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Theorem xrge0iifhom 24315
Description: The defined function from the closed unit interval and the extended non-negative reals is also a monoid homomorphism. (Contributed by Thierry Arnoux, 5-Apr-2017.)
Hypotheses
Ref Expression
xrge0iifhmeo.1  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  =  0 ,  +oo ,  -u ( log `  x
) ) )
xrge0iifhmeo.k  |-  J  =  ( (ordTop `  <_  )t  ( 0 [,]  +oo )
)
Assertion
Ref Expression
xrge0iifhom  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  e.  ( 0 [,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  ( ( F `  X
) + e ( F `  Y ) ) )
Distinct variable groups:    x, X    x, F    x, Y
Allowed substitution hint:    J( x)

Proof of Theorem xrge0iifhom
StepHypRef Expression
1 0xr 9123 . . . . . 6  |-  0  e.  RR*
2 1re 9082 . . . . . . 7  |-  1  e.  RR
32rexri 9129 . . . . . 6  |-  1  e.  RR*
4 0le1 9543 . . . . . 6  |-  0  <_  1
5 snunioc 24129 . . . . . 6  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  0  <_ 
1 )  ->  ( { 0 }  u.  ( 0 (,] 1
) )  =  ( 0 [,] 1 ) )
61, 3, 4, 5mp3an 1279 . . . . 5  |-  ( { 0 }  u.  (
0 (,] 1 ) )  =  ( 0 [,] 1 )
76eleq2i 2499 . . . 4  |-  ( Y  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
Y  e.  ( 0 [,] 1 ) )
8 elun 3480 . . . 4  |-  ( Y  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
( Y  e.  {
0 }  \/  Y  e.  ( 0 (,] 1
) ) )
97, 8bitr3i 243 . . 3  |-  ( Y  e.  ( 0 [,] 1 )  <->  ( Y  e.  { 0 }  \/  Y  e.  ( 0 (,] 1 ) ) )
10 elsni 3830 . . . 4  |-  ( Y  e.  { 0 }  ->  Y  =  0 )
1110orim1i 504 . . 3  |-  ( ( Y  e.  { 0 }  \/  Y  e.  ( 0 (,] 1
) )  ->  ( Y  =  0  \/  Y  e.  ( 0 (,] 1 ) ) )
129, 11sylbi 188 . 2  |-  ( Y  e.  ( 0 [,] 1 )  ->  ( Y  =  0  \/  Y  e.  ( 0 (,] 1 ) ) )
13 0elunit 11007 . . . . . . . 8  |-  0  e.  ( 0 [,] 1
)
14 iftrue 3737 . . . . . . . . 9  |-  ( x  =  0  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  =  +oo )
15 xrge0iifhmeo.1 . . . . . . . . 9  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  =  0 ,  +oo ,  -u ( log `  x
) ) )
16 pnfxr 10705 . . . . . . . . . 10  |-  +oo  e.  RR*
1716elexi 2957 . . . . . . . . 9  |-  +oo  e.  _V
1814, 15, 17fvmpt 5798 . . . . . . . 8  |-  ( 0  e.  ( 0 [,] 1 )  ->  ( F `  0 )  =  +oo )
1913, 18ax-mp 8 . . . . . . 7  |-  ( F `
 0 )  = 
+oo
2019oveq2i 6084 . . . . . 6  |-  ( ( F `  X ) + e ( F `
 0 ) )  =  ( ( F `
 X ) + e  +oo )
21 eqeq1 2441 . . . . . . . . . . 11  |-  ( x  =  X  ->  (
x  =  0  <->  X  =  0 ) )
22 fveq2 5720 . . . . . . . . . . . 12  |-  ( x  =  X  ->  ( log `  x )  =  ( log `  X
) )
2322negeqd 9292 . . . . . . . . . . 11  |-  ( x  =  X  ->  -u ( log `  x )  = 
-u ( log `  X
) )
2421, 23ifbieq2d 3751 . . . . . . . . . 10  |-  ( x  =  X  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  =  if ( X  =  0 ,  +oo , 
-u ( log `  X
) ) )
25 negex 9296 . . . . . . . . . . 11  |-  -u ( log `  X )  e. 
