |- ( ( z e. P. /\ 1P e. P. ) -> ( z +P. 1P ) e. P. )
6	4, 5	mpan2 425	. . . . . 6 |- ( z e. P. -> ( z +P. 1P ) e. P. )
7		archpr 7830	. . . . . 6 |- ( ( z +P. 1P ) e. P. -> E. x e. N. ( z +P. 1P ) <P <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. )
8	6, 7	syl 14	. . . . 5 |- ( z e. P. -> E. x e. N. ( z +P. 1P ) <P <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. )
9	8	adantr 276	. . . 4 |- ( ( z e. P. /\ w e. P. ) -> E. x e. N. ( z +P. 1P ) <P <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. )
10		nnprlu 7740	. . . . . . . . . 10 |- ( x e. N. -> <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. e. P. )
11	10	adantl 277	. . . . . . . . 9 |- ( ( ( z e. P. /\ w e. P. ) /\ x e. N. ) -> <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. e. P. )
12		addclpr 7724	. . . . . . . . 9 |- ( ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. e. P. /\ 1P e. P. ) -> ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. )
13	11, 4, 12	sylancl 413	. . . . . . . 8 |- ( ( ( z e. P. /\ w e. P. ) /\ x e. N. ) -> ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. )
14		simplr 528	. . . . . . . 8 |- ( ( ( z e. P. /\ w e. P. ) /\ x e. N. ) -> w e. P. )
15		ltaddpr 7784	. . . . . . . 8 |- ( ( ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. /\ w e. P. ) -> ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) <P ( ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) +P. w ) )
16	13, 14, 15	syl2anc 411	. . . . . . 7 |- ( ( ( z e. P. /\ w e. P. ) /\ x e. N. ) -> ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) <P ( ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) +P. w ) )
17		addcomprg 7765	. . . . . . . 8 |- ( ( w e. P. /\ ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. ) -> ( w +P. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) ) = ( ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) +P. w ) )
18	14, 13, 17	syl2anc 411	. . . . . . 7 |- ( ( ( z e. P. /\ w e. P. ) /\ x e. N. ) -> ( w +P. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) ) = ( ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) +P. w ) )
19	16, 18	breqtrrd 4111	. . . . . 6 |- ( ( ( z e. P. /\ w e. P. ) /\ x e. N. ) -> ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) <P ( w +P. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) ) )
20		ltaddpr 7784	. . . . . . . 8 |- ( ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. e. P. /\ 1P e. P. ) -> <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. <P ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) )
21	11, 4, 20	sylancl 413	. . . . . . 7 |- ( ( ( z e. P. /\ w e. P. ) /\ x e. N. ) -> <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. <P ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) )
22		ltsopr 7783	. . . . . . . . 9 |- <P Or P.
23		ltrelpr 7692	. . . . . . . . 9 |- <P C_ ( P. X. P. )
24	22, 23	sotri 5124	. . . . . . . 8 |- ( ( ( z +P. 1P ) <P <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. /\ <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. <P ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) ) -> ( z +P. 1P ) <P ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) )
25	24	expcom 116	. . . . . . 7 |- ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. <P ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) -> ( ( z +P. 1P ) <P <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. -> ( z +P. 1P ) <P ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) ) )
26	21, 25	syl 14	. . . . . 6 |- ( ( ( z e. P. /\ w e. P. ) /\ x e. N. ) -> ( ( z +P. 1P ) <P <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. -> ( z +P. 1P ) <P ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) ) )
27	22, 23	sotri 5124	. . . . . . 7 |- ( ( ( z +P. 1P ) <P ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) /\ ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) <P ( w +P. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) ) ) -> ( z +P. 1P ) <P ( w +P. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) ) )
28	27	expcom 116	. . . . . 6 |- ( ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) <P ( w +P. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) ) -> ( ( z +P. 1P ) <P ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) -> ( z +P. 1P ) <P ( w +P. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) ) ) )
29	19, 26, 28	sylsyld 58	. . . . 5 |- ( ( ( z e. P. /\ w e. P. ) /\ x e. N. ) -> ( ( z +P. 1P ) <P <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. -> ( z +P. 1P ) <P ( w +P. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) ) ) )
30	29	reximdva 2632	. . . 4 |- ( ( z e. P. /\ w e. P. ) -> ( E. x e. N. ( z +P. 1P ) <P <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. -> E. x e. N. ( z +P. 1P ) <P ( w +P. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) ) ) )
31	9, 30	mpd 13	. . 3 |- ( ( z e. P. /\ w e. P. ) -> E. x e. N. ( z +P. 1P ) <P ( w +P. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) ) )
32		simpl 109	. . . . 5 |- ( ( ( z e. P. /\ w e. P. ) /\ x e. N. ) -> ( z e. P. /\ w e. P. ) )
33	4	a1i 9	. . . . 5 |- ( ( ( z e. P. /\ w e. P. ) /\ x e. N. ) -> 1P e. P. )
34		ltsrprg 7934	. . . . 5 |- ( ( ( z e. P. /\ w e. P. ) /\ ( ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. /\ 1P e. P. ) ) -> ( [ <. z , w >. ] ~R <R [ <. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R <-> ( z +P. 1P ) <P ( w +P. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) ) ) )
35	32, 13, 33, 34	syl12anc 1269	. . . 4 |- ( ( ( z e. P. /\ w e. P. ) /\ x e. N. ) -> ( [ <. z , w >. ] ~R <R [ <. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R <-> ( z +P. 1P ) <P ( w +P. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) ) ) )
36	35	rexbidva 2527	. . 3 |- ( ( z e. P. /\ w e. P. ) -> ( E. x e. N. [ <. z , w >. ] ~R <R [ <. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R <-> E. x e. N. ( z +P. 1P ) <P ( w +P. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) ) ) )
37	31, 36	mpbird 167	. 2 |- ( ( z e. P. /\ w e. P. ) -> E. x e. N. [ <. z , w >. ] ~R <R [ <. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R )
38	1, 3, 37	ecoptocl 6769	1 |- ( A e. R. -> E. x e. N. A <R [ <. ( <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {cab 2215   E.wrex 2509   <.cop 3669   class class class wbr 4083  (class class class)co 6001   1oc1o 6555   [cec 6678   N.cnpi 7459   ~Q ceq 7466   <Q cltq 7472   P.cnp 7478   1Pc1p 7479   +P. cpp 7480   <P cltp 7482   ~R cer 7483   R.cnr 7484   <R cltr 7490

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-1o 6562  df-2o 6563  df-oadd 6566  df-omul 6567  df-er 6680  df-ec 6682  df-qs 6686  df-ni 7491  df-pli 7492  df-mi 7493  df-lti 7494  df-plpq 7531  df-mpq 7532  df-enq 7534  df-nqqs 7535  df-plqqs 7536  df-mqqs 7537  df-1nqqs 7538  df-rq 7539  df-ltnqqs 7540  df-enq0 7611  df-nq0 7612  df-0nq0 7613  df-plq0 7614  df-mq0 7615  df-inp 7653  df-i1p 7654  df-iplp 7655  df-iltp 7657  df-enr 7913  df-nr 7914  df-ltr 7917

This theorem is referenced by:  axarch  8078