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Theorem infmxrgelb 10669
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 10491 . . . . . . . 8  |-  <  Or  RR*
2 cnvso 5230 . . . . . . . 8  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
31, 2mpbi 199 . . . . . . 7  |-  `'  <  Or 
RR*
43a1i 10 . . . . . 6  |-  ( A 
C_  RR*  ->  `'  <  Or 
RR* )
5 xrinfmss2 10645 . . . . . 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 19 . . . . . 6  |-  ( A 
C_  RR*  ->  A  C_  RR* )
74, 5, 6suplub2 7228 . . . . 5  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B `'  <  sup ( A ,  RR* ,  `'  <  )  <->  E. x  e.  A  B `'  <  x ) )
8 simpr 447 . . . . . 6  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
93supex 7230 . . . . . 6  |-  sup ( A ,  RR* ,  `'  <  )  e.  _V
10 brcnvg 4878 . . . . . 6  |-  ( ( B  e.  RR*  /\  sup ( A ,  RR* ,  `'  <  )  e.  _V )  ->  ( B `'  <  sup ( A ,  RR* ,  `'  <  )  <->  sup ( A ,  RR* ,  `'  <  )  <  B ) )
118, 9, 10sylancl 643 . . . . 5  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B `'  <  sup ( A ,  RR* ,  `'  <  )  <->  sup ( A ,  RR* ,  `'  <  )  <  B ) )
12 vex 2804 . . . . . . 7  |-  x  e. 
_V
13 brcnvg 4878 . . . . . . 7  |-  ( ( B  e.  RR*  /\  x  e.  _V )  ->  ( B `'  <  x  <->  x  <  B ) )
148, 12, 13sylancl 643 . . . . . 6  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B `'  <  x  <->  x  <  B ) )
1514rexbidv 2577 . . . . 5  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  A  B `'  <  x  <->  E. x  e.  A  x  <  B ) )
167, 11, 153bitr3d 274 . . . 4  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( sup ( A ,  RR* ,  `'  <  )  <  B  <->  E. x  e.  A  x  <  B ) )
1716notbid 285 . . 3  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( -.  sup ( A ,  RR* ,  `'  <  )  <  B  <->  -.  E. x  e.  A  x  <  B ) )
18 ralnex 2566 . . 3  |-  ( A. x  e.  A  -.  x  <  B  <->  -.  E. x  e.  A  x  <  B )
1917, 18syl6bbr 254 . 2  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( -.  sup ( A ,  RR* ,  `'  <  )  <  B  <->  A. x  e.  A  -.  x  <  B ) )
20 id 19 . . 3  |-  ( B  e.  RR*  ->  B  e. 
RR* )
21 infmxrcl 10651 . . 3  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  `'  <  )  e.  RR* )
22 xrlenlt 8906 . . 3  |-  ( ( B  e.  RR*  /\  sup ( A ,  RR* ,  `'  <  )  e.  RR* )  ->  ( B  <_  sup ( A ,  RR* ,  `'  <  )  <->  -.  sup ( A ,  RR* ,  `'  <  )  <  B ) )
2320, 21, 22syl2anr 464 . 2  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B  <_  sup ( A ,  RR* ,  `'  <  )  <->  -. 
sup ( A ,  RR* ,  `'  <  )  <  B ) )
24 simplr 731 . . . 4  |-  ( ( ( A  C_  RR*  /\  B  e.  RR* )  /\  x  e.  A )  ->  B  e.  RR* )
25 simpl 443 . . . . 5  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  A  C_ 
RR* )
2625sselda 3193 . . . 4  |-  ( ( ( A  C_  RR*  /\  B  e.  RR* )  /\  x  e.  A )  ->  x  e.  RR* )
27 xrlenlt 8906 . . . 4  |-  ( ( B  e.  RR*  /\  x  e.  RR* )  ->  ( B  <_  x  <->  -.  x  <  B ) )
2824, 26, 27syl2anc 642 . . 3  |-  ( ( ( A  C_  RR*  /\  B  e.  RR* )  /\  x  e.  A )  ->  ( B  <_  x  <->  -.  x  <  B ) )
2928ralbidva 2572 . 2  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( A. x  e.  A  B  <_  x  <->  A. x  e.  A  -.  x  <  B ) )
3019, 23, 293bitr4d 276 1  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B  <_  sup ( A ,  RR* ,  `'  <  )  <->  A. x  e.  A  B  <_  x ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   class class class wbr 4039    Or wor 4329   `'ccnv 4704   supcsup 7209   RR*cxr 8882    < clt 8883    <_ cle 8884
This theorem is referenced by:  infmxrre  10670  ixxlb  10694  limsuple  11968  limsupval2  11970  imasdsf1olem  17953  nmogelb  18241  metdsf  18368  metdsge  18369  ovolgelb  18855  ovolge0  18856  ovolsslem  18859  ovolicc2  18897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056
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