Theorem archsr 7866

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 7811	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		breq1 4037	. . 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 2498	. 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 7638	. . . . . . 7 |- 1P e. P.
5		addclpr 7621	. . . . . . 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 7727	. . . . . 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 7637	. . . . . . . . . 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 7621	. . . . . . . . 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 7681	. . . . . . . 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 7662	. . . . . . . 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 4062	. . . . . 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 7681	. . . . . . . 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 7680	. . . . . . . . 9 |- <P Or P.
23		ltrelpr 7589	. . . . . . . . 9 |- <P C_ ( P. X. P. )
24	22, 23	sotri 5066	. . . . . . . 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 5066	. . . . . . 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 2599	. . . 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 7831	. . . . 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 1247	. . . 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 2494	. . 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 6690	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 1364   e. wcel 2167   {cab 2182   E.wrex 2476   <.cop 3626   class class class wbr 4034  (class class class)co 5925   1oc1o 6476   [cec 6599   N.cnpi 7356   ~Q ceq 7363   <Q cltq 7369   P.cnp 7375   1Pc1p 7376   +P. cpp 7377   <P cltp 7379   ~R cer 7380   R.cnr 7381   <R cltr 7387

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-i1p 7551  df-iplp 7552  df-iltp 7554  df-enr 7810  df-nr 7811  df-ltr 7814

This theorem is referenced by:  axarch  7975