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Theorem infssuzcldc 10554
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 10552 . . 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 3088 . . . . . . 7  |-  { n  e.  ( ZZ>= `  M )  |  ps }  C_  ( ZZ>=
`  M )
72, 6eqsstri 3038 . . . . . 6  |-  S  C_  ( ZZ>= `  M )
8 uzssz 8771 . . . . . 6  |-  ( ZZ>= `  M )  C_  ZZ
97, 8sstri 3017 . . . . 5  |-  S  C_  ZZ
10 zssre 8491 . . . . 5  |-  ZZ  C_  RR
119, 10sstri 3017 . . . 4  |-  S  C_  RR
1211a1i 9 . . 3  |-  ( ph  ->  S  C_  RR )
135, 12infrenegsupex 8815 . 2  |-  ( ph  -> inf ( S ,  RR ,  <  )  =  -u sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  ) )
141, 2, 3, 4infssuzex 10552 . . . . . 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 8817 . . . . 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 7420 . . . . . . . . . 10  |-  ( w  =  u  ->  -u w  =  -u u )
1716eleq1d 2151 . . . . . . . . 9  |-  ( w  =  u  ->  ( -u w  e.  S  <->  -u u  e.  S ) )
1817elrab 2757 . . . . . . . 8  |-  ( u  e.  { w  e.  RR  |  -u w  e.  S }  <->  ( u  e.  RR  /\  -u u  e.  S ) )
199sseli 3004 . . . . . . . . . 10  |-  ( -u u  e.  S  ->  -u u  e.  ZZ )
2019adantl 271 . . . . . . . . 9  |-  ( ( u  e.  RR  /\  -u u  e.  S )  ->  -u u  e.  ZZ )
21 simpl 107 . . . . . . . . . . 11  |-  ( ( u  e.  RR  /\  -u u  e.  S )  ->  u  e.  RR )
2221recnd 7261 . . . . . . . . . 10  |-  ( ( u  e.  RR  /\  -u u  e.  S )  ->  u  e.  CC )
23 znegclb 8517 . . . . . . . . . 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 165 . . . . . . . 8  |-  ( ( u  e.  RR  /\  -u u  e.  S )  ->  u  e.  ZZ )
2618, 25sylbi 119 . . . . . . 7  |-  ( u  e.  { w  e.  RR  |  -u w  e.  S }  ->  u  e.  ZZ )
2726ssriv 3012 . . . . . 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 8578 . . . 4  |-  ( ph  ->  sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )  e.  {
w  e.  RR  |  -u w  e.  S }
)
30 nfrab1 2538 . . . . . 6  |-  F/_ w { w  e.  RR  |  -u w  e.  S }
31 nfcv 2223 . . . . . 6  |-  F/_ w RR
32 nfcv 2223 . . . . . 6  |-  F/_ w  <
3330, 31, 32nfsup 6499 . . . . 5  |-  F/_ w sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )
3433nfneg 7424 . . . . . 6  |-  F/_ w -u
sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )
3534nfel1 2233 . . . . 5  |-  F/ w -u
sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )  e.  S
36 negeq 7420 . . . . . 6  |-  ( w  =  sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )  -> 
-u w  =  -u sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  ) )
3736eleq1d 2151 . . . . 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 2755 . . . 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 120 . . 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 112 . 2  |-  ( ph  -> 
-u sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )  e.  S
)
4113, 40eqeltrd 2159 1  |-  ( ph  -> inf ( S ,  RR ,  <  )  e.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103  DECID wdc 776    = wceq 1285    e. wcel 1434   {crab 2357    C_ wss 2982   ` cfv 4952  (class class class)co 5563   supcsup 6489  infcinf 6490   CCcc 7093   RRcr 7094    < clt 7267   -ucneg 7399   ZZcz 8484   ZZ>=cuz 8752   ...cfz 9157
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-po 4079  df-iso 4080  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-isom 4961  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-sup 6491  df-inf 6492  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-inn 8159  df-n0 8408  df-z 8485  df-uz 8753  df-fz 9158  df-fzo 9282
This theorem is referenced by:  lcmval  10652  lcmcllem  10656
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