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Theorem suprzclex 9382
Description: The supremum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 20-Dec-2021.)
Hypotheses
Ref Expression
suprzclex.ex  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  A  y  <  z ) ) )
suprzclex.ss  |-  ( ph  ->  A  C_  ZZ )
Assertion
Ref Expression
suprzclex  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  A )
Distinct variable groups:    x, A, y, z    ph, x, z
Allowed substitution hint:    ph( y)

Proof of Theorem suprzclex
Dummy variables  w  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lttri3 8068 . . . . . 6  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
21adantl 277 . . . . 5  |-  ( (
ph  /\  ( f  e.  RR  /\  g  e.  RR ) )  -> 
( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
3 suprzclex.ex . . . . 5  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  A  y  <  z ) ) )
42, 3supclti 7028 . . . 4  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
54ltm1d 8920 . . 3  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  -  1 )  <  sup ( A ,  RR ,  <  ) )
6 suprzclex.ss . . . . 5  |-  ( ph  ->  A  C_  ZZ )
7 zssre 9291 . . . . 5  |-  ZZ  C_  RR
86, 7sstrdi 3182 . . . 4  |-  ( ph  ->  A  C_  RR )
9 peano2rem 8255 . . . . 5  |-  ( sup ( A ,  RR ,  <  )  e.  RR  ->  ( sup ( A ,  RR ,  <  )  -  1 )  e.  RR )
104, 9syl 14 . . . 4  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  -  1 )  e.  RR )
113, 8, 10suprlubex 8940 . . 3  |-  ( ph  ->  ( ( sup ( A ,  RR ,  <  )  -  1 )  <  sup ( A ,  RR ,  <  )  <->  E. z  e.  A  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )
125, 11mpbid 147 . 2  |-  ( ph  ->  E. z  e.  A  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
)
136adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  A  C_  ZZ )
1413sselda 3170 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  e.  ZZ )
157, 14sselid 3168 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  e.  RR )
164adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  e.  RR )
1716adantr 276 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  sup ( A ,  RR ,  <  )  e.  RR )
18 simprl 529 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  z  e.  A )
1913, 18sseldd 3171 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  z  e.  ZZ )
20 zre 9288 . . . . . . . . . . 11  |-  ( z  e.  ZZ  ->  z  e.  RR )
2119, 20syl 14 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  z  e.  RR )
22 peano2re 8124 . . . . . . . . . 10  |-  ( z  e.  RR  ->  (
z  +  1 )  e.  RR )
2321, 22syl 14 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  ( z  +  1 )  e.  RR )
2423adantr 276 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  (
z  +  1 )  e.  RR )
253ad2antrr 488 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  ( y  < 
x  ->  E. z  e.  A  y  <  z ) ) )
268ad2antrr 488 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  A  C_  RR )
27 simpr 110 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  e.  A )
2825, 26, 27suprubex 8939 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  <_  sup ( A ,  RR ,  <  ) )
29 simprr 531 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  ( sup ( A ,  RR ,  <  )  -  1 )  <  z )
30 1red 8003 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  1  e.  RR )
3116, 30, 21ltsubaddd 8529 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  ( ( sup ( A ,  RR ,  <  )  -  1 )  <  z  <->  sup ( A ,  RR ,  <  )  <  ( z  +  1 ) ) )
3229, 31mpbid 147 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  <  ( z  +  1 ) )
3332adantr 276 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  sup ( A ,  RR ,  <  )  <  ( z  +  1 ) )
3415, 17, 24, 28, 33lelttrd 8113 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  <  ( z  +  1 ) )
3519adantr 276 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  z  e.  ZZ )
36 zleltp1 9339 . . . . . . . 8  |-  ( ( w  e.  ZZ  /\  z  e.  ZZ )  ->  ( w  <_  z  <->  w  <  ( z  +  1 ) ) )
3714, 35, 36syl2anc 411 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  (
w  <_  z  <->  w  <  ( z  +  1 ) ) )
3834, 37mpbird 167 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  <_  z )
3938ralrimiva 2563 . . . . 5  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  A. w  e.  A  w  <_  z )
40 breq2 4022 . . . . . . . . . . . . 13  |-  ( z  =  w  ->  (
y  <  z  <->  y  <  w ) )
4140cbvrexv 2719 . . . . . . . . . . . 12  |-  ( E. z  e.  A  y  <  z  <->  E. w  e.  A  y  <  w )
4241imbi2i 226 . . . . . . . . . . 11  |-  ( ( y  <  x  ->  E. z  e.  A  y  <  z )  <->  ( y  <  x  ->  E. w  e.  A  y  <  w ) )
4342ralbii 2496 . . . . . . . . . 10  |-  ( A. y  e.  RR  (
y  <  x  ->  E. z  e.  A  y  <  z )  <->  A. y  e.  RR  ( y  < 
x  ->  E. w  e.  A  y  <  w ) )
4443anbi2i 457 . . . . . . . . 9  |-  ( ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  A  y  <  z ) )  <-> 
( A. y  e.  A  -.  x  < 
y  /\  A. y  e.  RR  ( y  < 
x  ->  E. w  e.  A  y  <  w ) ) )
4544rexbii 2497 . . . . . . . 8  |-  ( E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  A  y  <  z ) )  <->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. w  e.  A  y  <  w ) ) )
463, 45sylib 122 . . . . . . 7  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. w  e.  A  y  <  w ) ) )
4746adantr 276 . . . . . 6  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  ( y  < 
x  ->  E. w  e.  A  y  <  w ) ) )
4813, 7sstrdi 3182 . . . . . 6  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  A  C_  RR )
4947, 48, 21suprleubex 8942 . . . . 5  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  ( sup ( A ,  RR ,  <  )  <_  z  <->  A. w  e.  A  w  <_  z ) )
5039, 49mpbird 167 . . . 4  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  <_  z )
5147, 48, 18suprubex 8939 . . . 4  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  z  <_  sup ( A ,  RR ,  <  ) )
5216, 21letri3d 8104 . . . 4  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  ( sup ( A ,  RR ,  <  )  =  z  <->  ( sup ( A ,  RR ,  <  )  <_  z  /\  z  <_  sup ( A ,  RR ,  <  ) ) ) )
5350, 51, 52mpbir2and 946 . . 3  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  =  z )
5453, 18eqeltrd 2266 . 2  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  e.  A )
5512, 54rexlimddv 2612 1  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   A.wral 2468   E.wrex 2469    C_ wss 3144   class class class wbr 4018  (class class class)co 5897   supcsup 7012   RRcr 7841   1c1 7843    + caddc 7845    < clt 8023    <_ cle 8024    - cmin 8159   ZZcz 9284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-0id 7950  ax-rnegex 7951  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-sup 7014  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-inn 8951  df-n0 9208  df-z 9285
This theorem is referenced by:  infssuzcldc  11987  gcddvds  11999
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