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Theorem zsupcl 11435
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.
StepHypRef Expression
1 zsupcl.m . . . 4  |-  ( ph  ->  M  e.  ZZ )
21zred 9025 . . 3  |-  ( ph  ->  M  e.  RR )
3 lttri3 7715 . . . . 5  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( u  =  v  <-> 
( -.  u  < 
v  /\  -.  v  <  u ) ) )
43adantl 273 . . . 4  |-  ( (
ph  /\  ( u  e.  RR  /\  v  e.  RR ) )  -> 
( u  =  v  <-> 
( -.  u  < 
v  /\  -.  v  <  u ) ) )
5 zssre 8913 . . . . 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 )
101, 6, 7, 8, 9zsupcllemex 11434 . . . . 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 3109 . . . . 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 ) ) ) )
125, 10, 11mpsyl 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 ) ) )
134, 12supclti 6800 . . 3  |-  ( ph  ->  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  )  e.  RR )
146elrab 2793 . . . . 5  |-  ( M  e.  { n  e.  ZZ  |  ps }  <->  ( M  e.  ZZ  /\  ch ) )
151, 7, 14sylanbrc 411 . . . 4  |-  ( ph  ->  M  e.  { n  e.  ZZ  |  ps }
)
164, 12supubti 6801 . . . 4  |-  ( ph  ->  ( M  e.  {
n  e.  ZZ  |  ps }  ->  -.  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  )  <  M
) )
1715, 16mpd 13 . . 3  |-  ( ph  ->  -.  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  )  <  M )
182, 13, 17nltled 7754 . 2  |-  ( ph  ->  M  <_  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  ) )
195a1i 9 . . . 4  |-  ( ph  ->  ZZ  C_  RR )
204, 10, 19supelti 6804 . . 3  |-  ( ph  ->  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  )  e.  ZZ )
21 eluz 9189 . . 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 ,  <  )
) )
221, 20, 21syl2anc 406 . 2  |-  ( ph  ->  ( sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  )  e.  ( ZZ>= `  M )  <->  M  <_  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  )
) )
2318, 22mpbird 166 1  |-  ( ph  ->  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  )  e.  ( ZZ>= `  M )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104  DECID wdc 786    = wceq 1299    e. wcel 1448   A.wral 2375   E.wrex 2376   {crab 2379    C_ wss 3021   class class class wbr 3875   ` cfv 5059   supcsup 6784   RRcr 7499    < clt 7672    <_ cle 7673   ZZcz 8906   ZZ>=cuz 9176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-sup 6786  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-inn 8579  df-n0 8830  df-z 8907  df-uz 9177  df-fz 9632  df-fzo 9761
This theorem is referenced by:  gcdsupcl  11442
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