Theorem archsr 6958

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 6904	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		breq1 3788	. . 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 2369	. 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 6744	. . . . . . 7 |- 1P e. P.
5		addclpr 6727	. . . . . . 7 |- ( ( z e. P. /\ 1P e. P. ) -> ( z +P. 1P ) e. P. )
6	4, 5	mpan2 415	. . . . . 6 |- ( z e. P. -> ( z +P. 1P ) e. P. )
7		archpr 6833	. . . . . 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 270	. . . 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 6743	. . . . . . . . . 10 |- ( x e. N. -> <. { l | l <Q [ <. x , 1o >. ] ~Q } , { u | [ <. x , 1o >. ] ~Q <Q u } >. e. P. )
11	10	adantl 271	. . . . . . . . 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 6727	. . . . . . . . 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 404	. . . . . . . 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 496	. . . . . . . 8 |- ( ( ( z e. P. /\ w e. P. ) /\ x e. N. ) -> w e. P. )
15		ltaddpr 6787	. . . . . . . 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 403	. . . . . . 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 6768	. . . . . . . 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 403	. . . . . . 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 3811	. . . . . 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 6787	. . . . . . . 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 404	. . . . . . 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 6786	. . . . . . . . 9 |- <P Or P.
23		ltrelpr 6695	. . . . . . . . 9 |- <P C_ ( P. X. P. )
24	22, 23	sotri 4740	. . . . . . . 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 114	. . . . . . 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 4740	. . . . . . 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 114	. . . . . 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 57	. . . . 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 2463	. . . 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 107	. . . . 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 6924	. . . . 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 1167	. . . 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 2365	. . 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 165	. 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 6216	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 102   <-> wb 103   = wceq 1284   e. wcel 1433   {cab 2067   E.wrex 2349   <.cop 3401   class class class wbr 3785  (class class class)co 5532   1oc1o 6017   [cec 6127   N.cnpi 6462   ~Q ceq 6469   <Q cltq 6475   P.cnp 6481   1Pc1p 6482   +P. cpp 6483   <P cltp 6485   ~R cer 6486   R.cnr 6487   <R cltr 6493

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-i1p 6657  df-iplp 6658  df-iltp 6660  df-enr 6903  df-nr 6904  df-ltr 6907

This theorem is referenced by:  axarch  7057