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Theorem xrge0iifhom 27571
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 9639 . . . . . 6  |-  0  e.  RR*
2 1re 9594 . . . . . . 7  |-  1  e.  RR
32rexri 9645 . . . . . 6  |-  1  e.  RR*
4 0le1 10075 . . . . . 6  |-  0  <_  1
5 snunioc 11647 . . . . . 6  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  0  <_ 
1 )  ->  ( { 0 }  u.  ( 0 (,] 1
) )  =  ( 0 [,] 1 ) )
61, 3, 4, 5mp3an 1324 . . . . 5  |-  ( { 0 }  u.  (
0 (,] 1 ) )  =  ( 0 [,] 1 )
76eleq2i 2545 . . . 4  |-  ( Y  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
Y  e.  ( 0 [,] 1 ) )
8 elun 3645 . . . 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 4052 . . . 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 11637 . . . . . . . 8  |-  0  e.  ( 0 [,] 1
)
14 iftrue 3945 . . . . . . . . 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 11321 . . . . . . . . 9  |- +oo  e.  _V
1714, 15, 16fvmpt 5949 . . . . . . . 8  |-  ( 0  e.  ( 0 [,] 1 )  ->  ( F `  0 )  = +oo )
1813, 17ax-mp 5 . . . . . . 7  |-  ( F `
 0 )  = +oo
1918oveq2i 6294 . . . . . 6  |-  ( ( F `  X ) +e ( F `
 0 ) )  =  ( ( F `
 X ) +e +oo )
20 eqeq1 2471 . . . . . . . . . . 11  |-  ( x  =  X  ->  (
x  =  0  <->  X  =  0 ) )
21 fveq2 5865 . . . . . . . . . . . 12  |-  ( x  =  X  ->  ( log `  x )  =  ( log `  X
) )
2221negeqd 9813 . . . . . . . . . . 11  |-  ( x  =  X  ->  -u ( log `  x )  = 
-u ( log `  X
) )
2320, 22ifbieq2d 3964 . . . . . . . . . 10  |-  ( x  =  X  ->  if ( x  =  0 , +oo ,  -u ( log `  x ) )  =  if ( X  =  0 , +oo ,  -u ( log `  X
) ) )
24 negex 9817 . . . . . . . . . . 11  |-  -u ( log `  X )  e. 
_V
2516, 24ifex 4008 . . . . . . . . . 10  |-  if ( X  =  0 , +oo ,  -u ( log `  X ) )  e.  _V
2623, 15, 25fvmpt 5949 . . . . . . . . 9  |-  ( X  e.  ( 0 [,] 1 )  ->  ( F `  X )  =  if ( X  =  0 , +oo ,  -u ( log `  X
) ) )
27 pnfxr 11320 . . . . . . . . . . 11  |- +oo  e.  RR*
2827a1i 11 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  X  =  0 )  -> +oo  e.  RR* )
29 elunitrn 27531 . . . . . . . . . . . . . . 15  |-  ( X  e.  ( 0 [,] 1 )  ->  X  e.  RR )
3029adantr 465 . . . . . . . . . . . . . 14  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  X  e.  RR )
31 elunitge0 27533 . . . . . . . . . . . . . . . 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 )
3433neqned 2670 . . . . . . . . . . . . . . 15  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  X  =/=  0 )
3530, 32, 34ne0gt0d 9720 . . . . . . . . . . . . . 14  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  0  <  X )
3630, 35elrpd 11253 . . . . . . . . . . . . 13  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  X  e.  RR+ )
3736relogcld 22752 . . . . . . . . . . . 12  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  ( log `  X )  e.  RR )
3837renegcld 9985 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -u ( log `  X )  e.  RR )
3938rexrd 9642 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -u ( log `  X )  e.  RR* )
4028, 39ifclda 3971 . . . . . . . . 9  |-  ( X  e.  ( 0 [,] 1 )  ->  if ( X  =  0 , +oo ,  -u ( log `  X ) )  e.  RR* )
4126, 40eqeltrd 2555 . . . . . . . 8  |-  ( X  e.  ( 0 [,] 1 )  ->  ( F `  X )  e.  RR* )
4241adantr 465 . . . . . . 7  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  X )  e.  RR* )
43 neeq1 2748 . . . . . . . . . 10  |-  ( +oo  =  if ( X  =  0 , +oo ,  -u ( log `  X
) )  ->  ( +oo  =/= -oo  <->  if ( X  =  0 , +oo ,  -u ( log `  X
) )  =/= -oo ) )
44 neeq1 2748 . . . . . . . . . 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 11325 . . . . . . . . . . 11  |- +oo  =/= -oo
4645a1i 11 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  X  =  0 )  -> +oo  =/= -oo )
4738renemnfd 9644 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -u ( log `  X )  =/= -oo )
4843, 44, 46, 47ifbothda 3974 . . . . . . . . 9  |-  ( X  e.  ( 0 [,] 1 )  ->  if ( X  =  0 , +oo ,  -u ( log `  X ) )  =/= -oo )
4926, 48eqnetrd 2760 . . . . . . . 8  |-  ( X  e.  ( 0 [,] 1 )  ->  ( F `  X )  =/= -oo )
5049adantr 465 . . . . . . 7  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  X )  =/= -oo )
51 xaddpnf1 11424 . . . . . . 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 2520 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( ( F `
 X ) +e ( F ` 
0 ) )  = +oo )
54 unitsscn 27530 . . . . . . . . 9  |-  ( 0 [,] 1 )  C_  CC
55 simpl 457 . . . . . . . . 9  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  X  e.  ( 0 [,] 1 ) )
5654, 55sseldi 3502 . . . . . . . 8  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  X  e.  CC )
