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Theorem infmxrgelb 11578
Description: The infimum of a set of extended reals is greater than or equal to a lower bound. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by Mario Carneiro, 6-Sep-2014.)
Assertion
Ref Expression
infmxrgelb  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B  <_  sup ( A ,  RR* ,  `'  <  )  <->  A. x  e.  A  B  <_  x ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem infmxrgelb
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xrltso 11399 . . . . . . . 8  |-  <  Or  RR*
2 cnvso 5362 . . . . . . . 8  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
31, 2mpbi 208 . . . . . . 7  |-  `'  <  Or 
RR*
43a1i 11 . . . . . 6  |-  ( A 
C_  RR*  ->  `'  <  Or 
RR* )
5 xrinfmss2 11554 . . . . . 6  |-  ( A 
C_  RR*  ->  E. y  e.  RR*  ( A. z  e.  A  -.  y `'  <  z  /\  A. z  e.  RR*  ( z `'  <  y  ->  E. x  e.  A  z `'  <  x ) ) )
6 id 22 . . . . . 6  |-  ( A 
C_  RR*  ->  A  C_  RR* )
74, 5, 6suplub2 7953 . . . . 5  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B `'  <  sup ( A ,  RR* ,  `'  <  )  <->  E. x  e.  A  B `'  <  x ) )
8 simpr 459 . . . . . 6  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
93supex 7955 . . . . . 6  |-  sup ( A ,  RR* ,  `'  <  )  e.  _V
10 brcnvg 5003 . . . . . 6  |-  ( ( B  e.  RR*  /\  sup ( A ,  RR* ,  `'  <  )  e.  _V )  ->  ( B `'  <  sup ( A ,  RR* ,  `'  <  )  <->  sup ( A ,  RR* ,  `'  <  )  <  B ) )
118, 9, 10sylancl 660 . . . . 5  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B `'  <  sup ( A ,  RR* ,  `'  <  )  <->  sup ( A ,  RR* ,  `'  <  )  <  B ) )
12 vex 3061 . . . . . . 7  |-  x  e. 
_V
13 brcnvg 5003 . . . . . . 7  |-  ( ( B  e.  RR*  /\  x  e.  _V )  ->  ( B `'  <  x  <->  x  <  B ) )
148, 12, 13sylancl 660 . . . . . 6  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B `'  <  x  <->  x  <  B ) )
1514rexbidv 2917 . . . . 5  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  A  B `'  <  x  <->  E. x  e.  A  x  <  B ) )
167, 11, 153bitr3d 283 . . . 4  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( sup ( A ,  RR* ,  `'  <  )  <  B  <->  E. x  e.  A  x  <  B ) )
1716notbid 292 . . 3  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( -.  sup ( A ,  RR* ,  `'  <  )  <  B  <->  -.  E. x  e.  A  x  <  B ) )
18 ralnex 2849 . . 3  |-  ( A. x  e.  A  -.  x  <  B  <->  -.  E. x  e.  A  x  <  B )
1917, 18syl6bbr 263 . 2  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( -.  sup ( A ,  RR* ,  `'  <  )  <  B  <->  A. x  e.  A  -.  x  <  B ) )
20 id 22 . . 3  |-  ( B  e.  RR*  ->  B  e. 
RR* )
21 infmxrcl 11560 . . 3  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  `'  <  )  e.  RR* )
22 xrlenlt 9681 . . 3  |-  ( ( B  e.  RR*  /\  sup ( A ,  RR* ,  `'  <  )  e.  RR* )  ->  ( B  <_  sup ( A ,  RR* ,  `'  <  )  <->  -.  sup ( A ,  RR* ,  `'  <  )  <  B ) )
2320, 21, 22syl2anr 476 . 2  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B  <_  sup ( A ,  RR* ,  `'  <  )  <->  -. 
sup ( A ,  RR* ,  `'  <  )  <  B ) )
24 simplr 754 . . . 4  |-  ( ( ( A  C_  RR*  /\  B  e.  RR* )  /\  x  e.  A )  ->  B  e.  RR* )
25 simpl 455 . . . . 5  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  A  C_ 
RR* )
2625sselda 3441 . . . 4  |-  ( ( ( A  C_  RR*  /\  B  e.  RR* )  /\  x  e.  A )  ->  x  e.  RR* )
27 xrlenlt 9681 . . . 4  |-  ( ( B  e.  RR*  /\  x  e.  RR* )  ->  ( B  <_  x  <->  -.  x  <  B ) )
2824, 26, 27syl2anc 659 . . 3  |-  ( ( ( A  C_  RR*  /\  B  e.  RR* )  /\  x  e.  A )  ->  ( B  <_  x  <->  -.  x  <  B ) )
2928ralbidva 2839 . 2  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( A. x  e.  A  B  <_  x  <->  A. x  e.  A  -.  x  <  B ) )
3019, 23, 293bitr4d 285 1  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B  <_  sup ( A ,  RR* ,  `'  <  )  <->  A. x  e.  A  B  <_  x ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1842   A.wral 2753   E.wrex 2754   _Vcvv 3058    C_ wss 3413   class class class wbr 4394    Or wor 4742   `'ccnv 4821   supcsup 7933   RR*cxr 9656    < clt 9657    <_ cle 9658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-sup 7934  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843
This theorem is referenced by:  infmxrre  11579  ixxlb  11603  limsuple  13448  limsupval2  13450  imasdsf1olem  21166  nmogelb  21513  metdsf  21642  metdsge  21643  ovolgelb  22181  ovolge0  22182  ovolsslem  22185  ovolicc2  22223  ismblfin  31407  infmxrss  36842
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