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Theorem infmxrlb 11516
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 11338 . . . . . . 7  |-  <  Or  RR*
2 cnvso 5539 . . . . . . 7  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
31, 2mpbi 208 . . . . . 6  |-  `'  <  Or 
RR*
43a1i 11 . . . . 5  |-  ( A 
C_  RR*  ->  `'  <  Or 
RR* )
5 xrinfmss2 11493 . . . . 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 7910 . . . 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 11499 . . . 4  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  `'  <  )  e.  RR* )
9 brcnvg 5176 . . . 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 3494 . . 3  |-  ( ( A  C_  RR*  /\  B  e.  A )  ->  B  e.  RR* )
14 xrlenlt 9643 . . 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 1762    C_ wss 3471   class class class wbr 4442    Or wor 4794   `'ccnv 4993   supcsup 7891   RR*cxr 9618    < clt 9619    <_ cle 9620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799
This theorem is referenced by:  infmxrre  11518  ixxlb  11542  limsupval2  13254  imasdsf1olem  20606  ovollb  21620  ovolsslem  21625  infxrmnf  27230  infmxrss  31026
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