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Theorem xrge0iifhom 26367
Description: The defined function from the closed unit interval and the extended nonnegative 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 9430 . . . . . 6  |-  0  e.  RR*
2 1re 9385 . . . . . . 7  |-  1  e.  RR
32rexri 9436 . . . . . 6  |-  1  e.  RR*
4 0le1 9863 . . . . . 6  |-  0  <_  1
5 snunioc 11413 . . . . . 6  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  0  <_ 
1 )  ->  ( { 0 }  u.  ( 0 (,] 1
) )  =  ( 0 [,] 1 ) )
61, 3, 4, 5mp3an 1314 . . . . 5  |-  ( { 0 }  u.  (
0 (,] 1 ) )  =  ( 0 [,] 1 )
76eleq2i 2507 . . . 4  |-  ( Y  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
Y  e.  ( 0 [,] 1 ) )
8 elun 3497 . . . 4  |-  ( Y  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
( Y  e.  {
0 }  \/  Y  e.  ( 0 (,] 1
) ) )
97, 8bitr3i 251 . . 3  |-  ( Y  e.  ( 0 [,] 1 )  <->  ( Y  e.  { 0 }  \/  Y  e.  ( 0 (,] 1 ) ) )
10 elsni 3902 . . . 4  |-  ( Y  e.  { 0 }  ->  Y  =  0 )
1110orim1i 517 . . 3  |-  ( ( Y  e.  { 0 }  \/  Y  e.  ( 0 (,] 1
) )  ->  ( Y  =  0  \/  Y  e.  ( 0 (,] 1 ) ) )
129, 11sylbi 195 . 2  |-  ( Y  e.  ( 0 [,] 1 )  ->  ( Y  =  0  \/  Y  e.  ( 0 (,] 1 ) ) )
13 0elunit 11403 . . . . . . . 8  |-  0  e.  ( 0 [,] 1
)
14 iftrue 3797 . . . . . . . . 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 pnfex 11093 . . . . . . . . 9  |- +oo  e.  _V
1714, 15, 16fvmpt 5774 . . . . . . . 8  |-  ( 0  e.  ( 0 [,] 1 )  ->  ( F `  0 )  = +oo )
1813, 17ax-mp 5 . . . . . . 7  |-  ( F `
 0 )  = +oo
1918oveq2i 6102 . . . . . 6  |-  ( ( F `  X ) +e ( F `
 0 ) )  =  ( ( F `
 X ) +e +oo )
20 eqeq1 2449 . . . . . . . . . . 11  |-  ( x  =  X  ->  (
x  =  0  <->  X  =  0 ) )
21 fveq2 5691 . . . . . . . . . . . 12  |-  ( x  =  X  ->  ( log `  x )  =  ( log `  X
) )
2221negeqd 9604 . . . . . . . . . . 11  |-  ( x  =  X  ->  -u ( log `  x )  = 
-u ( log `  X
) )
2320, 22ifbieq2d 3814 . . . . . . . . . 10  |-  ( x  =  X  ->  if ( x  =  0 , +oo ,  -u ( log `  x ) )  =  if ( X  =  0 , +oo ,  -u ( log `  X
) ) )
24 negex 9608 . . . . . . . . . . 11  |-  -u ( log `  X )  e. 
_V
2516, 24ifex 3858 . . . . . . . . . 10  |-  if ( X  =  0 , +oo ,  -u ( log `  X ) )  e.  _V
2623, 15, 25fvmpt 5774 . . . . . . . . 9  |-  ( X  e.  ( 0 [,] 1 )  ->  ( F `  X )  =  if ( X  =  0 , +oo ,  -u ( log `  X
) ) )
27 pnfxr 11092 . . . . . . . . . . 11  |- +oo  e.  RR*
2827a1i 11 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  X  =  0 )  -> +oo  e.  RR* )
29 elunitrn 26327 . . . . . . . . . . . . . . 15  |-  ( X  e.  ( 0 [,] 1 )  ->  X  e.  RR )
3029adantr 465 . . . . . . . . . . . . . 14  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  X  e.  RR )
31 elunitge0 26329 . . . . . . . . . . . . . . . 16  |-  ( X  e.  ( 0 [,] 1 )  ->  0  <_  X )
3231adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  0  <_  X )
33 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -.  X  =  0 )
3433neneqad 2681 . . . . . . . . . . . . . . 15  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  X  =/=  0 )
3530, 32, 34ne0gt0d 9511 . . . . . . . . . . . . . 14  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  0  <  X )
3630, 35elrpd 11025 . . . . . . . . . . . . 13  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  X  e.  RR+ )
3736relogcld 22072 . . . . . . . . . . . 12  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  ( log `  X )  e.  RR )
3837renegcld 9775 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -u ( log `  X )  e.  RR )
3938rexrd 9433 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -u ( log `  X )  e.  RR* )
4028, 39ifclda 3821 . . . . . . . . 9  |-  ( X  e.  ( 0 [,] 1 )  ->  if ( X  =  0 , +oo ,  -u ( log `  X ) )  e.  RR* )
4126, 40eqeltrd 2517 . . . . . . . 8  |-  ( X  e.  ( 0 [,] 1 )  ->  ( F `  X )  e.  RR* )
4241adantr 465 . . . . . . 7  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  X )  e.  RR* )
43 neeq1 2616 . . . . . . . . . 10  |-  ( +oo  =  if ( X  =  0 , +oo ,  -u ( log `  X
) )  ->  ( +oo  =/= -oo  <->  if ( X  =  0 , +oo ,  -u ( log `  X