_V
2617, 25ifex 3789 . . . . . . . . . 10  |-  if ( X  =  0 , 
+oo ,  -u ( log `  X ) )  e. 
_V
2724, 15, 26fvmpt 5798 . . . . . . . . 9  |-  ( X  e.  ( 0 [,] 1 )  ->  ( F `  X )  =  if ( X  =  0 ,  +oo ,  -u ( log `  X
) ) )
2816a1i 11 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  X  =  0 )  ->  +oo  e.  RR* )
29 elunitrn 24287 . . . . . . . . . . . . . . 15  |-  ( X  e.  ( 0 [,] 1 )  ->  X  e.  RR )
3029adantr 452 . . . . . . . . . . . . . 14  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  X  e.  RR )
31 elunitge0 24289 . . . . . . . . . . . . . . . 16  |-  ( X  e.  ( 0 [,] 1 )  ->  0  <_  X )
3231adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  0  <_  X )
33 simpr 448 . . . . . . . . . . . . . . . 16  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -.  X  =  0 )
3433neneqad 2668 . . . . . . . . . . . . . . 15  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  X  =/=  0 )
3530, 32, 34ne0gt0d 9202 . . . . . . . . . . . . . 14  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  0  <  X )
3630, 35elrpd 10638 . . . . . . . . . . . . 13  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  X  e.  RR+ )
3736relogcld 20510 . . . . . . . . . . . 12  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  ( log `  X )  e.  RR )
3837renegcld 9456 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -u ( log `  X )  e.  RR )
3938rexrd 9126 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -u ( log `  X )  e.  RR* )
4028, 39ifclda 3758 . . . . . . . . 9  |-  ( X  e.  ( 0 [,] 1 )  ->  if ( X  =  0 ,  +oo ,  -u ( log `  X ) )  e.  RR* )
4127, 40eqeltrd 2509 . . . . . . . 8  |-  ( X  e.  ( 0 [,] 1 )  ->  ( F `  X )  e.  RR* )
4241adantr 452 . . . . . . 7  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  X )  e.  RR* )
43 neeq1 2606 . . . . . . . . . 10  |-  (  +oo  =  if ( X  =  0 ,  +oo ,  -u ( log `  X
) )  ->  (  +oo  =/=  -oo  <->  if ( X  =  0 ,  +oo ,  -u ( log `  X
) )  =/=  -oo ) )
44 neeq1 2606 . . . . . . . . . 10  |-  ( -u ( log `  X )  =  if ( X  =  0 ,  +oo , 
-u ( log `  X
) )  ->  ( -u ( log `  X
)  =/=  -oo  <->  if ( X  =  0 ,  +oo ,  -u ( log `  X
) )  =/=  -oo ) )
45 pnfnemnf 10709 . . . . . . . . . . 11  |-  +oo  =/=  -oo
4645a1i 11 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  X  =  0 )  ->  +oo  =/=  -oo )
4738renemnfd 9128 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -u ( log `  X )  =/=  -oo )
4843, 44, 46, 47ifbothda 3761 . . . . . . . . 9  |-  ( X  e.  ( 0 [,] 1 )  ->  if ( X  =  0 ,  +oo ,  -u ( log `  X ) )  =/=  -oo )
4927, 48eqnetrd 2616 . . . . . . . 8  |-  ( X  e.  ( 0 [,] 1 )  ->  ( F `  X )  =/=  -oo )
5049adantr 452 . . . . . . 7  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  X )  =/=  -oo )
51 xaddpnf1 10804 . . . . . . 7  |-  ( ( ( F `  X
)  e.  RR*  /\  ( F `  X )  =/=  -oo )  ->  (
( F `  X
) + e  +oo )  =  +oo )
5242, 50, 51syl2anc 643 . . . . . 6  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( ( F `
 X ) + e  +oo )  = 
+oo )
5320, 52syl5eq 2479 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( ( F `
 X ) + e ( F ` 
0 ) )  = 
+oo )
54 unitsscn 24286 . . . . . . . . 9  |-  ( 0 [,] 1 )  C_  CC
55 simpl 444 . . . . . . . . 9  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  X  e.  ( 0 [,] 1 ) )
5654, 55sseldi 3338 . . . . . . . 8  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  X  e.  CC )
5756mul01d 9257 . . . . . . 7  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( X  x.  0 )  =  0 )
5857fveq2d 5724 . . . . . 6  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  0
) )  =  ( F `  0 ) )
5958, 19syl6eq 2483 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  0
