Theorem zsupcl 10737

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 non-empty 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 8778	. . 3 |- ( ph -> M e. RR )
3		lttri3 7486	. . . . 5 |- ( ( u e. RR /\ v e. RR ) -> ( u = v <-> ( -. u < v /\ -. v < u ) ) )
4	3	adantl 271	. . . 4 |- ( ( ph /\ ( u e. RR /\ v e. RR ) ) -> ( u = v <-> ( -. u < v /\ -. v < u ) ) )
5		zssre 8667	. . . . 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 10736	. . . . 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 3072	. . . . 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 64	. . . 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 6614	. . 3 |- ( ph -> sup ( { n e. ZZ | ps } , RR , < ) e. RR )
14	6	elrab 2761	. . . . 5 |- ( M e. { n e. ZZ | ps } <-> ( M e. ZZ /\ ch ) )
15	1, 7, 14	sylanbrc 408	. . . 4 |- ( ph -> M e. { n e. ZZ | ps } )
16	4, 12	supubti 6615	. . . 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 7525	. 2 |- ( ph -> M <_ sup ( { n e. ZZ | ps } , RR , < ) )
19	5	a1i 9	. . . 4 |- ( ph -> ZZ C_ RR )
20	4, 10, 19	supelti 6618	. . 3 |- ( ph -> sup ( { n e. ZZ | ps } , RR , < ) e. ZZ )
21		eluz 8941	. . 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 403	. 2 |- ( ph -> ( sup ( { n e. ZZ | ps } , RR , < ) e. ( ZZ>= ` M ) <-> M <_ sup ( { n e. ZZ | ps } , RR , < ) ) )
23	18, 22	mpbird 165	1 |- ( ph -> sup ( { n e. ZZ | ps } , RR , < ) e. ( ZZ>= ` M ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103  DECID wdc 778   = wceq 1287   e. wcel 1436   A.wral 2355   E.wrex 2356   {crab 2359   C_ wss 2986   class class class wbr 3814   ` cfv 4972   supcsup 6598   RRcr 7270   < clt 7443   <_ cle 7444   ZZcz 8660   ZZ>=cuz 8928

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-addcom 7366  ax-addass 7368  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-0id 7374  ax-rnegex 7375  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-apti 7381  ax-pre-ltadd 7382

This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-sup 6600  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-inn 8335  df-n0 8584  df-z 8661  df-uz 8929  df-fz 9334  df-fzo 9458

This theorem is referenced by:  gcdsupcl  10744