) )  =/= -oo ) )
44 neeq1 2616 . . . . . . . . . 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 11097 . . . . . . . . . . 11  |- +oo  =/= -oo
4645a1i 11 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  X  =  0 )  -> +oo  =/= -oo )
4738renemnfd 9435 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -u ( log `  X )  =/= -oo )
4843, 44, 46, 47ifbothda 3824 . . . . . . . . 9  |-  ( X  e.  ( 0 [,] 1 )  ->  if ( X  =  0 , +oo ,  -u ( log `  X ) )  =/= -oo )
4926, 48eqnetrd 2626 . . . . . . . 8  |-  ( X  e.  ( 0 [,] 1 )  ->  ( F `  X )  =/= -oo )
5049adantr 465 . . . . . . 7  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  X )  =/= -oo )
51 xaddpnf1 11196 . . . . . . 7  |-  ( ( ( F `  X
)  e.  RR*  /\  ( F `  X )  =/= -oo )  ->  (
( F `  X
) +e +oo )  = +oo )
5242, 50, 51syl2anc 661 . . . . . 6  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( ( F `
 X ) +e +oo )  = +oo )
5319, 52syl5eq 2487 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( ( F `
 X ) +e ( F ` 
0 ) )  = +oo )
54 unitsscn 26326 . . . . . . . . 9  |-  ( 0 [,] 1 )  C_  CC
55 simpl 457 . . . . . . . . 9  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  X  e.  ( 0 [,] 1 ) )
5654, 55sseldi 3354 . . . . . . . 8  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  X  e.  CC )
5756mul01d 9568 . . . . . . 7  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( X  x.  0 )  =  0 )
5857fveq2d 5695 . . . . . 6  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  0
) )  =  ( F `  0 ) )
5958, 18syl6eq 2491 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  0
) )  = +oo )
6053, 59eqtr4d 2478 . . . 4  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( ( F `
 X ) +e ( F ` 
0 ) )  =  ( F `  ( X  x.  0 ) ) )
61 simpr 461 . . . . . 6  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  Y  =  0 )
6261fveq2d 5695 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  Y )  =  ( F `  0 ) )
6362oveq2d 6107 . . . 4  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( ( F `
 X ) +e ( F `  Y ) )  =  ( ( F `  X ) +e
( F `  0
) ) )
6461oveq2d 6107 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( X  x.  Y )  =  ( X  x.  0 ) )
6564fveq2d 5695 . . . 4  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  Y
) )  =  ( F `  ( X  x.  0 ) ) )
6660, 63, 653eqtr4rd 2486 . . 3  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  Y
) )  =  ( ( F `  X
) +e ( F `  Y ) ) )
676eleq2i 2507 . . . . . 6  |-  ( X  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
X  e.  ( 0 [,] 1 ) )
68 elun 3497 . . . . . 6  |-  ( X  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
( X  e.  {
0 }  \/  X  e.  ( 0 (,] 1
) ) )
6967, 68bitr3i 251 . . . . 5  |-  ( X  e.  ( 0 [,] 1 )  <->  ( X  e.  { 0 }  \/  X  e.  ( 0 (,] 1 ) ) )
70 elsni 3902 . . . . . 6  |-  ( X  e.  { 0 }  ->  X  =  0 )
7170orim1i 517 . . . . 5  |-  ( ( X  e.  { 0 }  \/  X  e.  ( 0 (,] 1
) )  ->  ( X  =  0  \/  X  e.  ( 0 (,] 1 ) ) )
7269, 71sylbi 195 . . . 4  |-  ( X  e.  ( 0 [,] 1 )  ->  ( X  =  0  \/  X  e.  ( 0 (,] 1 ) ) )
7318oveq1i 6101 . . . . . . . 8  |-  ( ( F `  0 ) +e ( F `
 Y ) )  =  ( +oo +e ( F `  Y ) )
74 simpr 461 . . . . . . . . . 10  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  ( 0 (,] 1 ) )
7515xrge0iifcv 26364 . . . . . . . . . . . 12  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  =  -u ( log `  Y
) )
76 0le0 10411 . . . . . . . . . . . . . . . . 17  |-  0  <_  0
77 ltpnf 11102 . . . . . . . . . . . . . . . . . 18  |-  ( 1  e.  RR  ->  1  < +oo )
782, 77ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  1  < +oo
79 iocssioo 26061 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 0  e.  RR*  /\ +oo  e.  RR* )  /\  (
0  <_  0  /\  1  < +oo ) )  -> 
( 0 (,] 1
)  C_  ( 0 (,) +oo ) )
801, 27, 76, 78, 79mp4an 673 . . . . . . . . . . . . . . . 16  |-  ( 0 (,] 1 )  C_  ( 0 (,) +oo )
81 ioorp 11373 . . . . . . . . . . . . . . . 16  |-  ( 0 (,) +oo )  = 
RR+
8280, 81sseqtri 3388 . . . . . . . . . . . . . . 15  |-  ( 0 (,] 1 )  C_  RR+
