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Theorem xrge0iifhom 28104
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 9569 . . . . . 6  |-  0  e.  RR*
2 1re 9524 . . . . . . 7  |-  1  e.  RR
32rexri 9575 . . . . . 6  |-  1  e.  RR*
4 0le1 10011 . . . . . 6  |-  0  <_  1
5 snunioc 11587 . . . . . 6  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  0  <_ 
1 )  ->  ( { 0 }  u.  ( 0 (,] 1
) )  =  ( 0 [,] 1 ) )
61, 3, 4, 5mp3an 1322 . . . . 5  |-  ( { 0 }  u.  (
0 (,] 1 ) )  =  ( 0 [,] 1 )
76eleq2i 2470 . . . 4  |-  ( Y  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
Y  e.  ( 0 [,] 1 ) )
8 elun 3572 . . . 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 3982 . . . 4  |-  ( Y  e.  { 0 }  ->  Y  =  0 )
1110orim1i 515 . . 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 11577 . . . . . . . 8  |-  0  e.  ( 0 [,] 1
)
14 iftrue 3876 . . . . . . . . 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 11261 . . . . . . . . 9  |- +oo  e.  _V
1714, 15, 16fvmpt 5870 . . . . . . . 8  |-  ( 0  e.  ( 0 [,] 1 )  ->  ( F `  0 )  = +oo )
1813, 17ax-mp 5 . . . . . . 7  |-  ( F `
 0 )  = +oo
1918oveq2i 6225 . . . . . 6  |-  ( ( F `  X ) +e ( F `
 0 ) )  =  ( ( F `
 X ) +e +oo )
20 eqeq1 2396 . . . . . . . . . . 11  |-  ( x  =  X  ->  (
x  =  0  <->  X  =  0 ) )
21 fveq2 5787 . . . . . . . . . . . 12  |-  ( x  =  X  ->  ( log `  x )  =  ( log `  X
) )
2221negeqd 9745 . . . . . . . . . . 11  |-  ( x  =  X  ->  -u ( log `  x )  = 
-u ( log `  X
) )
2320, 22ifbieq2d 3895 . . . . . . . . . 10  |-  ( x  =  X  ->  if ( x  =  0 , +oo ,  -u ( log `  x ) )  =  if ( X  =  0 , +oo ,  -u ( log `  X
) ) )
24 negex 9749 . . . . . . . . . . 11  |-  -u ( log `  X )  e. 
_V
2516, 24ifex 3938 . . . . . . . . . 10  |-  if ( X  =  0 , +oo ,  -u ( log `  X ) )  e.  _V
2623, 15, 25fvmpt 5870 . . . . . . . . 9  |-  ( X  e.  ( 0 [,] 1 )  ->  ( F `  X )  =  if ( X  =  0 , +oo ,  -u ( log `  X
) ) )
27 pnfxr 11260 . . . . . . . . . . 11  |- +oo  e.  RR*
2827a1i 11 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  X  =  0 )  -> +oo  e.  RR* )
29 elunitrn 28064 . . . . . . . . . . . . . . 15  |-  ( X  e.  ( 0 [,] 1 )  ->  X  e.  RR )
3029adantr 463 . . . . . . . . . . . . . 14  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  X  e.  RR )
31 elunitge0 28066 . . . . . . . . . . . . . . . 16  |-  ( X  e.  ( 0 [,] 1 )  ->  0  <_  X )
3231adantr 463 . . . . . . . . . . . . . . 15  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  0  <_  X )
33 simpr 459 . . . . . . . . . . . . . . . 16  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -.  X  =  0 )
3433neqned 2595 . . . . . . . . . . . . . . 15  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  X  =/=  0 )
3530, 32, 34ne0gt0d 9651 . . . . . . . . . . . . . 14  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  0  <  X )
3630, 35elrpd 11192 . . . . . . . . . . . . 13  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  X  e.  RR+ )
3736relogcld 23114 . . . . . . . . . . . 12  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  ( log `  X )  e.  RR )
3837renegcld 9922 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -u ( log `  X )  e.  RR )
3938rexrd 9572 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -u ( log `  X )  e.  RR* )
4028, 39ifclda 3902 . . . . . . . . 9  |-  ( X  e.  ( 0 [,] 1 )  ->  if ( X  =  0 , +oo ,  -u ( log `  X ) )  e.  RR* )
4126, 40eqeltrd 2480 . . . . . . . 8  |-  ( X  e.  ( 0 [,] 1 )  ->  ( F `  X )  e.  RR* )
4241adantr 463 . . . . . . 7  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  X )  e.  RR* )