) )  =  +oo )
6053, 59eqtr4d 2470 . . . 4  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( ( F `
 X ) + e ( F ` 
0 ) )  =  ( F `  ( X  x.  0 ) ) )
61 simpr 448 . . . . . 6  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  Y  =  0 )
6261fveq2d 5724 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  Y )  =  ( F `  0 ) )
6362oveq2d 6089 . . . 4  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( ( F `
 X ) + e ( F `  Y ) )  =  ( ( F `  X ) + e
( F `  0
) ) )
6461oveq2d 6089 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( X  x.  Y )  =  ( X  x.  0 ) )
6564fveq2d 5724 . . . 4  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  Y
) )  =  ( F `  ( X  x.  0 ) ) )
6660, 63, 653eqtr4rd 2478 . . 3  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  Y
) )  =  ( ( F `  X
) + e ( F `  Y ) ) )
676eleq2i 2499 . . . . . 6  |-  ( X  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
X  e.  ( 0 [,] 1 ) )
68 elun 3480 . . . . . 6  |-  ( X  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
( X  e.  {
0 }  \/  X  e.  ( 0 (,] 1
) ) )
6967, 68bitr3i 243 . . . . 5  |-  ( X  e.  ( 0 [,] 1 )  <->  ( X  e.  { 0 }  \/  X  e.  ( 0 (,] 1 ) ) )
70 elsni 3830 . . . . . 6  |-  ( X  e.  { 0 }  ->  X  =  0 )
7170orim1i 504 . . . . 5  |-  ( ( X  e.  { 0 }  \/  X  e.  ( 0 (,] 1
) )  ->  ( X  =  0  \/  X  e.  ( 0 (,] 1 ) ) )
7269, 71sylbi 188 . . . 4  |-  ( X  e.  ( 0 [,] 1 )  ->  ( X  =  0  \/  X  e.  ( 0 (,] 1 ) ) )
7319oveq1i 6083 . . . . . . . 8  |-  ( ( F `  0 ) + e ( F `
 Y ) )  =  (  +oo + e ( F `  Y ) )
74 simpr 448 . . . . . . . . . 10  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  ( 0 (,] 1 ) )
7515xrge0iifcv 24312 . . . . . . . . . . . 12  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  =  -u ( log `  Y
) )
76 0le0 10073 . . . . . . . . . . . . . . . . 17  |-  0  <_  0
77 ltpnf 10713 . . . . . . . . . . . . . . . . . 18  |-  ( 1  e.  RR  ->  1  <  +oo )
782, 77ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  1  <  +oo
79 iocssioo 24124 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 0  e.  RR*  /\ 
+oo  e.  RR* )  /\  ( 0  <_  0  /\  1  <  +oo )
)  ->  ( 0 (,] 1 )  C_  ( 0 (,)  +oo ) )
801, 16, 76, 78, 79mp4an 655 . . . . . . . . . . . . . . . 16  |-  ( 0 (,] 1 )  C_  ( 0 (,)  +oo )
81 ioorp 10980 . . . . . . . . . . . . . . . 16  |-  ( 0 (,)  +oo )  =  RR+
8280, 81sseqtri 3372 . . . . . . . . . . . . . . 15  |-  ( 0 (,] 1 )  C_  RR+
8382sseli 3336 . . . . . . . . . . . . . 14  |-  ( Y  e.  ( 0 (,] 1 )  ->  Y  e.  RR+ )
8483relogcld 20510 . . . . . . . . . . . . 13  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( log `  Y )  e.  RR )
8584renegcld 9456 . . . . . . . . . . . 12  |-  ( Y  e.  ( 0 (,] 1 )  ->  -u ( log `  Y )  e.  RR )
8675, 85eqeltrd 2509 . . . . . . . . . . 11  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  e.  RR )
8786rexrd 9126 . . . . . . . . . 10  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  e.  RR* )
8874, 87syl 16 . . . . . . . . 9  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  Y )  e.  RR* )
8986renemnfd 9128 . . . . . . . . . 10  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  =/=  -oo )
9074, 89syl 16 . . . . . . . . 9  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  Y )  =/=  -oo )
91 xaddpnf2 10805 . . . . . . . . 9  |-  ( ( ( F `  Y
)  e.  RR*  /\  ( F `  Y )  =/=  -oo )  ->  (  +oo + e ( F `
 Y ) )  =  +oo )
9288, 90, 91syl2anc 643 . . . . . . . 8  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  (  +oo + e ( F `  Y ) )  = 
+oo )
9373, 92syl5eq 2479 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 0 ) + e ( F `  Y ) )  = 