8382sseli 3352 . . . . . . . . . . . . . 14  |-  ( Y  e.  ( 0 (,] 1 )  ->  Y  e.  RR+ )
8483relogcld 22072 . . . . . . . . . . . . 13  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( log `  Y )  e.  RR )
8584renegcld 9775 . . . . . . . . . . . 12  |-  ( Y  e.  ( 0 (,] 1 )  ->  -u ( log `  Y )  e.  RR )
8675, 85eqeltrd 2517 . . . . . . . . . . 11  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  e.  RR )
8786rexrd 9433 . . . . . . . . . 10  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  e.  RR* )
8874, 87syl 16 . . . . . . . . 9  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  Y )  e.  RR* )
8986renemnfd 9435 . . . . . . . . . 10  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  =/= -oo )
9074, 89syl 16 . . . . . . . . 9  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  Y )  =/= -oo )
91 xaddpnf2 11197 . . . . . . . . 9  |-  ( ( ( F `  Y
)  e.  RR*  /\  ( F `  Y )  =/= -oo )  ->  ( +oo +e ( F `
 Y ) )  = +oo )
9288, 90, 91syl2anc 661 . . . . . . . 8  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( +oo +e ( F `  Y ) )  = +oo )
9373, 92syl5eq 2487 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 0 ) +e ( F `  Y ) )  = +oo )
94 rpssre 11001 . . . . . . . . . . . . 13  |-  RR+  C_  RR
9582, 94sstri 3365 . . . . . . . . . . . 12  |-  ( 0 (,] 1 )  C_  RR
96 ax-resscn 9339 . . . . . . . . . . . 12  |-  RR  C_  CC
9795, 96sstri 3365 . . . . . . . . . . 11  |-  ( 0 (,] 1 )  C_  CC
9897, 74sseldi 3354 . . . . . . . . . 10  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  CC )
9998mul02d 9567 . . . . . . . . 9  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( 0  x.  Y )  =  0 )
10099fveq2d 5695 . . . . . . . 8  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( 0  x.  Y
) )  =  ( F `  0 ) )
101100, 18syl6eq 2491 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( 0  x.  Y
) )  = +oo )
10293, 101eqtr4d 2478 . . . . . 6  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 0 ) +e ( F `  Y ) )  =  ( F `  (
0  x.  Y ) ) )
103 simpl 457 . . . . . . . 8  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  =  0 )
104103fveq2d 5695 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  X )  =  ( F `  0 ) )
105104oveq1d 6106 . . . . . 6  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 X ) +e ( F `  Y ) )  =  ( ( F ` 
0 ) +e
( F `  Y
) ) )
106103oveq1d 6106 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  =  ( 0  x.  Y ) )
107106fveq2d 5695 . . . . . 6  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  ( F `  ( 0  x.  Y ) ) )
108102, 105, 1073eqtr4rd 2486 . . . . 5  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  ( ( F `  X
) +e ( F `  Y ) ) )
109 simpl 457 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  ( 0 (,] 1 ) )
11082, 109sseldi 3354 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  RR+ )
111110relogcld 22072 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  X
)  e.  RR )
112111renegcld 9775 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( log `  X
)  e.  RR )
113 simpr 461 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  ( 0 (,] 1 ) )
11482, 113sseldi 3354 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  RR+ )
115114relogcld 22072 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  Y
)  e.  RR )
116115renegcld 9775 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( log `  Y
)  e.  RR )
117 rexadd 11202 . . . . . . 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 661 . . . . . 6  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( -u ( log `  X ) +e -u ( log `  Y ) )  =  ( -u ( log `  X )  +  -u ( log `  Y ) ) )
11915xrge0iifcv 26364 . . . . . . 7  |-  ( X  e.  ( 0 (,] 1 )  ->  ( F `  X )  =  -u ( log `  X
) )
120119, 75oveqan12d 6110 . . . . . 6  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 X ) +e ( F `  Y ) )  =  ( -u ( log `  X ) +e -u ( log `  Y
) ) )
121110rpred 11027 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  RR )
122114rpred 11027 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  RR )