43 neeq1 2673 . . . . . . . . . 10  |-  ( +oo  =  if ( X  =  0 , +oo ,  -u ( log `  X
) )  ->  ( +oo  =/= -oo  <->  if ( X  =  0 , +oo ,  -u ( log `  X
) )  =/= -oo ) )
44 neeq1 2673 . . . . . . . . . 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 11265 . . . . . . . . . . 11  |- +oo  =/= -oo
4645a1i 11 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  X  =  0 )  -> +oo  =/= -oo )
4738renemnfd 9574 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -u ( log `  X )  =/= -oo )
4843, 44, 46, 47ifbothda 3905 . . . . . . . . 9  |-  ( X  e.  ( 0 [,] 1 )  ->  if ( X  =  0 , +oo ,  -u ( log `  X ) )  =/= -oo )
4926, 48eqnetrd 2685 . . . . . . . 8  |-  ( X  e.  ( 0 [,] 1 )  ->  ( F `  X )  =/= -oo )
5049adantr 463 . . . . . . 7  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  X )  =/= -oo )
51 xaddpnf1 11364 . . . . . . 7  |-  ( ( ( F `  X
)  e.  RR*  /\  ( F `  X )  =/= -oo )  ->  (
( F `  X
) +e +oo )  = +oo )
5242, 50, 51syl2anc 659 . . . . . 6  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( ( F `
 X ) +e +oo )  = +oo )
5319, 52syl5eq 2445 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( ( F `
 X ) +e ( F ` 
0 ) )  = +oo )
54 unitsscn 28063 . . . . . . . . 9  |-  ( 0 [,] 1 )  C_  CC
55 simpl 455 . . . . . . . . 9  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  X  e.  ( 0 [,] 1 ) )
5654, 55sseldi 3428 . . . . . . . 8  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  X  e.  CC )
5756mul01d 9708 . . . . . . 7  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( X  x.  0 )  =  0 )
5857fveq2d 5791 . . . . . 6  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  0
) )  =  ( F `  0 ) )
5958, 18syl6eq 2449 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  0
) )  = +oo )
6053, 59eqtr4d 2436 . . . 4  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( ( F `
 X ) +e ( F ` 
0 ) )  =  ( F `  ( X  x.  0 ) ) )
61 simpr 459 . . . . . 6  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  Y  =  0 )
6261fveq2d 5791 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  Y )  =  ( F `  0 ) )
6362oveq2d 6230 . . . 4  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( ( F `
 X ) +e ( F `  Y ) )  =  ( ( F `  X ) +e
( F `  0
) ) )
6461oveq2d 6230 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( X  x.  Y )  =  ( X  x.  0 ) )
6564fveq2d 5791 . . . 4  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  Y
) )  =  ( F `  ( X  x.  0 ) ) )
6660, 63, 653eqtr4rd 2444 . . 3  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  Y
) )  =  ( ( F `  X
) +e ( F `  Y ) ) )
676eleq2i 2470 . . . . . 6  |-  ( X  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
X  e.  ( 0 [,] 1 ) )
68 elun 3572 . . . . . 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 3982 . . . . . 6  |-  ( X  e.  { 0 }  ->  X  =  0 )
7170orim1i 515 . . . . 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 6224 . . . . . . . 8  |-  ( ( F `  0 ) +e ( F `
 Y ) )  =  ( +oo +e ( F `  Y ) )
74 simpr 459 . . . . . . . . . 10  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  ( 0 (,] 1 ) )
7515xrge0iifcv 28101 . . . . . . . . . . . 12  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  =  -u ( log `  Y
) )
76 0le0 10560 . . . . . . . . . . . . . . . . 17  |-  0  <_  0
77 ltpnf 11270 . . . . . . . . . . . . . . . . . 18  |-  ( 1  e.  RR  ->  1  < +oo )
782, 77ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  1  < +oo
79 iocssioo 11553 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 0  e.  RR*  /\ +oo  e.  RR* )  /\  (
0  <_  0  /\  1  < +oo ) )  -> 
( 0 (,] 1