+oo )
94 rpssre 10614 . . . . . . . . . . . . 13  |-  RR+  C_  RR
9582, 94sstri 3349 . . . . . . . . . . . 12  |-  ( 0 (,] 1 )  C_  RR
96 ax-resscn 9039 . . . . . . . . . . . 12  |-  RR  C_  CC
9795, 96sstri 3349 . . . . . . . . . . 11  |-  ( 0 (,] 1 )  C_  CC
9897, 74sseldi 3338 . . . . . . . . . 10  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  CC )
9998mul02d 9256 . . . . . . . . 9  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( 0  x.  Y )  =  0 )
10099fveq2d 5724 . . . . . . . 8  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( 0  x.  Y
) )  =  ( F `  0 ) )
101100, 19syl6eq 2483 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( 0  x.  Y
) )  =  +oo )
10293, 101eqtr4d 2470 . . . . . 6  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 0 ) + e ( F `  Y ) )  =  ( F `  (
0  x.  Y ) ) )
103 simpl 444 . . . . . . . 8  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  =  0 )
104103fveq2d 5724 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  X )  =  ( F `  0 ) )
105104oveq1d 6088 . . . . . 6  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 X ) + e ( F `  Y ) )  =  ( ( F ` 
0 ) + e
( F `  Y
) ) )
106103oveq1d 6088 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  =  ( 0  x.  Y ) )
107106fveq2d 5724 . . . . . 6  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  ( F `  ( 0  x.  Y ) ) )
108102, 105, 1073eqtr4rd 2478 . . . . 5  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  ( ( F `  X
) + e ( F `  Y ) ) )
109 simpl 444 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  ( 0 (,] 1 ) )
11082, 109sseldi 3338 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  RR+ )
111110relogcld 20510 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  X
)  e.  RR )
112111renegcld 9456 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( log `  X
)  e.  RR )
113 simpr 448 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  ( 0 (,] 1 ) )
11482, 113sseldi 3338 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  RR+ )
115114relogcld 20510 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  Y
)  e.  RR )
116115renegcld 9456 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( log `  Y
)  e.  RR )
117 rexadd 10810 . . . . . . 7  |-  ( (
-u ( log `  X
)  e.  RR  /\  -u ( log `  Y
)  e.  RR )  ->  ( -u ( log `  X ) + e -u ( log `  Y ) )  =  ( -u ( log `  X )  +  -u ( log `  Y ) ) )
118112, 116, 117syl2anc 643 . . . . . 6  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( -u ( log `  X ) + e -u ( log `  Y ) )  =  ( -u ( log `  X )  +  -u ( log `  Y ) ) )
11915xrge0iifcv 24312 . . . . . . 7  |-  ( X  e.  ( 0 (,] 1 )  ->  ( F `  X )  =  -u ( log `  X
) )
120119, 75oveqan12d 6092 . . . . . 6  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 X ) + e ( F `  Y ) )  =  ( -u ( log `  X ) + e -u ( log `  Y
) ) )
121110rpred 10640 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  RR )
122114rpred 10640 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  RR )
123121, 122remulcld 9108 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  e.  RR )
124110rpgt0d 10643 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  0  <  X
)
125114rpgt0d 10643 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  0  <  Y
)
126121, 122, 124, 125mulgt0d 9217 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  0  <  ( X  x.  Y )
)
127 iocssicc 24122 . . . . . . . . . . . 12  |-  ( 0 (,] 1 )  C_  ( 0 [,] 1
)
128127, 109sseldi 3338 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  ( 0 [,] 1 ) )
129127, 113sseldi 3338 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  ( 0 [,] 1 ) )