123121, 122remulcld 9414 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  e.  RR )
124110rpgt0d 11030 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  0  <  X
)
125114rpgt0d 11030 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  0  <  Y
)
126121, 122, 124, 125mulgt0d 9526 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  0  <  ( X  x.  Y )
)
127 iocssicc 26059 . . . . . . . . . . . 12  |-  ( 0 (,] 1 )  C_  ( 0 [,] 1
)
128127, 109sseldi 3354 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  ( 0 [,] 1 ) )
129127, 113sseldi 3354 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  ( 0 [,] 1 ) )
130 iimulcl 20509 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  e.  ( 0 [,] 1 ) )  ->  ( X  x.  Y )  e.  ( 0 [,] 1 ) )
131128, 129, 130syl2anc 661 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  e.  ( 0 [,] 1 ) )
132 0re 9386 . . . . . . . . . . . 12  |-  0  e.  RR
133132, 2elicc2i 11361 . . . . . . . . . . 11  |-  ( ( X  x.  Y )  e.  ( 0 [,] 1 )  <->  ( ( X  x.  Y )  e.  RR  /\  0  <_ 
( X  x.  Y
)  /\  ( X  x.  Y )  <_  1
) )
134133simp3bi 1005 . . . . . . . . . 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 11358 . . . . . . . . . 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 672 . . . . . . . . 9  |-  ( ( X  x.  Y )  e.  ( 0 (,] 1 )  <->  ( ( X  x.  Y )  e.  RR  /\  0  < 
( X  x.  Y
)  /\  ( X  x.  Y )  <_  1
) )
138123, 126, 135, 137syl3anbrc 1172 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  e.  ( 0 (,] 1 ) )
13915xrge0iifcv 26364 . . . . . . . 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 22074 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  ( X  x.  Y )
)  =  ( ( log `  X )  +  ( log `  Y
) ) )
142141negeqd 9604 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( log `  ( X  x.  Y )
)  =  -u (
( log `  X
)  +  ( log `  Y ) ) )
143111recnd 9412 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  X
)  e.  CC )
144115recnd 9412 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  Y
)  e.  CC )
145143, 144negdid 9732 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( ( log `  X )  +  ( log `  Y ) )  =  ( -u ( log `  X )  +  -u ( log `  Y
) ) )
146140, 142, 1453eqtrd 2479 . . . . . 6  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  (
-u ( log `  X
)  +  -u ( log `  Y ) ) )
147118, 120, 1463eqtr4rd 2486 . . . . 5  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  ( ( F `  X
) +e ( F `  Y ) ) )
148108, 147jaoian 782 . . . 4  |-  ( ( ( X  =  0  \/  X  e.  ( 0 (,] 1 ) )  /\  Y  e.  ( 0 (,] 1
) )  ->  ( F `  ( X  x.  Y ) )  =  ( ( F `  X ) +e
( F `  Y
) ) )
14972, 148sylan 471 . . 3  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  ( ( F `  X
) +e ( F `  Y ) ) )
15066, 149jaodan 783 . 2  |-  ( ( X  e.  ( 0 [,] 1 )  /\  ( Y  =  0  \/  Y  e.  (
0 (,] 1 ) ) )  ->  ( F `  ( X  x.  Y ) )  =  ( ( F `  X ) +e
( F `  Y
) ) )
15112, 150sylan2 474 1  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  e.  ( 0 [,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  ( ( F `  X
) +e ( F `  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606    u. cun 3326    C_ wss 3328   ifcif 3791   {csn 3877   class class class wbr 4292    e. cmpt 4350   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287   +oocpnf 9415   -oocmnf 9416   RR*cxr 9417    < clt 9418    <_ cle 9419   -ucneg 9596   RR+crp 10991   +ecxad 11087   (,)cioo 11300   (,]cioc 11301   [,]cicc 11303   ↾t crest 14359  ordTopcordt 14437   logclog 22006
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-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ioc 11305  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-shft 12556  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-sum 13164  df-ef 13353  df-sin 13355  df-cos 13356  df-pi 13358  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-tms 19897  df-cncf 20454  df-limc 21341  df-dv 21342  df-log 22008
This theorem is referenced by:  xrge0iifmhm  26369  xrge0pluscn  26370
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