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Theorem xrge0iifcv 28740
Description: The defined function's value in the real. (Contributed by Thierry Arnoux, 1-Apr-2017.)
Hypothesis
Ref Expression
xrge0iifhmeo.1  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  =  0 , +oo ,  -u ( log `  x
) ) )
Assertion
Ref Expression
xrge0iifcv  |-  ( X  e.  ( 0 (,] 1 )  ->  ( F `  X )  =  -u ( log `  X
) )
Distinct variable group:    x, X
Allowed substitution hint:    F( x)

Proof of Theorem xrge0iifcv
StepHypRef Expression
1 iocssicc 11722 . . . 4  |-  ( 0 (,] 1 )  C_  ( 0 [,] 1
)
21sseli 3428 . . 3  |-  ( X  e.  ( 0 (,] 1 )  ->  X  e.  ( 0 [,] 1
) )
3 eqeq1 2455 . . . . 5  |-  ( x  =  X  ->  (
x  =  0  <->  X  =  0 ) )
4 fveq2 5865 . . . . . 6  |-  ( x  =  X  ->  ( log `  x )  =  ( log `  X
) )
54negeqd 9869 . . . . 5  |-  ( x  =  X  ->  -u ( log `  x )  = 
-u ( log `  X
) )
63, 5ifbieq2d 3906 . . . 4  |-  ( x  =  X  ->  if ( x  =  0 , +oo ,  -u ( log `  x ) )  =  if ( X  =  0 , +oo ,  -u ( log `  X
) ) )
7 xrge0iifhmeo.1 . . . 4  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  =  0 , +oo ,  -u ( log `  x
) ) )
8 pnfex 11413 . . . . 5  |- +oo  e.  _V
9 negex 9873 . . . . 5  |-  -u ( log `  X )  e. 
_V
108, 9ifex 3949 . . . 4  |-  if ( X  =  0 , +oo ,  -u ( log `  X ) )  e.  _V
116, 7, 10fvmpt 5948 . . 3  |-  ( X  e.  ( 0 [,] 1 )  ->  ( F `  X )  =  if ( X  =  0 , +oo ,  -u ( log `  X
) ) )
122, 11syl 17 . 2  |-  ( X  e.  ( 0 (,] 1 )  ->  ( F `  X )  =  if ( X  =  0 , +oo ,  -u ( log `  X
) ) )
13 0xr 9687 . . . . . . 7  |-  0  e.  RR*
14 1re 9642 . . . . . . 7  |-  1  e.  RR
15 elioc2 11697 . . . . . . 7  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  ( X  e.  ( 0 (,] 1 )  <->  ( X  e.  RR  /\  0  < 
X  /\  X  <_  1 ) ) )
1613, 14, 15mp2an 678 . . . . . 6  |-  ( X  e.  ( 0 (,] 1 )  <->  ( X  e.  RR  /\  0  < 
X  /\  X  <_  1 ) )
1716simp2bi 1024 . . . . 5  |-  ( X  e.  ( 0 (,] 1 )  ->  0  <  X )
1817gt0ne0d 10178 . . . 4  |-  ( X  e.  ( 0 (,] 1 )  ->  X  =/=  0 )
1918neneqd 2629 . . 3  |-  ( X  e.  ( 0 (,] 1 )  ->  -.  X  =  0 )
2019iffalsed 3892 . 2  |-  ( X  e.  ( 0 (,] 1 )  ->  if ( X  =  0 , +oo ,  -u ( log `  X ) )  =  -u ( log `  X
) )
2112, 20eqtrd 2485 1  |-  ( X  e.  ( 0 (,] 1 )  ->  ( F `  X )  =  -u ( log `  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ w3a 985    = wceq 1444    e. wcel 1887   ifcif 3881   class class class wbr 4402    |-> cmpt 4461   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540   +oocpnf 9672   RR*cxr 9674    < clt 9675    <_ cle 9676   -ucneg 9861   (,]cioc 11636   [,]cicc 11638   logclog 23504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-i2m1 9607  ax-1ne0 9608  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-neg 9863  df-ioc 11640  df-icc 11642
This theorem is referenced by:  xrge0iifiso  28741  xrge0iifhom  28743
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