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Theorem zssinfcl 10551
Description: The infimum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 16-Jan-2022.)
Hypotheses
Ref Expression
zssinfcl.ex  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  B  -.  y  <  x  /\  A. y  e.  RR  (
x  <  y  ->  E. z  e.  B  z  <  y ) ) )
zssinfcl.ss  |-  ( ph  ->  B  C_  ZZ )
zssinfcl.zz  |-  ( ph  -> inf ( B ,  RR ,  <  )  e.  ZZ )
Assertion
Ref Expression
zssinfcl  |-  ( ph  -> inf ( B ,  RR ,  <  )  e.  B
)
Distinct variable groups:    x, B, y, z    ph, x, y, z

Proof of Theorem zssinfcl
Dummy variables  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zssinfcl.zz . . . . 5  |-  ( ph  -> inf ( B ,  RR ,  <  )  e.  ZZ )
21zred 8602 . . . 4  |-  ( ph  -> inf ( B ,  RR ,  <  )  e.  RR )
3 1red 7248 . . . 4  |-  ( ph  ->  1  e.  RR )
42, 3readdcld 7262 . . 3  |-  ( ph  ->  (inf ( B ,  RR ,  <  )  +  1 )  e.  RR )
52ltp1d 8127 . . 3  |-  ( ph  -> inf ( B ,  RR ,  <  )  <  (inf ( B ,  RR ,  <  )  +  1 ) )
6 lttri3 7310 . . . . 5  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
76adantl 271 . . . 4  |-  ( (
ph  /\  ( f  e.  RR  /\  g  e.  RR ) )  -> 
( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
8 zssinfcl.ex . . . 4  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  B  -.  y  <  x  /\  A. y  e.  RR  (
x  <  y  ->  E. z  e.  B  z  <  y ) ) )
97, 8infglbti 6532 . . 3  |-  ( ph  ->  ( ( (inf ( B ,  RR ,  <  )  +  1 )  e.  RR  /\ inf ( B ,  RR ,  <  )  <  (inf ( B ,  RR ,  <  )  +  1 ) )  ->  E. z  e.  B  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )
104, 5, 9mp2and 424 . 2  |-  ( ph  ->  E. z  e.  B  z  <  (inf ( B ,  RR ,  <  )  +  1 ) )
112adantr 270 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )  -> inf ( B ,  RR ,  <  )  e.  RR )
12 zssinfcl.ss . . . . . . . 8  |-  ( ph  ->  B  C_  ZZ )
1312adantr 270 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )  ->  B  C_  ZZ )
14 simprl 498 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )  ->  z  e.  B
)
1513, 14sseldd 3009 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )  ->  z  e.  ZZ )
1615zred 8602 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )  ->  z  e.  RR )
177, 8inflbti 6531 . . . . . . 7  |-  ( ph  ->  ( z  e.  B  ->  -.  z  < inf ( B ,  RR ,  <  ) ) )
1817imp 122 . . . . . 6  |-  ( (
ph  /\  z  e.  B )  ->  -.  z  < inf ( B ,  RR ,  <  ) )
1918adantrr 463 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )  ->  -.  z  < inf ( B ,  RR ,  <  ) )
2011, 16, 19nltled 7349 . . . 4  |-  ( (
ph  /\  ( z  e.  B  /\  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )  -> inf ( B ,  RR ,  <  )  <_ 
z )
21 simprr 499 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )  ->  z  <  (inf ( B ,  RR ,  <  )  +  1 ) )
221adantr 270 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )  -> inf ( B ,  RR ,  <  )  e.  ZZ )
23 zleltp1 8539 . . . . . 6  |-  ( ( z  e.  ZZ  /\ inf ( B ,  RR ,  <  )  e.  ZZ )  ->  ( z  <_ inf ( B ,  RR ,  <  )  <->  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )
2415, 22, 23syl2anc 403 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )  ->  ( z  <_ inf ( B ,  RR ,  <  )  <->  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )
2521, 24mpbird 165 . . . 4  |-  ( (
ph  /\  ( z  e.  B  /\  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )  ->  z  <_ inf ( B ,  RR ,  <  ) )
2611, 16letri3d 7345 . . . 4  |-  ( (
ph  /\  ( z  e.  B  /\  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )  ->  (inf ( B ,  RR ,  <  )  =  z  <->  (inf ( B ,  RR ,  <  )  <_  z  /\  z  <_ inf ( B ,  RR ,  <  ) ) ) )
2720, 25, 26mpbir2and 886 . . 3  |-  ( (
ph  /\  ( z  e.  B  /\  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )  -> inf ( B ,  RR ,  <  )  =  z )
2827, 14eqeltrd 2159 . 2  |-  ( (
ph  /\  ( z  e.  B  /\  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )  -> inf ( B ,  RR ,  <  )  e.  B )
2910, 28rexlimddv 2486 1  |-  ( ph  -> inf ( B ,  RR ,  <  )  e.  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2353   E.wrex 2354    C_ wss 2982   class class class wbr 3805  (class class class)co 5563  infcinf 6490   RRcr 7094   1c1 7096    + caddc 7098    < clt 7267    <_ cle 7268   ZZcz 8484
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-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-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-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  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
This theorem is referenced by: (None)
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