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Theorem uzinfmiOLD 11239
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.) Obsolete version of uzinfi 11238 as of 4-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
uzinfi.1  |-  M  e.  ZZ
Assertion
Ref Expression
uzinfmiOLD  |-  sup (
( ZZ>= `  M ) ,  RR ,  `'  <  )  =  M

Proof of Theorem uzinfmiOLD
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uzssz 11178 . . . 4  |-  ( ZZ>= `  M )  C_  ZZ
2 zssre 10944 . . . 4  |-  ZZ  C_  RR
31, 2sstri 3441 . . 3  |-  ( ZZ>= `  M )  C_  RR
4 uzinfi.1 . . . . 5  |-  M  e.  ZZ
5 uzid 11173 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
64, 5ax-mp 5 . . . 4  |-  M  e.  ( ZZ>= `  M )
7 eluzle 11171 . . . . 5  |-  ( k  e.  ( ZZ>= `  M
)  ->  M  <_  k )
87rgen 2747 . . . 4  |-  A. k  e.  ( ZZ>= `  M ) M  <_  k
9 breq1 4405 . . . . . 6  |-  ( j  =  M  ->  (
j  <_  k  <->  M  <_  k ) )
109ralbidv 2827 . . . . 5  |-  ( j  =  M  ->  ( A. k  e.  ( ZZ>=
`  M ) j  <_  k  <->  A. k  e.  ( ZZ>= `  M ) M  <_  k ) )
1110rspcev 3150 . . . 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 678 . . 3  |-  E. j  e.  ( ZZ>= `  M ) A. k  e.  ( ZZ>=
`  M ) j  <_  k
13 lbinfmOLD 10560 . . 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 678 . 2  |-  sup (
( ZZ>= `  M ) ,  RR ,  `'  <  )  =  ( iota_ j  e.  ( ZZ>= `  M ) A. k  e.  ( ZZ>=
`  M ) j  <_  k )
15 lbreu 10556 . . . . 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 678 . . . 4  |-  E! j  e.  ( ZZ>= `  M
) A. k  e.  ( ZZ>= `  M )
j  <_  k
1710riota2 6274 . . . 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 678 . . 3  |-  ( A. k  e.  ( ZZ>= `  M ) M  <_ 
k  <->  ( iota_ j  e.  ( ZZ>= `  M ) A. k  e.  ( ZZ>=
`  M ) j  <_  k )  =  M )
198, 18mpbi 212 . 2  |-  ( iota_ j  e.  ( ZZ>= `  M
) A. k  e.  ( ZZ>= `  M )
j  <_  k )  =  M
2014, 19eqtri 2473 1  |-  sup (
( ZZ>= `  M ) ,  RR ,  `'  <  )  =  M
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    = wceq 1444    e. wcel 1887   A.wral 2737   E.wrex 2738   E!wreu 2739    C_ wss 3404   class class class wbr 4402   `'ccnv 4833   ` cfv 5582   iota_crio 6251   supcsup 7954   RRcr 9538    < clt 9675    <_ cle 9676   ZZcz 10937   ZZ>=cuz 11159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-pre-lttri 9613  ax-pre-lttrn 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-neg 9863  df-z 10938  df-uz 11160
This theorem is referenced by:  nninfmOLD  11242  nn0infmOLD  11243
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