Theorem archsr 8113

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 8058	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		breq1 4117	. . 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 2545	. 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 7885	. . . . . . 7 |- 1P e. P.
5		addclpr 7868	. . . . . . 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 7974	. . . . . 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 7884	. . . . . . . . . 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 7868	. . . . . . . . 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 529	. . . . . . . 8 |- ( ( ( z e. P. /\ w e. P. ) /\ x e. N. ) -> w e. P. )
15		ltaddpr 7928	. . . . . . . 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 7909	. . . . . . . 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 4142	. . . . . 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 7928	. . . . . . . 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 7927	. . . . . . . . 9 |- <P Or P.
23		ltrelpr 7836	. . . . . . . . 9 |- <P C_ ( P. X. P. )
24	22, 23	sotri 5163	. . . . . . . 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 5163	. . . . . . 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 2646	. . . 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 8078	. . . . 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 1272	. . . 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 2541	. . 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 6869	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 1398   e. wcel 2205   {cab 2220   E.wrex 2523   <.cop 3697   class class class wbr 4114  (class class class)co 6058   1oc1o 6653   [cec 6778   N.cnpi 7603   ~Q ceq 7610   <Q cltq 7616   P.cnp 7622   1Pc1p 7623   +P. cpp 7624   <P cltp 7626   ~R cer 7627   R.cnr 7628   <R cltr 7634

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-enq0 7755  df-nq0 7756  df-0nq0 7757  df-plq0 7758  df-mq0 7759  df-inp 7797  df-i1p 7798  df-iplp 7799  df-iltp 7801  df-enr 8057  df-nr 8058  df-ltr 8061

This theorem is referenced by:  axarch  8222