Theorem zsupcl 12087

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 9442	. . 3 |- ( ph -> M e. RR )
3		lttri3 8101	. . . . 5 |- ( ( u e. RR /\ v e. RR ) -> ( u = v <-> ( -. u < v /\ -. v < u ) ) )
4	3	adantl 277	. . . 4 |- ( ( ph /\ ( u e. RR /\ v e. RR ) ) -> ( u = v <-> ( -. u < v /\ -. v < u ) ) )
5		zssre 9327	. . . . 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 12086	. . . . 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 3245	. . . . 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 7059	. . 3 |- ( ph -> sup ( { n e. ZZ | ps } , RR , < ) e. RR )
14	6	elrab 2917	. . . . 5 |- ( M e. { n e. ZZ | ps } <-> ( M e. ZZ /\ ch ) )
15	1, 7, 14	sylanbrc 417	. . . 4 |- ( ph -> M e. { n e. ZZ | ps } )
16	4, 12	supubti 7060	. . . 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 8142	. 2 |- ( ph -> M <_ sup ( { n e. ZZ | ps } , RR , < ) )
19	5	a1i 9	. . . 4 |- ( ph -> ZZ C_ RR )
20	4, 10, 19	supelti 7063	. . 3 |- ( ph -> sup ( { n e. ZZ | ps } , RR , < ) e. ZZ )
21		eluz 9608	. . 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 411	. 2 |- ( ph -> ( sup ( { n e. ZZ | ps } , RR , < ) e. ( ZZ>= ` M ) <-> M <_ sup ( { n e. ZZ | ps } , RR , < ) ) )
23	18, 22	mpbird 167	1 |- ( ph -> sup ( { n e. ZZ | ps } , RR , < ) e. ( ZZ>= ` M ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 835   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   {crab 2476   C_ wss 3154   class class class wbr 4030   ` cfv 5255   supcsup 7043   RRcr 7873   < clt 8056   <_ cle 8057   ZZcz 9320   ZZ>=cuz 9595

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-sup 7045  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596  df-fz 10078  df-fzo 10212

This theorem is referenced by:  suprzubdc  12092  gcdsupcl  12098