)  C_  ( 0 (,) +oo ) )
801, 27, 76, 78, 79mp4an 671 . . . . . . . . . . . . . . . 16  |-  ( 0 (,] 1 )  C_  ( 0 (,) +oo )
81 ioorp 11541 . . . . . . . . . . . . . . . 16  |-  ( 0 (,) +oo )  = 
RR+
8280, 81sseqtri 3462 . . . . . . . . . . . . . . 15  |-  ( 0 (,] 1 )  C_  RR+
8382sseli 3426 . . . . . . . . . . . . . 14  |-  ( Y  e.  ( 0 (,] 1 )  ->  Y  e.  RR+ )
8483relogcld 23114 . . . . . . . . . . . . 13  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( log `  Y )  e.  RR )
8584renegcld 9922 . . . . . . . . . . . 12  |-  ( Y  e.  ( 0 (,] 1 )  ->  -u ( log `  Y )  e.  RR )
8675, 85eqeltrd 2480 . . . . . . . . . . 11  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  e.  RR )
8786rexrd 9572 . . . . . . . . . 10  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  e.  RR* )
8874, 87syl 16 . . . . . . . . 9  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  Y )  e.  RR* )
8986renemnfd 9574 . . . . . . . . . 10  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  =/= -oo )
9074, 89syl 16 . . . . . . . . 9  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  Y )  =/= -oo )
91 xaddpnf2 11365 . . . . . . . . 9  |-  ( ( ( F `  Y
)  e.  RR*  /\  ( F `  Y )  =/= -oo )  ->  ( +oo +e ( F `
 Y ) )  = +oo )
9288, 90, 91syl2anc 659 . . . . . . . 8  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( +oo +e ( F `  Y ) )  = +oo )
9373, 92syl5eq 2445 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 0 ) +e ( F `  Y ) )  = +oo )
94 rpssre 11167 . . . . . . . . . . . . 13  |-  RR+  C_  RR
9582, 94sstri 3439 . . . . . . . . . . . 12  |-  ( 0 (,] 1 )  C_  RR
96 ax-resscn 9478 . . . . . . . . . . . 12  |-  RR  C_  CC
9795, 96sstri 3439 . . . . . . . . . . 11  |-  ( 0 (,] 1 )  C_  CC
9897, 74sseldi 3428 . . . . . . . . . 10  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  CC )
9998mul02d 9707 . . . . . . . . 9  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( 0  x.  Y )  =  0 )
10099fveq2d 5791 . . . . . . . 8  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( 0  x.  Y
) )  =  ( F `  0 ) )
101100, 18syl6eq 2449 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( 0  x.  Y
) )  = +oo )
10293, 101eqtr4d 2436 . . . . . 6  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 0 ) +e ( F `  Y ) )  =  ( F `  (
0  x.  Y ) ) )
103 simpl 455 . . . . . . . 8  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  =  0 )
104103fveq2d 5791 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  X )  =  ( F `  0 ) )
105104oveq1d 6229 . . . . . 6  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 X ) +e ( F `  Y ) )  =  ( ( F ` 
0 ) +e
( F `  Y
) ) )
106103oveq1d 6229 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  =  ( 0  x.  Y ) )
107106fveq2d 5791 . . . . . 6  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  ( F `  ( 0  x.  Y ) ) )
108102, 105, 1073eqtr4rd 2444 . . . . 5  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  ( ( F `  X
) +e ( F `  Y ) ) )
109 simpl 455 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  ( 0 (,] 1 ) )
11082, 109sseldi 3428 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  RR+ )
111110relogcld 23114 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  X
)  e.  RR )
112111renegcld 9922 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( log `  X
)  e.  RR )
113 simpr 459 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  ( 0 (,] 1 ) )
11482, 113sseldi 3428 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  RR+ )
115114relogcld 23114 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  Y
)  e.  RR )