5756mul01d 9777 . . . . . . 7  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( X  x.  0 )  =  0 )
5857fveq2d 5869 . . . . . 6  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  0
) )  =  ( F `  0 ) )
5958, 18syl6eq 2524 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  0
) )  = +oo )
6053, 59eqtr4d 2511 . . . 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 5869 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  Y )  =  ( F `  0 ) )
6362oveq2d 6299 . . . 4  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( ( F `
 X ) +e ( F `  Y ) )  =  ( ( F `  X ) +e
( F `  0
) ) )
6461oveq2d 6299 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( X  x.  Y )  =  ( X  x.  0 ) )
6564fveq2d 5869 . . . 4  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  Y
) )  =  ( F `  ( X  x.  0 ) ) )
6660, 63, 653eqtr4rd 2519 . . 3  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  Y
) )  =  ( ( F `  X
) +e ( F `  Y ) ) )
676eleq2i 2545 . . . . . 6  |-  ( X  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
X  e.  ( 0 [,] 1 ) )
68 elun 3645 . . . . . 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 4052 . . . . . 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 6293 . . . . . . . 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 27568 . . . . . . . . . . . 12  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  =  -u ( log `  Y
) )
76 0le0 10624 . . . . . . . . . . . . . . . . 17  |-  0  <_  0
77 ltpnf 11330 . . . . . . . . . . . . . . . . . 18  |-  ( 1  e.  RR  ->  1  < +oo )
782, 77ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  1  < +oo
79 iocssioo 11613 . . . . . . . . . . . . . . . . 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 11601 . . . . . . . . . . . . . . . 16  |-  ( 0 (,) +oo )  = 
RR+
8280, 81sseqtri 3536 . . . . . . . . . . . . . . 15  |-  ( 0 (,] 1 )  C_  RR+
8382sseli 3500 . . . . . . . . . . . . . 14  |-  ( Y  e.  ( 0 (,] 1 )  ->  Y  e.  RR+ )
8483relogcld 22752 . . . . . . . . . . . . 13  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( log `  Y )  e.  RR )
8584renegcld 9985 . . . . . . . . . . . 12  |-  ( Y  e.  ( 0 (,] 1 )  ->  -u ( log `  Y )  e.  RR )
8675, 85eqeltrd 2555 . . . . . . . . . . 11  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  e.  RR )
8786rexrd 9642 . . . . . . . . . 10  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  e.  RR* )
8874, 87syl 16 . . . . . . . . 9  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  Y )  e.  RR* )
8986renemnfd 9644 . . . . . . . . . 10  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  =/= -oo )
9074, 89syl 16 . . . . . . . . 9  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  Y )  =/= -oo )
91 xaddpnf2 11425 . . . . . . . . 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 2520 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 0 ) +e ( F `  Y ) )  = +oo )
94 rpssre 11229 . . . . . . . . . . . . 13  |-  RR+  C_  RR
9582, 94sstri 3513 . . . . . . . . . . . 12  |-  ( 0 (,] 1 )  C_  RR
96 ax-resscn 9548 . . . . . . . . . . . 12  |-  RR  C_  CC
9795, 96sstri 3513 . . . . . . . . . . 11  |-  ( 0 (,] 1 )  C_  CC
9897, 74sseldi 3502 . . . . . . . . . 10  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  CC )
9998mul02d 9776 . . . . . . . . 9  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( 0  x.  Y )  =  0 )
10099fveq2d 5869 . . . . . . . 8  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( 0  x.  Y
) )  =  ( F `  0 ) )
101100, 18syl6eq 2524 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( 0  x.  Y
) )  = +oo )
10293, 101eqtr4d 2511 . . . . . 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 5869 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  X )  =  ( F `  0 ) )
105104oveq1d 6298 . . . . . 6  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 X ) +e ( F `  Y ) )  =  ( ( F ` 
0 ) +e
( F `  Y
) ) )
106103oveq1d 6298 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  =  ( 0  x.  Y ) )
107106fveq2d 5869 . . . . . 6  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  ( F `  ( 0  x.  Y ) ) )
108102, 105, 1073eqtr4rd 2519 . . . . 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 3502 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  RR+ )
111110relogcld 22752 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  X
)  e.  RR )
112111renegcld 9985 . . . . . . 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 3502 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  RR+ )
115114relogcld 22752 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  Y
)  e.  RR )
116115renegcld 9985 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( log `  Y
)  e.  RR )
