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Theorem suprzclex 9303
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 7992 . . . . . 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 6973 . . . 4  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
54ltm1d 8841 . . 3  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  -  1 )  <  sup ( A ,  RR ,  <  ) )
6 suprzclex.ss . . . . 5  |-  ( ph  ->  A  C_  ZZ )
7 zssre 9212 . . . . 5  |-  ZZ  C_  RR
86, 7sstrdi 3159 . . . 4  |-  ( ph  ->  A  C_  RR )
9 peano2rem 8179 . . . . 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 8861 . . 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 3147 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  e.  ZZ )
157, 14sselid 3145 . . . . . . . 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 526 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  z  e.  A )
1913, 18sseldd 3148 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  z  e.  ZZ )
20 zre 9209 . . . . . . . . . . 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 8048 . . . . . . . . . 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 485 . . . . . . . . 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 485 . . . . . . . . 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 8860 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  <_  sup ( A ,  RR ,  <  ) )
29 simprr 527 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  ( sup ( A ,  RR ,  <  )  -  1 )  <  z )
30 1red 7928 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  1  e.  RR )
3116, 30, 21ltsubaddd 8453 . . . . . . . . . 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 8037 . . . . . . 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 9260 . . . . . . . 8  |-  ( ( w  e.  ZZ  /\  z  e.  ZZ )  ->  ( w  <_  z  <->  w  <  ( z  +  1 ) ) )
3714, 35, 36syl2anc 409 . . . . . . 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 2543 . . . . 5  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  A. w  e.  A  w  <_  z )
40 breq2 3991 . . . . . . . . . . . . 13  |-  ( z  =  w  ->  (
y  <  z  <->  y  <  w ) )
4140cbvrexv 2697 . . . . . . . . . . . 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 2476 . . . . . . . . . 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 454 . . . . . . . . 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 2477 . . . . . . . 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 3159 . . . . . 6  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  A  C_  RR )
4947, 48, 21suprleubex 8863 . . . . 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 8860 . . . 4  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  z  <_  sup ( A ,  RR ,  <  ) )
5216, 21letri3d 8028 . . . 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 939 . . 3  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  =  z )
5453, 18eqeltrd 2247 . 2  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  e.  A )
5512, 54rexlimddv 2592 1  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   A.wral 2448   E.wrex 2449    C_ wss 3121   class class class wbr 3987  (class class class)co 5851   supcsup 6957   RRcr 7766   1c1 7768    + caddc 7770    < clt 7947    <_ cle 7948    - cmin 8083   ZZcz 9205
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-addcom 7867  ax-addass 7869  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-0id 7875  ax-rnegex 7876  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-apti 7882  ax-pre-ltadd 7883
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-id 4276  df-po 4279  df-iso 4280  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-sup 6959  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086  df-inn 8872  df-n0 9129  df-z 9206
This theorem is referenced by:  infssuzcldc  11899  gcddvds  11911
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