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Theorem infssuzcldc 11571
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 11569 . . 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 3152 . . . . . . 7  |-  { n  e.  ( ZZ>= `  M )  |  ps }  C_  ( ZZ>=
`  M )
72, 6eqsstri 3099 . . . . . 6  |-  S  C_  ( ZZ>= `  M )
8 uzssz 9313 . . . . . 6  |-  ( ZZ>= `  M )  C_  ZZ
97, 8sstri 3076 . . . . 5  |-  S  C_  ZZ
10 zssre 9029 . . . . 5  |-  ZZ  C_  RR
119, 10sstri 3076 . . . 4  |-  S  C_  RR
1211a1i 9 . . 3  |-  ( ph  ->  S  C_  RR )
135, 12infrenegsupex 9357 . 2  |-  ( ph  -> inf ( S ,  RR ,  <  )  =  -u sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  ) )
141, 2, 3, 4infssuzex 11569 . . . . . 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 9359 . . . . 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 7923 . . . . . . . . . 10  |-  ( w  =  u  ->  -u w  =  -u u )
1716eleq1d 2186 . . . . . . . . 9  |-  ( w  =  u  ->  ( -u w  e.  S  <->  -u u  e.  S ) )
1817elrab 2813 . . . . . . . 8  |-  ( u  e.  { w  e.  RR  |  -u w  e.  S }  <->  ( u  e.  RR  /\  -u u  e.  S ) )
199sseli 3063 . . . . . . . . . 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 7762 . . . . . . . . . 10  |-  ( ( u  e.  RR  /\  -u u  e.  S )  ->  u  e.  CC )
23 znegclb 9055 . . . . . . . . . 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 3071 . . . . . 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 9117 . . . 4  |-  ( ph  ->  sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )  e.  {
w  e.  RR  |  -u w  e.  S }
)
30 nfrab1 2587 . . . . . 6  |-  F/_ w { w  e.  RR  |  -u w  e.  S }
31 nfcv 2258 . . . . . 6  |-  F/_ w RR
32 nfcv 2258 . . . . . 6  |-  F/_ w  <
3330, 31, 32nfsup 6847 . . . . 5  |-  F/_ w sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )
3433nfneg 7927 . . . . . 6  |-  F/_ w -u
sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )
3534nfel1 2269 . . . . 5  |-  F/ w -u
sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )  e.  S
36 negeq 7923 . . . . . 6  |-  ( w  =  sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  )  -> 
-u w  =  -u sup ( { w  e.  RR  |  -u w  e.  S } ,  RR ,  <  ) )
3736eleq1d 2186 . . . . 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 2811 . . . 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 2194 1  |-  ( ph  -> inf ( S ,  RR ,  <  )  e.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104  DECID wdc 804    = wceq 1316    e. wcel 1465   {crab 2397    C_ wss 3041   ` cfv 5093  (class class class)co 5742   supcsup 6837  infcinf 6838   CCcc 7586   RRcr 7587    < clt 7768   -ucneg 7902   ZZcz 9022   ZZ>=cuz 9294   ...cfz 9758
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-po 4188  df-iso 4189  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-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-isom 5102  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-sup 6839  df-inf 6840  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8689  df-n0 8946  df-z 9023  df-uz 9295  df-fz 9759  df-fzo 9888
This theorem is referenced by:  lcmval  11671  lcmcllem  11675
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