117 rexadd 11430 . . . . . . 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 27568 . . . . . . 7  |-  ( X  e.  ( 0 (,] 1 )  ->  ( F `  X )  =  -u ( log `  X
) )
120119, 75oveqan12d 6302 . . . . . 6  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 X ) +e ( F `  Y ) )  =  ( -u ( log `  X ) +e -u ( log `  Y
) ) )
121110rpred 11255 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  RR )
122114rpred 11255 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  RR )
123121, 122remulcld 9623 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  e.  RR )
124110rpgt0d 11258 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  0  <  X
)
125114rpgt0d 11258 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  0  <  Y
)
126121, 122, 124, 125mulgt0d 9735 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  0  <  ( X  x.  Y )
)
127 iocssicc 11611 . . . . . . . . . . . 12  |-  ( 0 (,] 1 )  C_  ( 0 [,] 1
)
128127, 109sseldi 3502 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  ( 0 [,] 1 ) )
129127, 113sseldi 3502 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  ( 0 [,] 1 ) )
130 iimulcl 21188 . . . . . . . . . . 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 9595 . . . . . . . . . . . 12  |-  0  e.  RR
133132, 2elicc2i 11589 . . . . . . . . . . 11  |-  ( ( X  x.  Y )  e.  ( 0 [,] 1 )  <->  ( ( X  x.  Y )  e.  RR  /\  0  <_ 
( X  x.  Y
)  /\  ( X  x.  Y )  <_  1
) )
134133simp3bi 1013 . . . . . . . . . 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 11586 . . . . . . . . . 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 1180 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  e.  ( 0 (,] 1 ) )
13915xrge0iifcv 27568 . . . . . . . 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 22754 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  ( X  x.  Y )
)  =  ( ( log `  X )  +  ( log `  Y
) ) )
142141negeqd 9813 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( log `  ( X  x.  Y )
)  =  -u (
( log `  X
)  +  ( log `  Y ) ) )
143111recnd 9621 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  X
)  e.  CC )
144115recnd 9621 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  Y
)  e.  CC )
145143, 144negdid 9942 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( ( log `  X )  +  ( log `  Y ) )  =  ( -u ( log `  X )  +  -u ( log `  Y
) ) )
146140, 142, 1453eqtrd 2512 . . . . . 6  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  (
-u ( log `  X
)  +  -u ( log `  Y ) ) )
147118, 120, 1463eqtr4rd 2519 . . . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662    u. cun 3474    C_ wss 3476   ifcif 3939   {csn 4027   class class class wbr 4447    |-> cmpt 4505   ` cfv 5587  (class class class)co 6283   CCcc 9489   RRcr 9490   0cc0 9491   1c1 9492    + caddc 9494    x. cmul 9496   +oocpnf 9624   -oocmnf 9625   RR*cxr 9626    < clt 9627    <_ cle 9628   -ucneg 9805   RR+crp 11219   +ecxad 11315   (,)cioo 11528   (,]cioc 11529   [,]cicc 11531   ↾t crest 14675  ordTopcordt 14753   logclog 22686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-fi 7870  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-q 11182  df-rp 11220  df-xneg 11317  df-xadd 11318  df-xmul 11319  df-ioo 11532  df-ioc 11533  df-ico 11534  df-icc 11535  df-fz 11672  df-fzo 11792  df-fl 11896  df-mod 11964  df-seq 12075  df-exp 12134  df-fac 12321  df-bc 12348  df-hash 12373  df-shft 12862  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-limsup 13256  df-clim 13273  df-rlim 13274  df-sum 13471  df-ef 13664  df-sin 13666  df-cos 13667  df-pi 13669  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-starv 14569  df-sca 14570  df-vsca 14571  df-ip 14572  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-hom 14578  df-cco 14579  df-rest 14677  df-topn 14678  df-0g 14696  df-gsum 14697  df-topgen 14698  df-pt 14699  df-prds 14702  df-xrs 14756  df-qtop 14761  df-imas 14762  df-xps 14764  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-submnd 15784  df-mulg 15867  df-cntz 16157  df-cmn 16603  df-psmet 18198  df-xmet 18199  df-met 18200  df-bl 18201  df-mopn 18202  df-fbas 18203  df-fg 18204  df-cnfld 18208  df-top 19182  df-bases 19184  df-topon 19185  df-topsp 19186  df-cld 19302  df-ntr 19303  df-cls 19304  df-nei 19381  df-lp 19419  df-perf 19420  df-cn 19510  df-cnp 19511  df-haus 19598  df-tx 19814  df-hmeo 20007  df-fil 20098  df-fm 20190  df-flim 20191  df-flf 20192  df-xms 20574  df-ms 20575  df-tms 20576  df-cncf 21133  df-limc 22021  df-dv 22022  df-log 22688
This theorem is referenced by:  xrge0iifmhm  27573  xrge0pluscn  27574
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