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Theorem uzinfmi 11161
Description: Extract the lower bound of an upper set of integers as its infimum. Note that the " `'  < " argument turns supremum into infimum (for which we do not currently have a separate notation). (Contributed by NM, 7-Oct-2005.)
Hypothesis
Ref Expression
uzinfm.1  |-  M  e.  ZZ
Assertion
Ref Expression
uzinfmi  |-  sup (
( ZZ>= `  M ) ,  RR ,  `'  <  )  =  M

Proof of Theorem uzinfmi
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uzssz 11101 . . . 4  |-  ( ZZ>= `  M )  C_  ZZ
2 zssre 10871 . . . 4  |-  ZZ  C_  RR
31, 2sstri 3513 . . 3  |-  ( ZZ>= `  M )  C_  RR
4 uzinfm.1 . . . . 5  |-  M  e.  ZZ
5 uzid 11096 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
64, 5ax-mp 5 . . . 4  |-  M  e.  ( ZZ>= `  M )
7 eluzle 11094 . . . . 5  |-  ( k  e.  ( ZZ>= `  M
)  ->  M  <_  k )
87rgen 2824 . . . 4  |-  A. k  e.  ( ZZ>= `  M ) M  <_  k
9 breq1 4450 . . . . . 6  |-  ( j  =  M  ->  (
j  <_  k  <->  M  <_  k ) )
109ralbidv 2903 . . . . 5  |-  ( j  =  M  ->  ( A. k  e.  ( ZZ>=
`  M ) j  <_  k  <->  A. k  e.  ( ZZ>= `  M ) M  <_  k ) )
1110rspcev 3214 . . . 4  |-  ( ( M  e.  ( ZZ>= `  M )  /\  A. k  e.  ( ZZ>= `  M ) M  <_ 
k )  ->  E. j  e.  ( ZZ>= `  M ) A. k  e.  ( ZZ>=
`  M ) j  <_  k )
126, 8, 11mp2an 672 . . 3  |-  E. j  e.  ( ZZ>= `  M ) A. k  e.  ( ZZ>=
`  M ) j  <_  k
13 lbinfm 10496 . . 3  |-  ( ( ( ZZ>= `  M )  C_  RR  /\  E. j  e.  ( ZZ>= `  M ) A. k  e.  ( ZZ>=
`  M ) j  <_  k )  ->  sup ( ( ZZ>= `  M
) ,  RR ,  `'  <  )  =  (
iota_ j  e.  ( ZZ>=
`  M ) A. k  e.  ( ZZ>= `  M ) j  <_ 
k ) )
143, 12, 13mp2an 672 . 2  |-  sup (
( ZZ>= `  M ) ,  RR ,  `'  <  )  =  ( iota_ j  e.  ( ZZ>= `  M ) A. k  e.  ( ZZ>=
`  M ) j  <_  k )
15 lbreu 10493 . . . . 5  |-  ( ( ( ZZ>= `  M )  C_  RR  /\  E. j  e.  ( ZZ>= `  M ) A. k  e.  ( ZZ>=
`  M ) j  <_  k )  ->  E! j  e.  ( ZZ>=
`  M ) A. k  e.  ( ZZ>= `  M ) j  <_ 
k )
163, 12, 15mp2an 672 . . . 4  |-  E! j  e.  ( ZZ>= `  M
) A. k  e.  ( ZZ>= `  M )
j  <_  k
1710riota2 6268 . . . 4  |-  ( ( M  e.  ( ZZ>= `  M )  /\  E! j  e.  ( ZZ>= `  M ) A. k  e.  ( ZZ>= `  M )
j  <_  k )  ->  ( A. k  e.  ( ZZ>= `  M ) M  <_  k  <->  ( iota_ j  e.  ( ZZ>= `  M
) A. k  e.  ( ZZ>= `  M )
j  <_  k )  =  M ) )
186, 16, 17mp2an 672 . . 3  |-  ( A. k  e.  ( ZZ>= `  M ) M  <_ 
k  <->  ( iota_ j  e.  ( ZZ>= `  M ) A. k  e.  ( ZZ>=
`  M ) j  <_  k )  =  M )
198, 18mpbi 208 . 2  |-  ( iota_ j  e.  ( ZZ>= `  M
) A. k  e.  ( ZZ>= `  M )
j  <_  k )  =  M
2014, 19eqtri 2496 1  |-  sup (
( ZZ>= `  M ) ,  RR ,  `'  <  )  =  M
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   E!wreu 2816    C_ wss 3476   class class class wbr 4447   `'ccnv 4998   ` cfv 5588   iota_crio 6244   supcsup 7900   RRcr 9491    < clt 9628    <_ cle 9629   ZZcz 10864   ZZ>=cuz 11082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-pre-lttri 9566  ax-pre-lttrn 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-neg 9808  df-z 10865  df-uz 11083
This theorem is referenced by:  nninfm  11162  nn0infm  11163
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