130 iimulcl 18954 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  e.  ( 0 [,] 1 ) )  ->  ( X  x.  Y )  e.  ( 0 [,] 1 ) )
131128, 129, 130syl2anc 643 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  e.  ( 0 [,] 1 ) )
132 0re 9083 . . . . . . . . . . . 12  |-  0  e.  RR
133132, 2elicc2i 10968 . . . . . . . . . . 11  |-  ( ( X  x.  Y )  e.  ( 0 [,] 1 )  <->  ( ( X  x.  Y )  e.  RR  /\  0  <_ 
( X  x.  Y
)  /\  ( X  x.  Y )  <_  1
) )
134133simp3bi 974 . . . . . . . . . 10  |-  ( ( X  x.  Y )  e.  ( 0 [,] 1 )  ->  ( X  x.  Y )  <_  1 )
135131, 134syl 16 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  <_  1
)
136 elioc2 10965 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  (
( X  x.  Y
)  e.  ( 0 (,] 1 )  <->  ( ( X  x.  Y )  e.  RR  /\  0  < 
( X  x.  Y
)  /\  ( X  x.  Y )  <_  1
) ) )
1371, 2, 136mp2an 654 . . . . . . . . 9  |-  ( ( X  x.  Y )  e.  ( 0 (,] 1 )  <->  ( ( X  x.  Y )  e.  RR  /\  0  < 
( X  x.  Y
)  /\  ( X  x.  Y )  <_  1
) )
138123, 126, 135, 137syl3anbrc 1138 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  e.  ( 0 (,] 1 ) )
13915xrge0iifcv 24312 . . . . . . . 8  |-  ( ( X  x.  Y )  e.  ( 0 (,] 1 )  ->  ( F `  ( X  x.  Y ) )  = 
-u ( log `  ( X  x.  Y )
) )
140138, 139syl 16 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  -u ( log `  ( X  x.  Y ) ) )
141110, 114relogmuld 20512 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  ( X  x.  Y )
)  =  ( ( log `  X )  +  ( log `  Y
) ) )
142141negeqd 9292 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( log `  ( X  x.  Y )
)  =  -u (
( log `  X
)  +  ( log `  Y ) ) )
143111recnd 9106 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  X
)  e.  CC )
144115recnd 9106 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  Y
)  e.  CC )
145143, 144negdid 9416 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( ( log `  X )  +  ( log `  Y ) )  =  ( -u ( log `  X )  +  -u ( log `  Y
) ) )
146140, 142, 1453eqtrd 2471 . . . . . 6  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  (
-u ( log `  X
)  +  -u ( log `  Y ) ) )
147118, 120, 1463eqtr4rd 2478 . . . . 5  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  ( ( F `  X
) + e ( F `  Y ) ) )
148108, 147jaoian 760 . . . 4  |-  ( ( ( X  =  0  \/  X  e.  ( 0 (,] 1 ) )  /\  Y  e.  ( 0 (,] 1
) )  ->  ( F `  ( X  x.  Y ) )  =  ( ( F `  X ) + e
( F `  Y
) ) )
14972, 148sylan 458 . . 3  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  ( ( F `  X
) + e ( F `  Y ) ) )
15066, 149jaodan 761 . 2  |-  ( ( X  e.  ( 0 [,] 1 )  /\  ( Y  =  0  \/  Y  e.  (
0 (,] 1 ) ) )  ->  ( F `  ( X  x.  Y ) )  =  ( ( F `  X ) + e
( F `  Y
) ) )
15112, 150sylan2 461 1  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  e.  ( 0 [,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  ( ( F `  X
) + e ( F `  Y ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598    u. cun 3310    C_ wss 3312   ifcif 3731   {csn 3806   class class class wbr 4204    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111    < clt 9112    <_ cle 9113   -ucneg 9284   RR+crp 10604   + ecxad 10700   (,)cioo 10908   (,]cioc 10909   [,]cicc 10911   ↾t crest 13640  ordTopcordt 13713   logclog 20444
This theorem is referenced by:  xrge0iifmhm  24317  xrge0pluscn  24318
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-ef 12662  df-sin 12664  df-cos 12665  df-pi 12667  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cn 17283  df-cnp 17284  df-haus 17371  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-tms 18344  df-cncf 18900  df-limc 19745  df-dv 19746  df-log 20446
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