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Theorem zssinfcl 10811
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 8794 . . . 4  |-  ( ph  -> inf ( B ,  RR ,  <  )  e.  RR )
3 1red 7440 . . . 4  |-  ( ph  ->  1  e.  RR )
42, 3readdcld 7454 . . 3  |-  ( ph  ->  (inf ( B ,  RR ,  <  )  +  1 )  e.  RR )
52ltp1d 8319 . . 3  |-  ( ph  -> inf ( B ,  RR ,  <  )  <  (inf ( B ,  RR ,  <  )  +  1 ) )
6 lttri3 7502 . . . . 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 6657 . . 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 3015 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )  ->  z  e.  ZZ )
1615zred 8794 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )  ->  z  e.  RR )
177, 8inflbti 6656 . . . . . . 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 7541 . . . 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 8731 . . . . . 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 7537 . . . 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 888 . . 3  |-  ( (
ph  /\  ( z  e.  B  /\  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )  -> inf ( B ,  RR ,  <  )  =  z )
2827, 14eqeltrd 2161 . 2  |-  ( (
ph  /\  ( z  e.  B  /\  z  <  (inf ( B ,  RR ,  <  )  +  1 ) ) )  -> inf ( B ,  RR ,  <  )  e.  B )
2910, 28rexlimddv 2489 1  |-  ( ph  -> inf ( B ,  RR ,  <  )  e.  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1287    e. wcel 1436   A.wral 2355   E.wrex 2356    C_ wss 2988   class class class wbr 3820  (class class class)co 5607  infcinf 6615   RRcr 7286   1c1 7288    + caddc 7290    < clt 7459    <_ cle 7460   ZZcz 8676
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 3931  ax-pow 3983  ax-pr 4009  ax-un 4233  ax-setind 4325  ax-cnex 7373  ax-resscn 7374  ax-1cn 7375  ax-1re 7376  ax-icn 7377  ax-addcl 7378  ax-addrcl 7379  ax-mulcl 7380  ax-addcom 7382  ax-addass 7384  ax-distr 7386  ax-i2m1 7387  ax-0lt1 7388  ax-0id 7390  ax-rnegex 7391  ax-cnre 7393  ax-pre-ltirr 7394  ax-pre-ltwlin 7395  ax-pre-lttrn 7396  ax-pre-apti 7397  ax-pre-ltadd 7398
This theorem depends on definitions:  df-bi 115  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-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-br 3821  df-opab 3875  df-id 4093  df-xp 4416  df-rel 4417  df-cnv 4418  df-co 4419  df-dm 4420  df-iota 4943  df-fun 4980  df-fv 4986  df-riota 5563  df-ov 5610  df-oprab 5611  df-mpt2 5612  df-sup 6616  df-inf 6617  df-pnf 7461  df-mnf 7462  df-xr 7463  df-ltxr 7464  df-le 7465  df-sub 7592  df-neg 7593  df-inn 8351  df-n0 8600  df-z 8677
This theorem is referenced by: (None)
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