Theorem archsr 7930

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 7875	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		breq1 4062	. . 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 2509	. 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 7702	. . . . . . 7 |- 1P e. P.
5		addclpr 7685	. . . . . . 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 7791	. . . . . 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 7701	. . . . . . . . . 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 7685	. . . . . . . . 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 7745	. . . . . . . 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 7726	. . . . . . . 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 4087	. . . . . 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 7745	. . . . . . . 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 7744	. . . . . . . . 9 |- <P Or P.
23		ltrelpr 7653	. . . . . . . . 9 |- <P C_ ( P. X. P. )
24	22, 23	sotri 5097	. . . . . . . 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 5097	. . . . . . 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 2610	. . . 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 7895	. . . . 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 1248	. . . 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 2505	. . 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 6732	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 1373   e. wcel 2178   {cab 2193   E.wrex 2487   <.cop 3646   class class class wbr 4059  (class class class)co 5967   1oc1o 6518   [cec 6641   N.cnpi 7420   ~Q ceq 7427   <Q cltq 7433   P.cnp 7439   1Pc1p 7440   +P. cpp 7441   <P cltp 7443   ~R cer 7444   R.cnr 7445   <R cltr 7451

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-eprel 4354  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-1o 6525  df-2o 6526  df-oadd 6529  df-omul 6530  df-er 6643  df-ec 6645  df-qs 6649  df-ni 7452  df-pli 7453  df-mi 7454  df-lti 7455  df-plpq 7492  df-mpq 7493  df-enq 7495  df-nqqs 7496  df-plqqs 7497  df-mqqs 7498  df-1nqqs 7499  df-rq 7500  df-ltnqqs 7501  df-enq0 7572  df-nq0 7573  df-0nq0 7574  df-plq0 7575  df-mq0 7576  df-inp 7614  df-i1p 7615  df-iplp 7616  df-iltp 7618  df-enr 7874  df-nr 7875  df-ltr 7878

This theorem is referenced by:  axarch  8039