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

Theorem infssuzcldc 11629
Description: The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by Jim Kingdon, 20-Jan-2022.)
Hypotheses
Ref Expression
infssuzledc.m  |-  ( ph  ->  M  e.  ZZ )
infssuzledc.s  |-  S  =  { n  e.  (
ZZ>= `  M )  |  ps }
infssuzledc.a  |-  ( ph  ->  A  e.  S )
infssuzledc.dc  |-  ( (
ph  /\  n  e.  ( M ... A ) )  -> DECID  ps )
Assertion
Ref Expression
infssuzcldc  |-  ( ph  -> inf ( S ,  RR ,  <  )  e.  S
)
Distinct variable groups:    A, n    n, M    ph, n
Allowed substitution hints:    ps( n)    S( n)

Proof of Theorem infssuzcldc
Dummy variables  y  w  x  z  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 infssuzledc.m . . . 4  |-  ( ph  ->  M  e.  ZZ )
2 infssuzledc.s . . . 4  |-  S  =  { n  e.  (
ZZ>= `  M )  |  ps }
3 infssuzledc.a . . . 4  |-  ( ph  ->  A  e.  S )
4 infssuzledc.dc . . . 4  |-  ( (
ph  /\  n  e.  ( M ... A ) )  -> DECID  ps )
51, 2, 3, 4infssuzex 11627 . . 3  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  S  -.  y  <  x  /\  A. y  e.  RR  (
x  <  y  ->  E. w  e.  S  w  <  y ) ) )
6 ssrab2 3177 . . . . . . 7  |-  { n  e.  ( ZZ>= `  M )  |  ps }  C_  ( ZZ>=
`  M )
72, 6eqsstri 3124 . . . . . 6  |-  S  C_  ( ZZ>= `  M )
8 uzssz 9338 . . . . . 6  |-  ( ZZ>= `  M )  C_  ZZ
97, 8sstri 3101 . . . . 5  |-  S  C_  ZZ
10 zssre 9054 . . . . 5  |-  ZZ  C_  RR
119, 10sstri 3101 . . . 4  |-  S  C_  RR
1211a1i 9 . . 3  |-  ( ph  ->  S  C_  RR )
135, 12infrenegsupex 9382 . 2  |-  ( ph  -> inf ( S ,  RR ,  <  )  =  -u sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  ) )
141, 2, 3, 4infssuzex 11627 . . . . . 6  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  S  -.  y  <  x  /\  A. y  e.  RR  (
x  <  y  ->  E. z  e.  S  z  <  y ) ) )
1514, 12infsupneg 9384 . . . . 5  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  {
w  e.  RR  |  -u w  e.  S }  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  { w  e.  RR  |  -u w  e.  S } y  < 
z ) ) )
16 negeq 7948 . . . . . . . . . 10  |-  ( w  =  u  ->  -u w  =  -u u )
1716eleq1d 2206 . . . . . . . . 9  |-  ( w  =  u  ->  ( -u w  e.  S  <->  -u u  e.  S ) )
1817elrab 2835 . . . . . . . 8  |-  ( u  e.  { w  e.  RR  |  -u w  e.  S }  <->  ( u  e.  RR  /\  -u u  e.  S ) )
199sseli 3088 . . . . . . . . . 10  |-  ( -u u  e.  S  ->  -u u  e.  ZZ )
2019adantl 275 . . . . . . . . 9  |-  ( ( u  e.  RR  /\  -u u  e.  S )  ->  -u u  e.  ZZ )
21 simpl 108 . . . . . . . . . . 11  |-  ( ( u  e.  RR  /\  -u u  e.  S )  ->  u  e.  RR )
2221recnd 7787 . . . . . . . . . 10  |-  ( ( u  e.  RR  /\  -u u  e.  S )  ->  u  e.  CC )
23 znegclb 9080 . . . . . . . . . 10  |-  ( u  e.  CC  ->  (
u  e.  ZZ  <->  -u u  e.  ZZ ) )
2422, 23syl 14 . . . . . . . . 9  |-  ( ( u  e.  RR  /\  -u u  e.  S )  ->  ( u  e.  ZZ  <->  -u u  e.  ZZ ) )
2520, 24mpbird 166 . . . . . . . 8  |-  ( ( u  e.  RR  /\  -u u  e.  S )  ->  u  e.  ZZ )
2618, 25sylbi 120 . . . . . . 7  |-  ( u  e.  { w  e.  RR  |  -u w  e.  S }  ->  u  e.  ZZ )
2726ssriv 3096 . . . . . 6  |-  { w  e.  RR  |  -u w  e.  S }  C_  ZZ
2827a1i 9 . . . . 5  |-  ( ph  ->  { w  e.  RR  |  -u w  e.  S }  C_  ZZ )
2915, 28suprzclex 9142 . . . 4  |-  ( ph  ->  sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )  e.  {
w  e.  RR  |  -u w  e.  S }
)
30 nfrab1 2608 . . . . . 6  |-  F/_ w { w  e.  RR  |  -u w  e.  S }
31 nfcv 2279 . . . . . 6  |-  F/_ w RR
32 nfcv 2279 . . . . . 6  |-  F/_ w  <
3330, 31, 32nfsup 6872 . . . . 5  |-  F/_ w sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )
3433nfneg 7952 . . . . . 6  |-  F/_ w -u
sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )
3534nfel1 2290 . . . . 5  |-  F/ w -u
sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )  e.  S
36 negeq 7948 . . . . . 6  |-  ( w  =  sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )  -> 
-u w  =  -u sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  ) )
3736eleq1d 2206 . . . . 5  |-  ( w  =  sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )  ->  ( -u w  e.  S  <->  -u sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )  e.  S ) )
3833, 31, 35, 37elrabf 2833 . . . 4  |-  ( sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )  e.  {
w  e.  RR  |  -u w  e.  S }  <->  ( sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )  e.  RR  /\  -u sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )  e.  S
) )
3929, 38sylib 121 . . 3  |-  ( ph  ->  ( sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )  e.  RR  /\  -u sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )  e.  S ) )
4039simprd 113 . 2  |-  ( ph  -> 
-u sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )  e.  S
)
4113, 40eqeltrd 2214 1  |-  ( ph  -> inf ( S ,  RR ,  <  )  e.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104  DECID wdc 819    = wceq 1331    e. wcel 1480   {crab 2418    C_ wss 3066   ` cfv 5118  (class class class)co 5767   supcsup 6862  infcinf 6863   CCcc 7611   RRcr 7612    < clt 7793   -ucneg 7927   ZZcz 9047   ZZ>=cuz 9319   ...cfz 9783
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-isom 5127  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-sup 6864  df-inf 6865  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-fz 9784  df-fzo 9913
This theorem is referenced by:  lcmval  11729  lcmcllem  11733
  Copyright terms: Public domain W3C validator