116115renegcld 9922 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( log `  Y
)  e.  RR )
117 rexadd 11370 . . . . . . 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 659 . . . . . 6  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( -u ( log `  X ) +e -u ( log `  Y ) )  =  ( -u ( log `  X )  +  -u ( log `  Y ) ) )
11915xrge0iifcv 28101 . . . . . . 7  |-  ( X  e.  ( 0 (,] 1 )  ->  ( F `  X )  =  -u ( log `  X
) )
120119, 75oveqan12d 6233 . . . . . 6  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 X ) +e ( F `  Y ) )  =  ( -u ( log `  X ) +e -u ( log `  Y
) ) )
121110rpred 11195 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  RR )
122114rpred 11195 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  RR )
123121, 122remulcld 9553 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  e.  RR )
124110rpgt0d 11198 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  0  <  X
)
125114rpgt0d 11198 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  0  <  Y
)
126121, 122, 124, 125mulgt0d 9666 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  0  <  ( X  x.  Y )
)
127 iocssicc 11551 . . . . . . . . . . . 12  |-  ( 0 (,] 1 )  C_  ( 0 [,] 1
)
128127, 109sseldi 3428 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  ( 0 [,] 1 ) )
129127, 113sseldi 3428 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  ( 0 [,] 1 ) )
130 iimulcl 21541 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  e.  ( 0 [,] 1 ) )  ->  ( X  x.  Y )  e.  ( 0 [,] 1 ) )
131128, 129, 130syl2anc 659 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  e.  ( 0 [,] 1 ) )
132 0re 9525 . . . . . . . . . . . 12  |-  0  e.  RR
133132, 2elicc2i 11529 . . . . . . . . . . 11  |-  ( ( X  x.  Y )  e.  ( 0 [,] 1 )  <->  ( ( X  x.  Y )  e.  RR  /\  0  <_ 
( X  x.  Y
)  /\  ( X  x.  Y )  <_  1
) )
134133simp3bi 1011 . . . . . . . . . 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 11526 . . . . . . . . . 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 670 . . . . . . . . 9  |-  ( ( X  x.  Y )  e.  ( 0 (,] 1 )  <->  ( ( X  x.  Y )  e.  RR  /\  0  < 
( X  x.  Y
)  /\  ( X  x.  Y )  <_  1
) )
138123, 126, 135, 137syl3anbrc 1178 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  e.  ( 0 (,] 1 ) )
13915xrge0iifcv 28101 . . . . . . . 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 23116 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  ( X  x.  Y )
)  =  ( ( log `  X )  +  ( log `  Y
) ) )
142141negeqd 9745 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( log `  ( X  x.  Y )
)  =  -u (
( log `  X
)  +  ( log `  Y ) ) )
143111recnd 9551 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  X
)  e.  CC )
144115recnd 9551 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  Y
)  e.  CC )
145143, 144negdid 9875 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( ( log `  X )  +  ( log `  Y ) )  =  ( -u ( log `  X )  +  -u ( log `  Y
) ) )
146140, 142, 1453eqtrd 2437 . . . . . 6  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  (
-u ( log `  X
)  +  -u ( log `  Y ) ) )
147118, 120, 1463eqtr4rd 2444 . . . . 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 469 . . 