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Theorem zsupcl 11674
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 9196 . . 3  |-  ( ph  ->  M  e.  RR )
3 lttri3 7867 . . . . 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 9084 . . . . 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 11673 . . . . 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 3166 . . . . 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 6892 . . 3  |-  ( ph  ->  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  )  e.  RR )
146elrab 2843 . . . . 5  |-  ( M  e.  { n  e.  ZZ  |  ps }  <->  ( M  e.  ZZ  /\  ch ) )
151, 7, 14sylanbrc 414 . . . 4  |-  ( ph  ->  M  e.  { n  e.  ZZ  |  ps }
)
164, 12supubti 6893 . . . 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 7906 . 2  |-  ( ph  ->  M  <_  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  ) )
195a1i 9 . . . 4  |-  ( ph  ->  ZZ  C_  RR )
204, 10, 19supelti 6896 . . 3  |-  ( ph  ->  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  )  e.  ZZ )
21 eluz 9362 . . 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 409 . 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 820    = wceq 1332    e. wcel 1481   A.wral 2417   E.wrex 2418   {crab 2421    C_ wss 3075   class class class wbr 3936   ` cfv 5130   supcsup 6876   RRcr 7642    < clt 7823    <_ cle 7824   ZZcz 9077   ZZ>=cuz 9349
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-addcom 7743  ax-addass 7745  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-0id 7751  ax-rnegex 7752  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-sup 6878  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-inn 8744  df-n0 9001  df-z 9078  df-uz 9350  df-fz 9821  df-fzo 9950
This theorem is referenced by:  gcdsupcl  11681
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