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Theorem infmxrgelb 11297
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 11118 . . . . . . . 8  |-  <  Or  RR*
2 cnvso 5376 . . . . . . . 8  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
31, 2mpbi 208 . . . . . . 7  |-  `'  <  Or 
RR*
43a1i 11 . . . . . 6  |-  ( A 
C_  RR*  ->  `'  <  Or 
RR* )
5 xrinfmss2 11273 . . . . . 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 7711 . . . . 5  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B `'  <  sup ( A ,  RR* ,  `'  <  )  <->  E. x  e.  A  B `'  <  x ) )
8 simpr 461 . . . . . 6  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
93supex 7713 . . . . . 6  |-  sup ( A ,  RR* ,  `'  <  )  e.  _V
10 brcnvg 5020 . . . . . 6  |-  ( ( B  e.  RR*  /\  sup ( A ,  RR* ,  `'  <  )  e.  _V )  ->  ( B `'  <  sup ( A ,  RR* ,  `'  <  )  <->  sup ( A ,  RR* ,  `'  <  )  <  B ) )
118, 9, 10sylancl 662 . . . . 5  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B `'  <  sup ( A ,  RR* ,  `'  <  )  <->  sup ( A ,  RR* ,  `'  <  )  <  B ) )
12 vex 2975 . . . . . . 7  |-  x  e. 
_V
13 brcnvg 5020 . . . . . . 7  |-  ( ( B  e.  RR*  /\  x  e.  _V )  ->  ( B `'  <  x  <->  x  <  B ) )
148, 12, 13sylancl 662 . . . . . 6  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B `'  <  x  <->  x  <  B ) )
1514rexbidv 2736 . . . . 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 294 . . 3  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( -.  sup ( A ,  RR* ,  `'  <  )  <  B  <->  -.  E. x  e.  A  x  <  B ) )
18 ralnex 2725 . . 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 11279 . . 3  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  `'  <  )  e.  RR* )
22 xrlenlt 9442 . . 3  |-  ( ( B  e.  RR*  /\  sup ( A ,  RR* ,  `'  <  )  e.  RR* )  ->  ( B  <_  sup ( A ,  RR* ,  `'  <  )  <->  -.  sup ( A ,  RR* ,  `'  <  )  <  B ) )
2320, 21, 22syl2anr 478 . 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 457 . . . . 5  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  A  C_ 
RR* )
2625sselda 3356 . . . 4  |-  ( ( ( A  C_  RR*  /\  B  e.  RR* )  /\  x  e.  A )  ->  x  e.  RR* )
27 xrlenlt 9442 . . . 4  |-  ( ( B  e.  RR*  /\  x  e.  RR* )  ->  ( B  <_  x  <->  -.  x  <  B ) )
2824, 26, 27syl2anc 661 . . 3  |-  ( ( ( A  C_  RR*  /\  B  e.  RR* )  /\  x  e.  A )  ->  ( B  <_  x  <->  -.  x  <  B ) )
2928ralbidva 2731 . 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 369    e. wcel 1756   A.wral 2715   E.wrex 2716   _Vcvv 2972    C_ wss 3328   class class class wbr 4292    Or wor 4640   `'ccnv 4839   supcsup 7690   RR*cxr 9417    < clt 9418    <_ cle 9419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598
This theorem is referenced by:  infmxrre  11298  ixxlb  11322  limsuple  12956  limsupval2  12958  imasdsf1olem  19948  nmogelb  20295  metdsf  20424  metdsge  20425  ovolgelb  20963  ovolge0  20964  ovolsslem  20967  ovolicc2  21005  ismblfin  28432
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