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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.
StepHypRef Expression
1 zsupcl.m . . . 4  |-  ( ph  ->  M  e.  ZZ )
21zred 8778 . . 3  |-  ( ph  ->  M  e.  RR )
3 lttri3 7486 . . . . 5  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( u  =  v  <-> 
( -.  u  < 
v  /\  -.  v  <  u ) ) )
43adantl 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 )
101, 6, 7, 8, 9zsupcllemex 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 ) ) ) )
125, 10, 11mpsyl 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 ) ) )
134, 12supclti 6614 . . 3  |-  ( ph  ->  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  )  e.  RR )
146elrab 2761 . . . . 5  |-  ( M  e.  { n  e.  ZZ  |  ps }  <->  ( M  e.  ZZ  /\  ch ) )
151, 7, 14sylanbrc 408 . . . 4  |-  ( ph  ->  M  e.  { n  e.  ZZ  |  ps }
)
164, 12supubti 6615 . . . 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 7525 . 2  |-  ( ph  ->  M  <_  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  ) )
195a1i 9 . . . 4  |-  ( ph  ->  ZZ  C_  RR )
204, 10, 19supelti 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 ,  <  )
) )
221, 20, 21syl2anc 403 . 2  |-  ( ph  ->  ( sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  )  e.  ( ZZ>= `  M )  <->  M  <_  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  )
) )
2318, 22mpbird 165 1  |-  ( ph  ->  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  )  e.  ( ZZ>= `  M )
)
Colors of variables: wff set class
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
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