ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  zsupcl Unicode version

Theorem zsupcl 10818
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 8801 . . 3  |-  ( ph  ->  M  e.  RR )
3 lttri3 7509 . . . . 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 8690 . . . . 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 10817 . . . . 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 3075 . . . . 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 6637 . . 3  |-  ( ph  ->  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  )  e.  RR )
146elrab 2762 . . . . 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 6638 . . . 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 7548 . 2  |-  ( ph  ->  M  <_  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  ) )
195a1i 9 . . . 4  |-  ( ph  ->  ZZ  C_  RR )
204, 10, 19supelti 6641 . . 3  |-  ( ph  ->  sup ( { n  e.  ZZ  |  ps } ,  RR ,  <  )  e.  ZZ )
21 eluz 8964 . . 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 2988   class class class wbr 3820   ` cfv 4981   supcsup 6621   RRcr 7293    < clt 7466    <_ cle 7467   ZZcz 8683   ZZ>=cuz 8951
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 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-cnex 7380  ax-resscn 7381  ax-1cn 7382  ax-1re 7383  ax-icn 7384  ax-addcl 7385  ax-addrcl 7386  ax-mulcl 7387  ax-addcom 7389  ax-addass 7391  ax-distr 7393  ax-i2m1 7394  ax-0lt1 7395  ax-0id 7397  ax-rnegex 7398  ax-cnre 7400  ax-pre-ltirr 7401  ax-pre-ltwlin 7402  ax-pre-lttrn 7403  ax-pre-apti 7404  ax-pre-ltadd 7405
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 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-id 4094  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-fv 4989  df-riota 5569  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-1st 5868  df-2nd 5869  df-sup 6623  df-pnf 7468  df-mnf 7469  df-xr 7470  df-ltxr 7471  df-le 7472  df-sub 7599  df-neg 7600  df-inn 8358  df-n0 8607  df-z 8684  df-uz 8952  df-fz 9357  df-fzo 9482
This theorem is referenced by:  gcdsupcl  10825
  Copyright terms: Public domain W3C validator