Theorem zsupcl 11880

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 9313	. . 3 |- ( ph -> M e. RR )
3		lttri3 7978	. . . . 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 9198	. . . . 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 11879	. . . . 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 3207	. . . . 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 6963	. . 3 |- ( ph -> sup ( { n e. ZZ | ps } , RR , < ) e. RR )
14	6	elrab 2882	. . . . 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 6964	. . . 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 8019	. 2 |- ( ph -> M <_ sup ( { n e. ZZ | ps } , RR , < ) )
19	5	a1i 9	. . . 4 |- ( ph -> ZZ C_ RR )
20	4, 10, 19	supelti 6967	. . 3 |- ( ph -> sup ( { n e. ZZ | ps } , RR , < ) e. ZZ )
21		eluz 9479	. . 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 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   {crab 2448   C_ wss 3116   class class class wbr 3982   ` cfv 5188   supcsup 6947   RRcr 7752   < clt 7933   <_ cle 7934   ZZcz 9191   ZZ>=cuz 9466

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-fz 9945  df-fzo 10078

This theorem is referenced by:  suprzubdc  11885  gcdsupcl  11891