Theorem zsupcl 11865

Description: Closure of supremum for decidable integer properties. The property which defines the set we are taking the supremum of must (a) be true at M (which corresponds to the nonempty condition of classical supremum theorems), (b) decidable at each value after M, and (c) be false after j (which corresponds to the upper bound condition found in classical supremum theorems). (Contributed by Jim Kingdon, 7-Dec-2021.)

Hypotheses

Ref	Expression
zsupcl.m	|- ( ph -> M e. ZZ )
zsupcl.sbm	|- ( n = M -> ( ps <-> ch ) )
zsupcl.mtru	|- ( ph -> ch )
zsupcl.dc	|- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> DECID ps )
zsupcl.bnd	|- ( ph -> E. j e. ( ZZ>= ` M ) A. n e. ( ZZ>= ` j ) -. ps )

Assertion

Ref	Expression
zsupcl	|- ( ph -> sup ( { n e. ZZ | ps } , RR , < ) e. ( ZZ>= ` M ) )

Distinct variable groups:   ph, j, n   ps, j   ch, j, n   j, M, n

Allowed substitution hint:   ps( n)

Proof of Theorem zsupcl

Dummy variables x y z u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zsupcl.m	. . . 4 |- ( ph -> M e. ZZ )
2	1	zred 9304	. . 3 |- ( ph -> M e. RR )
3		lttri3 7969	. . . . 5 |- ( ( u e. RR /\ v e. RR ) -> ( u = v <-> ( -. u < v /\ -. v < u ) ) )
4	3	adantl 275	. . . 4 |- ( ( ph /\ ( u e. RR /\ v e. RR ) ) -> ( u = v <-> ( -. u < v /\ -. v < u ) ) )
5		zssre 9189	. . . . 5 |- ZZ C_ RR
6		zsupcl.sbm	. . . . . 6 |- ( n = M -> ( ps <-> ch ) )
7		zsupcl.mtru	. . . . . 6 |- ( ph -> ch )
8		zsupcl.dc	. . . . . 6 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> DECID ps )
9		zsupcl.bnd	. . . . . 6 |- ( ph -> E. j e. ( ZZ>= ` M ) A. n e. ( ZZ>= ` j ) -. ps )
10	1, 6, 7, 8, 9	zsupcllemex 11864	. . . . 5 |- ( ph -> E. x e. ZZ ( A. y e. { n e. ZZ | ps } -. x < y /\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ps } y < z ) ) )
11		ssrexv 3202	. . . . 5 |- ( ZZ C_ RR -> ( E. x e. ZZ ( A. y e. { n e. ZZ | ps } -. x < y /\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ps } y < z ) ) -> E. x e. RR ( A. y e. { n e. ZZ | ps } -. x < y /\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ps } y < z ) ) ) )
12	5, 10, 11	mpsyl 65	. . . 4 |- ( ph -> E. x e. RR ( A. y e. { n e. ZZ | ps } -. x < y /\ A. y e. RR ( y < x -> E. z e. { n e. ZZ | ps } y < z ) ) )
13	4, 12	supclti 6954	. . 3 |- ( ph -> sup ( { n e. ZZ | ps } , RR , < ) e. RR )
14	6	elrab 2877	. . . . 5 |- ( M e. { n e. ZZ | ps } <-> ( M e. ZZ /\ ch ) )
15	1, 7, 14	sylanbrc 414	. . . 4 |- ( ph -> M e. { n e. ZZ | ps } )
16	4, 12	supubti 6955	. . . 4 |- ( ph -> ( M e. { n e. ZZ | ps } -> -. sup ( { n e. ZZ | ps } , RR , < ) < M ) )
17	15, 16	mpd 13	. . 3 |- ( ph -> -. sup ( { n e. ZZ | ps } , RR , < ) < M )
18	2, 13, 17	nltled 8010	. 2 |- ( ph -> M <_ sup ( { n e. ZZ | ps } , RR , < ) )
19	5	a1i 9	. . . 4 |- ( ph -> ZZ C_ RR )
20	4, 10, 19	supelti 6958	. . 3 |- ( ph -> sup ( { n e. ZZ | ps } , RR , < ) e. ZZ )
21		eluz 9470	. . 3 |- ( ( M e. ZZ /\ sup ( { n e. ZZ | ps } , RR , < ) e. ZZ ) -> ( sup ( { n e. ZZ | ps } , RR , < ) e. ( ZZ>= ` M ) <-> M <_ sup ( { n e. ZZ | ps } , RR , < ) ) )
22	1, 20, 21	syl2anc 409	. 2 |- ( ph -> ( sup ( { n e. ZZ | ps } , RR , < ) e. ( ZZ>= ` M ) <-> M <_ sup ( { n e. ZZ | ps } , RR , < ) ) )
23	18, 22	mpbird 166	1 |- ( ph -> sup ( { n e. ZZ | ps } , RR , < ) e. ( ZZ>= ` M ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104  DECID wdc 824   = wceq 1342   e. wcel 2135   A.wral 2442   E.wrex 2443   {crab 2446   C_ wss 3111   class class class wbr 3976   ` cfv 5182   supcsup 6938   RRcr 7743   < clt 7924   <_ cle 7925   ZZcz 9182   ZZ>=cuz 9457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-addcom 7844  ax-addass 7846  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-0id 7852  ax-rnegex 7853  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-sup 6940  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-inn 8849  df-n0 9106  df-z 9183  df-uz 9458  df-fz 9936  df-fzo 10068

This theorem is referenced by:  suprzubdc  11870  gcdsupcl  11876