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Theorem suprzclex 9117
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 7812 . . . . . 6  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
21adantl 275 . . . . 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 6853 . . . 4  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
54ltm1d 8658 . . 3  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  -  1 )  <  sup ( A ,  RR ,  <  ) )
6 suprzclex.ss . . . . 5  |-  ( ph  ->  A  C_  ZZ )
7 zssre 9029 . . . . 5  |-  ZZ  C_  RR
86, 7sstrdi 3079 . . . 4  |-  ( ph  ->  A  C_  RR )
9 peano2rem 7997 . . . . 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 8678 . . 3  |-  ( ph  ->  ( ( sup ( A ,  RR ,  <  )  -  1 )  <  sup ( A ,  RR ,  <  )  <->  E. z  e.  A  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )
125, 11mpbid 146 . 2  |-  ( ph  ->  E. z  e.  A  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
)
136adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  A  C_  ZZ )
1413sselda 3067 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  e.  ZZ )
157, 14sseldi 3065 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  e.  RR )
164adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  e.  RR )
1716adantr 274 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  sup ( A ,  RR ,  <  )  e.  RR )
18 simprl 505 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  z  e.  A )
1913, 18sseldd 3068 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  z  e.  ZZ )
20 zre 9026 . . . . . . . . . . 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 7866 . . . . . . . . . 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 274 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  (
z  +  1 )  e.  RR )
253ad2antrr 479 . . . . . . . . 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 479 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  A  C_  RR )
27 simpr 109 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  e.  A )
2825, 26, 27suprubex 8677 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  <_  sup ( A ,  RR ,  <  ) )
29 simprr 506 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  ( sup ( A ,  RR ,  <  )  -  1 )  <  z )
30 1red 7749 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  1  e.  RR )
3116, 30, 21ltsubaddd 8271 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  ( ( sup ( A ,  RR ,  <  )  -  1 )  <  z  <->  sup ( A ,  RR ,  <  )  <  ( z  +  1 ) ) )
3229, 31mpbid 146 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  <  ( z  +  1 ) )
3332adantr 274 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  sup ( A ,  RR ,  <  )  <  ( z  +  1 ) )
3415, 17, 24, 28, 33lelttrd 7855 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  <  ( z  +  1 ) )
3519adantr 274 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  z  e.  ZZ )
36 zleltp1 9077 . . . . . . . 8  |-  ( ( w  e.  ZZ  /\  z  e.  ZZ )  ->  ( w  <_  z  <->  w  <  ( z  +  1 ) ) )
3714, 35, 36syl2anc 408 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  (
w  <_  z  <->  w  <  ( z  +  1 ) ) )
3834, 37mpbird 166 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  <_  z )
3938ralrimiva 2482 . . . . 5  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  A. w  e.  A  w  <_  z )
40 breq2 3903 . . . . . . . . . . . . 13  |-  ( z  =  w  ->  (
y  <  z  <->  y  <  w ) )
4140cbvrexv 2632 . . . . . . . . . . . 12  |-  ( E. z  e.  A  y  <  z  <->  E. w  e.  A  y  <  w )
4241imbi2i 225 . . . . . . . . . . 11  |-  ( ( y  <  x  ->  E. z  e.  A  y  <  z )  <->  ( y  <  x  ->  E. w  e.  A  y  <  w ) )
4342ralbii 2418 . . . . . . . . . 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 452 . . . . . . . . 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 2419 . . . . . . . 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 121 . . . . . . 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 274 . . . . . 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 3079 . . . . . 6  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  A  C_  RR )
4947, 48, 21suprleubex 8680 . . . . 5  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  ( sup ( A ,  RR ,  <  )  <_  z  <->  A. w  e.  A  w  <_  z ) )
5039, 49mpbird 166 . . . 4  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  <_  z )
5147, 48, 18suprubex 8677 . . . 4  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  z  <_  sup ( A ,  RR ,  <  ) )
5216, 21letri3d 7847 . . . 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 913 . . 3  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  =  z )
5453, 18eqeltrd 2194 . 2  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  e.  A )
5512, 54rexlimddv 2531 1  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   A.wral 2393   E.wrex 2394    C_ wss 3041   class class class wbr 3899  (class class class)co 5742   supcsup 6837   RRcr 7587   1c1 7589    + caddc 7591    < clt 7768    <_ cle 7769    - cmin 7901   ZZcz 9022
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-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-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-br 3900  df-opab 3960  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-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-sup 6839  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
This theorem is referenced by:  infssuzcldc  11571  gcddvds  11579
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