Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  uzfissfz Structured version   Visualization version   Unicode version

Theorem uzfissfz 37636
Description: For any finite subset of the upper integers, there is a finite set of sequential integers that includes it. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
uzfissfz.m  |-  ( ph  ->  M  e.  ZZ )
uzfissfz.z  |-  Z  =  ( ZZ>= `  M )
uzfissfz.a  |-  ( ph  ->  A  C_  Z )
uzfissfz.fi  |-  ( ph  ->  A  e.  Fin )
Assertion
Ref Expression
uzfissfz  |-  ( ph  ->  E. k  e.  Z  A  C_  ( M ... k ) )
Distinct variable groups:    A, k    k, M    k, Z
Allowed substitution hint:    ph( k)

Proof of Theorem uzfissfz
Dummy variables  j  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uzfissfz.m . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
2 uzid 11197 . . . . . 6  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
31, 2syl 17 . . . . 5  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
4 uzfissfz.z . . . . . . 7  |-  Z  =  ( ZZ>= `  M )
54a1i 11 . . . . . 6  |-  ( ph  ->  Z  =  ( ZZ>= `  M ) )
65eqcomd 2477 . . . . 5  |-  ( ph  ->  ( ZZ>= `  M )  =  Z )
73, 6eleqtrd 2551 . . . 4  |-  ( ph  ->  M  e.  Z )
87adantr 472 . . 3  |-  ( (
ph  /\  A  =  (/) )  ->  M  e.  Z )
9 id 22 . . . . 5  |-  ( A  =  (/)  ->  A  =  (/) )
10 0ss 3766 . . . . . 6  |-  (/)  C_  ( M ... M )
1110a1i 11 . . . . 5  |-  ( A  =  (/)  ->  (/)  C_  ( M ... M ) )
129, 11eqsstrd 3452 . . . 4  |-  ( A  =  (/)  ->  A  C_  ( M ... M ) )
1312adantl 473 . . 3  |-  ( (
ph  /\  A  =  (/) )  ->  A  C_  ( M ... M ) )
14 oveq2 6316 . . . . 5  |-  ( k  =  M  ->  ( M ... k )  =  ( M ... M
) )
1514sseq2d 3446 . . . 4  |-  ( k  =  M  ->  ( A  C_  ( M ... k )  <->  A  C_  ( M ... M ) ) )
1615rspcev 3136 . . 3  |-  ( ( M  e.  Z  /\  A  C_  ( M ... M ) )  ->  E. k  e.  Z  A  C_  ( M ... k ) )
178, 13, 16syl2anc 673 . 2  |-  ( (
ph  /\  A  =  (/) )  ->  E. k  e.  Z  A  C_  ( M ... k ) )
18 uzfissfz.a . . . . 5  |-  ( ph  ->  A  C_  Z )
1918adantr 472 . . . 4  |-  ( (
ph  /\  -.  A  =  (/) )  ->  A  C_  Z )
20 uzssz 11202 . . . . . . . . 9  |-  ( ZZ>= `  M )  C_  ZZ
214, 20eqsstri 3448 . . . . . . . 8  |-  Z  C_  ZZ
2221a1i 11 . . . . . . 7  |-  ( ph  ->  Z  C_  ZZ )
2318, 22sstrd 3428 . . . . . 6  |-  ( ph  ->  A  C_  ZZ )
2423adantr 472 . . . . 5  |-  ( (
ph  /\  -.  A  =  (/) )  ->  A  C_  ZZ )
259necon3bi 2669 . . . . . 6  |-  ( -.  A  =  (/)  ->  A  =/=  (/) )
2625adantl 473 . . . . 5  |-  ( (
ph  /\  -.  A  =  (/) )  ->  A  =/=  (/) )
27 uzfissfz.fi . . . . . 6  |-  ( ph  ->  A  e.  Fin )
2827adantr 472 . . . . 5  |-  ( (
ph  /\  -.  A  =  (/) )  ->  A  e.  Fin )
29 suprfinzcl 11073 . . . . 5  |-  ( ( A  C_  ZZ  /\  A  =/=  (/)  /\  A  e. 
