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Theorem zsupcl 11552
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 9131 . . 3  |-  ( ph  ->  M  e.  RR )
3 lttri3 7812 . . . . 5  |-  ( ( u  e.  RR  /\  v  e.  RR )  ->  ( u  =  v  <-> 
( -.  u  < 
v  /\  -.  v  <  u ) ) )
43adantl 275 . . . 4  |-  ( (
ph  /\  ( u  e.  RR  /\  v  e.  RR ) )  -> 
( u  =  v  <-> 
( -.  u  < 
v  /\  -.  v  <  u ) ) )
5 zssre 9019 . . . . 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 11551 . . . . 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 3132 . . . . 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 6853 . . 3  |-  ( ph  ->  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  )  e.  RR )
146elrab 2813 . . . . 5  |-  ( M  e.  { n  e.  ZZ  |  ps }  <->  ( M  e.  ZZ  /\  ch ) )
151, 7, 14sylanbrc 413 . . . 4  |-  ( ph  ->  M  e.  { n  e.  ZZ  |  ps }
)
164, 12supubti 6854 . . . 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 7851 . 2  |-  ( ph  ->  M  <_  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  ) )
195a1i 9 . . . 4  |-  ( ph  ->  ZZ  C_  RR )
204, 10, 19supelti 6857 . . 3  |-  ( ph  ->  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  )  e.  ZZ )
21 eluz 9295 . . 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 408 . 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 804    = wceq 1316    e. wcel 1465   A.wral 2393   E.wrex 2394   {crab 2397    C_ wss 3041   class class class wbr 3899   ` cfv 5093   supcsup 6837   RRcr 7587    < clt 7768    <_ cle 7769   ZZcz 9012   ZZ>=cuz 9282
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-sup 6839  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8685  df-n0 8936  df-z 9013  df-uz 9283  df-fz 9746  df-fzo 9875
This theorem is referenced by:  gcdsupcl  11559
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