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Theorem suprzclex 8754
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 7486 . . . . . 6  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
21adantl 271 . . . . 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 6614 . . . 4  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
54ltm1d 8305 . . 3  |-  ( ph  ->  ( sup ( A ,  RR ,  <  )  -  1 )  <  sup ( A ,  RR ,  <  ) )
6 suprzclex.ss . . . . 5  |-  ( ph  ->  A  C_  ZZ )
7 zssre 8667 . . . . 5  |-  ZZ  C_  RR
86, 7syl6ss 3024 . . . 4  |-  ( ph  ->  A  C_  RR )
9 peano2rem 7670 . . . . 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 8325 . . 3  |-  ( ph  ->  ( ( sup ( A ,  RR ,  <  )  -  1 )  <  sup ( A ,  RR ,  <  )  <->  E. z  e.  A  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )
125, 11mpbid 145 . 2  |-  ( ph  ->  E. z  e.  A  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
)
136adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  A  C_  ZZ )
1413sselda 3012 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  e.  ZZ )
157, 14sseldi 3010 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  e.  RR )
164adantr 270 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  e.  RR )
1716adantr 270 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  sup ( A ,  RR ,  <  )  e.  RR )
18 simprl 498 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  z  e.  A )
1913, 18sseldd 3013 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  z  e.  ZZ )
20 zre 8664 . . . . . . . . . . 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 7539 . . . . . . . . . 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 270 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  (
z  +  1 )  e.  RR )
253ad2antrr 472 . . . . . . . . 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 472 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  A  C_  RR )
27 simpr 108 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  e.  A )
2825, 26, 27suprubex 8324 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  <_  sup ( A ,  RR ,  <  ) )
29 simprr 499 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  ( sup ( A ,  RR ,  <  )  -  1 )  <  z )
30 1red 7424 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  1  e.  RR )
3116, 30, 21ltsubaddd 7936 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  ( ( sup ( A ,  RR ,  <  )  -  1 )  <  z  <->  sup ( A ,  RR ,  <  )  <  ( z  +  1 ) ) )
3229, 31mpbid 145 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  <  ( z  +  1 ) )
3332adantr 270 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  sup ( A ,  RR ,  <  )  <  ( z  +  1 ) )
3415, 17, 24, 28, 33lelttrd 7529 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  <  ( z  +  1 ) )
3519adantr 270 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  z  e.  ZZ )
36 zleltp1 8715 . . . . . . . 8  |-  ( ( w  e.  ZZ  /\  z  e.  ZZ )  ->  ( w  <_  z  <->  w  <  ( z  +  1 ) ) )
3714, 35, 36syl2anc 403 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  (
w  <_  z  <->  w  <  ( z  +  1 ) ) )
3834, 37mpbird 165 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  A  /\  ( sup ( A ,  RR ,  <  )  - 
1 )  <  z
) )  /\  w  e.  A )  ->  w  <_  z )
3938ralrimiva 2442 . . . . 5  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  A. w  e.  A  w  <_  z )
40 breq2 3818 . . . . . . . . . . . . 13  |-  ( z  =  w  ->  (
y  <  z  <->  y  <  w ) )
4140cbvrexv 2586 . . . . . . . . . . . 12  |-  ( E. z  e.  A  y  <  z  <->  E. w  e.  A  y  <  w )
4241imbi2i 224 . . . . . . . . . . 11  |-  ( ( y  <  x  ->  E. z  e.  A  y  <  z )  <->  ( y  <  x  ->  E. w  e.  A  y  <  w ) )
4342ralbii 2380 . . . . . . . . . 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 445 . . . . . . . . 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 2381 . . . . . . . 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 120 . . . . . . 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 270 . . . . . 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, 7syl6ss 3024 . . . . . 6  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  A  C_  RR )
4947, 48, 21suprleubex 8327 . . . . 5  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  ( sup ( A ,  RR ,  <  )  <_  z  <->  A. w  e.  A  w  <_  z ) )
5039, 49mpbird 165 . . . 4  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  <_  z )
5147, 48, 18suprubex 8324 . . . 4  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  z  <_  sup ( A ,  RR ,  <  ) )
5216, 21letri3d 7521 . . . 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 888 . . 3  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  =  z )
5453, 18eqeltrd 2161 . 2  |-  ( (
ph  /\  ( z  e.  A  /\  ( sup ( A ,  RR ,  <  )  -  1 )  <  z ) )  ->  sup ( A ,  RR ,  <  )  e.  A )
5512, 54rexlimddv 2489 1  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  A )
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 2986   class class class wbr 3814  (class class class)co 5594   supcsup 6598   RRcr 7270   1c1 7272    + caddc 7274    < clt 7443    <_ cle 7444    - cmin 7574   ZZcz 8660
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 3925  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-addcom 7366  ax-addass 7368  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-0id 7374  ax-rnegex 7375  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-apti 7381  ax-pre-ltadd 7382
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 2616  df-sbc 2829  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-br 3815  df-opab 3869  df-id 4087  df-po 4090  df-iso 4091  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-iota 4937  df-fun 4974  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-sup 6600  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-inn 8335  df-n0 8584  df-z 8661
This theorem is referenced by:  infssuzcldc  10741  gcddvds  10749
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