Fin )  ->  sup ( A ,  RR ,  <  )  e.  A )
3024, 26, 28, 29syl3anc 1292 . . . 4  |-  ( (
ph  /\  -.  A  =  (/) )  ->  sup ( A ,  RR ,  <  )  e.  A )
3119, 30sseldd 3419 . . 3  |-  ( (
ph  /\  -.  A  =  (/) )  ->  sup ( A ,  RR ,  <  )  e.  Z )
321ad2antrr 740 . . . . . . . 8  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  M  e.  ZZ )
3321, 31sseldi 3416 . . . . . . . . 9  |-  ( (
ph  /\  -.  A  =  (/) )  ->  sup ( A ,  RR ,  <  )  e.  ZZ )
3433adantr 472 . . . . . . . 8  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  sup ( A ,  RR ,  <  )  e.  ZZ )
3524sselda 3418 . . . . . . . 8  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  j  e.  ZZ )
3632, 34, 353jca 1210 . . . . . . 7  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  ( M  e.  ZZ  /\ 
sup ( A ,  RR ,  <  )  e.  ZZ  /\  j  e.  ZZ ) )
3718sselda 3418 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  A )  ->  j  e.  Z )
384a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  A )  ->  Z  =  ( ZZ>= `  M
) )
3937, 38eleqtrd 2551 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  A )  ->  j  e.  ( ZZ>= `  M )
)
40 eluzle 11195 . . . . . . . . 9  |-  ( j  e.  ( ZZ>= `  M
)  ->  M  <_  j )
4139, 40syl 17 . . . . . . . 8  |-  ( (
ph  /\  j  e.  A )  ->  M  <_  j )
4241adantlr 729 . . . . . . 7  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  M  <_  j )
43 zssre 10968 . . . . . . . . . 10  |-  ZZ  C_  RR
4423, 43syl6ss 3430 . . . . . . . . 9  |-  ( ph  ->  A  C_  RR )
4544ad2antrr 740 . . . . . . . 8  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  A  C_  RR )
4626adantr 472 . . . . . . . 8  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  A  =/=  (/) )
47 fimaxre2 10574 . . . . . . . . . 10  |-  ( ( A  C_  RR  /\  A  e.  Fin )  ->  E. x  e.  RR  A. y  e.  A  y  <_  x
)
4844, 27, 47syl2anc 673 . . . . . . . . 9  |-  ( ph  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )
4948ad2antrr 740 . . . . . . . 8  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  E. x  e.  RR  A. y  e.  A  y  <_  x )
50 simpr 468 . . . . . . . 8  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  j  e.  A )
51 suprub 10592 . . . . . . . 8  |-  ( ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  /\  j  e.  A )  ->  j  <_  sup ( A ,  RR ,  <  ) )
5245, 46, 49, 50, 51syl31anc 1295 . . . . . . 7  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  j  <_  sup ( A ,  RR ,  <  ) )
5336, 42, 52jca32 544 . . . . . 6  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  ( ( M  e.  ZZ  /\  sup ( A ,  RR ,  <  )  e.  ZZ  /\  j  e.  ZZ )  /\  ( M  <_  j  /\  j  <_  sup ( A ,  RR ,  <  ) ) ) )
54 elfz2 11817 . . . . . 6  |-  ( j  e.  ( M ... sup ( A ,  RR ,  <  ) )  <->  ( ( M  e.  ZZ  /\  sup ( A ,  RR ,  <  )  e.  ZZ  /\  j  e.  ZZ )  /\  ( M  <_  j  /\  j  <_  sup ( A ,  RR ,  <  ) ) ) )
5553, 54sylibr 217 . . . . 5  |-  ( ( ( ph  /\  -.  A  =  (/) )  /\  j  e.  A )  ->  j  e.  ( M ... sup ( A ,  RR ,  <  ) ) )
5655ralrimiva 2809 . . . 4  |-  ( (
ph  /\  -.  A  =  (/) )  ->  A. j  e.  A  j  e.  ( M ... sup ( A ,  RR ,  <  ) ) )
57 dfss3 3408 . . . 4  |-  ( A 
C_  ( M ... sup ( A ,  RR ,  <  ) )  <->  A. j  e.  A  j  e.  ( M ... sup ( A ,  RR ,  <  ) ) )
5856, 57sylibr 217 . . 3  |-  ( (
ph  /\  -.  A  =  (/) )  ->  A  C_  ( M ... sup ( A ,  RR ,  <  ) ) )
59 oveq2 6316 . . . . 5  |-  ( k  =  sup ( A ,  RR ,  <  )  ->  ( M ... k )  =  ( M ... sup ( A ,  RR ,  <  ) ) )
6059sseq2d 3446 . . . 4  |-  ( k  =  sup ( A ,  RR ,  <  )  ->  ( A  C_  ( M ... k )  <-> 
A  C_  ( M ... sup ( A ,  RR ,  <  ) ) ) )
6160rspcev 3136 . . 3  |-  ( ( sup ( A ,  RR ,  <  )  e.  Z  /\  A  C_  ( M ... sup ( A ,  RR ,  <  ) ) )  ->  E. k  e.  Z  A  C_  ( M ... k ) )
6231, 58, 61syl2anc 673 . 2  |-  ( (
ph  /\  -.  A  =  (/) )  ->  E. k  e.  Z  A  C_  ( M ... k ) )
6317, 62pm2.61dan 808 1  |-  ( ph  ->  E. k  e.  Z  A  C_  ( M ... k ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757    C_ wss 3390   (/)c0 3722   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Fincfn 7587   supcsup 7972   RRcr 9556    < clt 9693    <_ cle 9694   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811
This theorem is referenced by:  sge0uzfsumgt  38400  sge0seq  38402  carageniuncllem2  38462  caratheodorylem2  38467
  Copyright terms: Public domain W3C validator