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Theorem infmxrgelb 10649
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
Dummy variables  y  z are mutually distinct and distinct from all other variables.

Proof of Theorem infmxrgelb
StepHypRef Expression
1 xrltso 10472 . . . . . . . 8  |-  <  Or  RR*
2 cnvso 5214 . . . . . . . 8  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
31, 2mpbi 201 . . . . . . 7  |-  `'  <  Or 
RR*
43a1i 12 . . . . . 6  |-  ( A 
C_  RR*  ->  `'  <  Or 
RR* )
5 xrinfmss2 10625 . . . . . 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 21 . . . . . 6  |-  ( A 
C_  RR*  ->  A  C_  RR* )
74, 5, 6suplub2 7209 . . . . 5  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B `'  <  sup ( A ,  RR* ,  `'  <  )  <->  E. x  e.  A  B `'  <  x ) )
8 simpr 449 . . . . . 6  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
93supex 7211 . . . . . 6  |-  sup ( A ,  RR* ,  `'  <  )  e.  _V
10 brcnvg 4863 . . . . . 6  |-  ( ( B  e.  RR*  /\  sup ( A ,  RR* ,  `'  <  )  e.  _V )  ->  ( B `'  <  sup ( A ,  RR* ,  `'  <  )  <->  sup ( A ,  RR* ,  `'  <  )  <  B ) )
118, 9, 10sylancl 645 . . . . 5  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B `'  <  sup ( A ,  RR* ,  `'  <  )  <->  sup ( A ,  RR* ,  `'  <  )  <  B ) )
12 vex 2794 . . . . . . 7  |-  x  e. 
_V
13 brcnvg 4863 . . . . . . 7  |-  ( ( B  e.  RR*  /\  x  e.  _V )  ->  ( B `'  <  x  <->  x  <  B ) )
148, 12, 13sylancl 645 . . . . . 6  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B `'  <  x  <->  x  <  B ) )
1514rexbidv 2567 . . . . 5  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( E. x  e.  A  B `'  <  x  <->  E. x  e.  A  x  <  B ) )
167, 11, 153bitr3d 276 . . . 4  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( sup ( A ,  RR* ,  `'  <  )  <  B  <->  E. x  e.  A  x  <  B ) )
1716notbid 287 . . 3  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( -.  sup ( A ,  RR* ,  `'  <  )  <  B  <->  -.  E. x  e.  A  x  <  B ) )
18 ralnex 2556 . . 3  |-  ( A. x  e.  A  -.  x  <  B  <->  -.  E. x  e.  A  x  <  B )
1917, 18syl6bbr 256 . 2  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( -.  sup ( A ,  RR* ,  `'  <  )  <  B  <->  A. x  e.  A  -.  x  <  B ) )
20 id 21 . . 3  |-  ( B  e.  RR*  ->  B  e. 
RR* )
21 infmxrcl 10631 . . 3  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  `'  <  )  e.  RR* )
22 xrlenlt 8887 . . 3  |-  ( ( B  e.  RR*  /\  sup ( A ,  RR* ,  `'  <  )  e.  RR* )  ->  ( B  <_  sup ( A ,  RR* ,  `'  <  )  <->  -.  sup ( A ,  RR* ,  `'  <  )  <  B ) )
2320, 21, 22syl2anr 466 . 2  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( B  <_  sup ( A ,  RR* ,  `'  <  )  <->  -. 
sup ( A ,  RR* ,  `'  <  )  <  B ) )
24 simplr 733 . . . 4  |-  ( ( ( A  C_  RR*  /\  B  e.  RR* )  /\  x  e.  A )  ->  B  e.  RR* )
25 simpl 445 . . . . 5  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  A  C_ 
RR* )
2625sselda 3183 . . . 4  |-  ( ( ( A  C_  RR*  /\  B  e.  RR* )  /\  x  e.  A )  ->  x  e.  RR* )
27 xrlenlt 8887 . . . 4  |-  ( ( B  e.  RR*  /\  x  e.  RR* )  ->  ( B  <_  x  <->  -.  x  <  B ) )
2824, 26, 27syl2anc 644 . . 3  |-  ( ( ( A  C_  RR*  /\  B  e.  RR* )  /\  x  e.  A )  ->  ( B  <_  x  <->  -.  x  <  B ) )
2928ralbidva 2562 . 2  |-  ( ( A  C_  RR*  /\  B  e.  RR* )  ->  ( A. x  e.  A  B  <_  x  <->  A. x  e.  A  -.  x  <  B ) )
3019, 23, 293bitr4d 278 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 5    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1687   A.wral 2546   E.wrex 2547   _Vcvv 2791    C_ wss 3155   class class class wbr 4026    Or wor 4314   `'ccnv 4689   supcsup 7190   RR*cxr 8863    < clt 8864    <_ cle 8865
This theorem is referenced by:  infmxrre  10650  ixxlb  10674  limsuple  11948  limsupval2  11950  imasdsf1olem  17933  nmogelb  18221  metdsf  18348  metdsge  18349  ovolgelb  18835  ovolge0  18836  ovolsslem  18839  ovolicc2  18877
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-cnex 8790  ax-resscn 8791  ax-1cn 8792  ax-icn 8793  ax-addcl 8794  ax-addrcl 8795  ax-mulcl 8796  ax-mulrcl 8797  ax-mulcom 8798  ax-addass 8799  ax-mulass 8800  ax-distr 8801  ax-i2m1 8802  ax-1ne0 8803  ax-1rid 8804  ax-rnegex 8805  ax-rrecex 8806  ax-cnre 8807  ax-pre-lttri 8808  ax-pre-lttrn 8809  ax-pre-ltadd 8810  ax-pre-mulgt0 8811  ax-pre-sup 8812
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rmo 2554  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3831  df-br 4027  df-opab 4081  df-mpt 4082  df-id 4310  df-po 4315  df-so 4316  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-iota 6254  df-riota 6301  df-er 6657  df-en 6861  df-dom 6862  df-sdom 6863  df-sup 7191  df-pnf 8866  df-mnf 8867  df-xr 8868  df-ltxr 8869  df-le 8870  df-sub 9036  df-neg 9037
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