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Theorem infmxrlb 11295
Description: A member of a set of extended reals is greater than or equal to the set's infimum. Note that we did not introduce a notation for the infimum, so we represent it as the supremum for the opposite order relation. (Contributed by Mario Carneiro, 16-Mar-2014.)
Assertion
Ref Expression
infmxrlb  |-  ( ( A  C_  RR*  /\  B  e.  A )  ->  sup ( A ,  RR* ,  `'  <  )  <_  B )

Proof of Theorem infmxrlb
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xrltso 11117 . . . . . . 7  |-  <  Or  RR*
2 cnvso 5375 . . . . . . 7  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
31, 2mpbi 208 . . . . . 6  |-  `'  <  Or 
RR*
43a1i 11 . . . . 5  |-  ( A 
C_  RR*  ->  `'  <  Or 
RR* )
5 xrinfmss2 11272 . . . . 5  |-  ( A 
C_  RR*  ->  E. x  e.  RR*  ( A. y  e.  A  -.  x `'  <  y  /\  A. y  e.  RR*  ( y `'  <  x  ->  E. z  e.  A  y `'  <  z ) ) )
64, 5supub 7708 . . . 4  |-  ( A 
C_  RR*  ->  ( B  e.  A  ->  -.  sup ( A ,  RR* ,  `'  <  ) `'  <  B
) )
76imp 429 . . 3  |-  ( ( A  C_  RR*  /\  B  e.  A )  ->  -.  sup ( A ,  RR* ,  `'  <  ) `'  <  B )
8 infmxrcl 11278 . . . 4  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  `'  <  )  e.  RR* )
9 brcnvg 5019 . . . 4  |-  ( ( sup ( A ,  RR* ,  `'  <  )  e.  RR*  /\  B  e.  A )  ->  ( sup ( A ,  RR* ,  `'  <  ) `'  <  B  <-> 
B  <  sup ( A ,  RR* ,  `'  <  ) ) )
108, 9sylan 471 . . 3  |-  ( ( A  C_  RR*  /\  B  e.  A )  ->  ( sup ( A ,  RR* ,  `'  <  ) `'  <  B  <-> 
B  <  sup ( A ,  RR* ,  `'  <  ) ) )
117, 10mtbid 300 . 2  |-  ( ( A  C_  RR*  /\  B  e.  A )  ->  -.  B  <  sup ( A ,  RR* ,  `'  <  )
)
128adantr 465 . . 3  |-  ( ( A  C_  RR*  /\  B  e.  A )  ->  sup ( A ,  RR* ,  `'  <  )  e.  RR* )
13 ssel2 3350 . . 3  |-  ( ( A  C_  RR*  /\  B  e.  A )  ->  B  e.  RR* )
14 xrlenlt 9441 . . 3  |-  ( ( sup ( A ,  RR* ,  `'  <  )  e.  RR*  /\  B  e. 
RR* )  ->  ( sup ( A ,  RR* ,  `'  <  )  <_  B  <->  -.  B  <  sup ( A ,  RR* ,  `'  <  ) ) )
1512, 13, 14syl2anc 661 . 2  |-  ( ( A  C_  RR*  /\  B  e.  A )  ->  ( sup ( A ,  RR* ,  `'  <  )  <_  B  <->  -.  B  <  sup ( A ,  RR* ,  `'  <  ) ) )
1611, 15mpbird 232 1  |-  ( ( A  C_  RR*  /\  B  e.  A )  ->  sup ( A ,  RR* ,  `'  <  )  <_  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1756    C_ wss 3327   class class class wbr 4291    Or wor 4639   `'ccnv 4838   supcsup 7689   RR*cxr 9416    < clt 9417    <_ cle 9418
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 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-po 4640  df-so 4641  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597
This theorem is referenced by:  infmxrre  11297  ixxlb  11321  limsupval2  12957  imasdsf1olem  19947  ovollb  20961  ovolsslem  20966  infxrmnf  26046
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