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 472 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 366    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836    =/= wne 2587    u. cun 3400    C_ wss 3402   ifcif 3870   {csn 3957   class class class wbr 4380    |-> cmpt 4438   ` cfv 5509  (class class class)co 6214   CCcc 9419   RRcr 9420   0cc0 9421   1c1 9422    + caddc 9424    x. cmul 9426   +oocpnf 9554   -oocmnf 9555   RR*cxr 9556    < clt 9557    <_ cle 9558   -ucneg 9737   RR+crp 11157   +ecxad 11255   (,)cioo 11468   (,]cioc 11469   [,]cicc 11471   ↾t crest 14847  ordTopcordt 14925   logclog 23046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-rep 4491  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509  ax-inf2 7990  ax-cnex 9477  ax-resscn 9478  ax-1cn 9479  ax-icn 9480  ax-addcl 9481  ax-addrcl 9482  ax-mulcl 9483  ax-mulrcl 9484  ax-mulcom 9485  ax-addass 9486  ax-mulass 9487  ax-distr 9488  ax-i2m1 9489  ax-1ne0 9490  ax-1rid 9491  ax-rnegex 9492  ax-rrecex 9493  ax-cnre 9494  ax-pre-lttri 9495  ax-pre-lttrn 9496  ax-pre-ltadd 9497  ax-pre-mulgt0 9498  ax-pre-sup 9499  ax-addf 9500  ax-mulf 9501
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-nel 2590  df-ral 2747  df-rex 2748  df-reu 2749  df-rmo 2750  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-pss 3418  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-tp 3962  df-op 3964  df-uni 4177  df-int 4213  df-iun 4258  df-iin 4259  df-br 4381  df-opab 4439  df-mpt 4440  df-tr 4474  df-eprel 4718  df-id 4722  df-po 4727  df-so 4728  df-fr 4765  df-se 4766  df-we 4767  df-ord 4808  df-on 4809  df-lim 4810  df-suc 4811  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-fv 5517  df-isom 5518  df-riota 6176  df-ov 6217  df-oprab 6218  df-mpt2 6219  df-of 6457  df-om 6618  df-1st 6717  df-2nd 6718  df-supp 6836  df-recs 6978  df-rdg 7012  df-1o 7066  df-2o 7067  df-oadd 7070  df-er 7247  df-map 7358  df-pm 7359  df-ixp 7407  df-en 7454  df-dom 7455  df-sdom 7456  df-fin 7457  df-fsupp 7763  df-fi 7804  df-sup 7834  df-oi 7868  df-card 8251  df-cda 8479  df-pnf 9559  df-mnf 9560  df-xr 9561  df-ltxr 9562  df-le 9563  df-sub 9738  df-neg 9739  df-div 10142  df-nn 10471  df-2 10529  df-3 10530  df-4 10531  df-5 10532  df-6 10533  df-7 10534  df-8 10535  df-9 10536  df-10 10537  df-n0 10731  df-z 10800  df-dec 10914  df-uz 11020  df-q 11120  df-rp 11158  df-xneg 11257  df-xadd 11258  df-xmul 11259  df-ioo 11472  df-ioc 11473  df-ico 11474  df-icc 11475  df-fz 11612  df-fzo 11736  df-fl 11847  df-mod 11916  df-seq 12030  df-exp 12089  df-fac 12275  df-bc 12302  df-hash 12327  df-shft 12921  df-cj 12953  df-re 12954  df-im 12955  df-sqrt 13089  df-abs 13090  df-limsup 13315  df-clim 13332  df-rlim 13333  df-sum 13530  df-ef 13824  df-sin 13826  df-cos 13827  df-pi 13829  df-struct 14655  df-ndx 14656  df-slot 14657  df-base 14658  df-sets 14659  df-ress 14660  df-plusg 14734  df-mulr 14735  df-starv 14736  df-sca 14737  df-vsca 14738  df-ip 14739  df-tset 14740  df-ple 14741  df-ds 14743  df-unif 14744  df-hom 14745  df-cco 14746  df-rest 14849  df-topn 14850  df-0g 14868  df-gsum 14869  df-topgen 14870  df-pt 14871  df-prds 14874  df-xrs 14928  df-qtop 14933  df-imas 14934  df-xps 14936  df-mre 15012  df-mrc 15013  df-acs 15015  df-mgm 16008  df-sgrp 16047  df-mnd 16057  df-submnd 16103  df-mulg 16196  df-cntz 16491  df-cmn 16936  df-psmet 18543  df-xmet 18544  df-met 18545  df-bl 18546  df-mopn 18547  df-fbas 18548  df-fg 18549  df-cnfld 18553  df-top 19503  df-bases 19505  df-topon 19506  df-topsp 19507  df-cld 19624  df-ntr 19625  df-cls 19626  df-nei 19704  df-lp 19742  df-perf 19743  df-cn 19833  df-cnp 19834  df-haus 19921  df-tx 20167  df-hmeo 20360  df-fil 20451  df-fm 20543  df-flim 20544  df-flf 20545  df-xms 20927  df-ms 20928  df-tms 20929  df-cncf 21486  df-limc 22374  df-dv 22375  df-log 23048
This theorem is referenced by:  xrge0iifmhm  28106  xrge0pluscn  28107
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