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Theorem xrge0iifhom 23321
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 8880 . . . . . . 7  |-  0  e.  RR*
2 1re 8839 . . . . . . . 8  |-  1  e.  RR
3 rexr 8879 . . . . . . . 8  |-  ( 1  e.  RR  ->  1  e.  RR* )
42, 3ax-mp 8 . . . . . . 7  |-  1  e.  RR*
5 0le1 9299 . . . . . . 7  |-  0  <_  1
6 snunioc 23269 . . . . . . 7  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  0  <_ 
1 )  ->  ( { 0 }  u.  ( 0 (,] 1
) )  =  ( 0 [,] 1 ) )
71, 4, 5, 6mp3an 1277 . . . . . 6  |-  ( { 0 }  u.  (
0 (,] 1 ) )  =  ( 0 [,] 1 )
87eleq2i 2349 . . . . 5  |-  ( Y  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
Y  e.  ( 0 [,] 1 ) )
9 elun 3318 . . . . 5  |-  ( Y  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
( Y  e.  {
0 }  \/  Y  e.  ( 0 (,] 1
) ) )
108, 9bitr3i 242 . . . 4  |-  ( Y  e.  ( 0 [,] 1 )  <->  ( Y  e.  { 0 }  \/  Y  e.  ( 0 (,] 1 ) ) )
11 elsni 3666 . . . . 5  |-  ( Y  e.  { 0 }  ->  Y  =  0 )
1211orim1i 503 . . . 4  |-  ( ( Y  e.  { 0 }  \/  Y  e.  ( 0 (,] 1
) )  ->  ( Y  =  0  \/  Y  e.  ( 0 (,] 1 ) ) )
1310, 12sylbi 187 . . 3  |-  ( Y  e.  ( 0 [,] 1 )  ->  ( Y  =  0  \/  Y  e.  ( 0 (,] 1 ) ) )
1413anim2i 552 . 2  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  e.  ( 0 [,] 1 ) )  ->  ( X  e.  ( 0 [,] 1
)  /\  ( Y  =  0  \/  Y  e.  ( 0 (,] 1
) ) ) )
15 0elunit 10756 . . . . . . . 8  |-  0  e.  ( 0 [,] 1
)
16 iftrue 3573 . . . . . . . . 9  |-  ( x  =  0  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  =  +oo )
17 xrge0iifhmeo.1 . . . . . . . . 9  |-  F  =  ( x  e.  ( 0 [,] 1 ) 
|->  if ( x  =  0 ,  +oo ,  -u ( log `  x
) ) )
18 pnfxr 10457 . . . . . . . . . 10  |-  +oo  e.  RR*
1918elexi 2799 . . . . . . . . 9  |-  +oo  e.  _V
2016, 17, 19fvmpt 5604 . . . . . . . 8  |-  ( 0  e.  ( 0 [,] 1 )  ->  ( F `  0 )  =  +oo )
2115, 20ax-mp 8 . . . . . . 7  |-  ( F `
 0 )  = 
+oo
2221oveq2i 5871 . . . . . 6  |-  ( ( F `  X ) + e ( F `
 0 ) )  =  ( ( F `
 X ) + e  +oo )
23 simpl 443 . . . . . . . 8  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  X  e.  ( 0 [,] 1 ) )
24 eqeq1 2291 . . . . . . . . . . 11  |-  ( x  =  X  ->  (
x  =  0  <->  X  =  0 ) )
25 fveq2 5527 . . . . . . . . . . . 12  |-  ( x  =  X  ->  ( log `  x )  =  ( log `  X
) )
2625negeqd 9048 . . . . . . . . . . 11  |-  ( x  =  X  ->  -u ( log `  x )  = 
-u ( log `  X
) )
2724, 26ifbieq2d 3587 . . . . . . . . . 10  |-  ( x  =  X  ->  if ( x  =  0 ,  +oo ,  -u ( log `  x ) )  =  if ( X  =  0 ,  +oo , 
-u ( log `  X
) ) )
28 negex 9052 . . . . . . . . . . 11  |-  -u ( log `  X )  e. 
_V
2919, 28ifex 3625 . . . . . . . . . 10  |-  if ( X  =  0 , 
+oo ,  -u ( log `  X ) )  e. 
_V
3027, 17, 29fvmpt 5604 . . . . . . . . 9  |-  ( X  e.  ( 0 [,] 1 )  ->  ( F `  X )  =  if ( X  =  0 ,  +oo ,  -u ( log `  X
) ) )
31 eleq1 2345 . . . . . . . . . 10  |-  (  +oo  =  if ( X  =  0 ,  +oo ,  -u ( log `  X
) )  ->  (  +oo  e.  RR*  <->  if ( X  =  0 ,  +oo ,  -u ( log `  X
) )  e.  RR* ) )
32 eleq1 2345 . . . . . . . . . 10  |-  ( -u ( log `  X )  =  if ( X  =  0 ,  +oo , 
-u ( log `  X
) )  ->  ( -u ( log `  X
)  e.  RR*  <->  if ( X  =  0 ,  +oo ,  -u ( log `  X
) )  e.  RR* ) )
3318a1i 10 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  X  =  0 )  ->  +oo  e.  RR* )
34 0re 8840 . . . . . . . . . . . . . . . . . 18  |-  0  e.  RR
35 elicc2 10717 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0  e.  RR  /\  1  e.  RR )  ->  ( X  e.  ( 0 [,] 1 )  <-> 
( X  e.  RR  /\  0  <_  X  /\  X  <_  1 ) ) )
3634, 2, 35mp2an 653 . . . . . . . . . . . . . . . . 17  |-  ( X  e.  ( 0 [,] 1 )  <->  ( X  e.  RR  /\  0  <_  X  /\  X  <_  1
) )
3736biimpi 186 . . . . . . . . . . . . . . . 16  |-  ( X  e.  ( 0 [,] 1 )  ->  ( X  e.  RR  /\  0  <_  X  /\  X  <_ 
1 ) )
3837simp1d 967 . . . . . . . . . . . . . . 15  |-  ( X  e.  ( 0 [,] 1 )  ->  X  e.  RR )
3938adantr 451 . . . . . . . . . . . . . 14  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  X  e.  RR )
4037simp2d 968 . . . . . . . . . . . . . . . 16  |-  ( X  e.  ( 0 [,] 1 )  ->  0  <_  X )
4140adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  0  <_  X )
42 simpr 447 . . . . . . . . . . . . . . . . 17  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -.  X  =  0 )
43 eqcom 2287 . . . . . . . . . . . . . . . . . 18  |-  ( 0  =  X  <->  X  = 
0 )
4443necon3abii 2478 . . . . . . . . . . . . . . . . 17  |-  ( 0  =/=  X  <->  -.  X  =  0 )
4542, 44sylibr 203 . . . . . . . . . . . . . . . 16  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  0  =/=  X )
4645necomd 2531 . . . . . . . . . . . . . . 15  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  X  =/=  0 )
4739, 41, 46ne0gt0d 8958 . . . . . . . . . . . . . 14  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  0  <  X )
4839, 47elrpd 10390 . . . . . . . . . . . . 13  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  X  e.  RR+ )
4948relogcld 19976 . . . . . . . . . . . 12  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  ( log `  X )  e.  RR )
5049renegcld 9212 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -u ( log `  X )  e.  RR )
5150rexrd 8883 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -u ( log `  X )  e.  RR* )
5231, 32, 33, 51ifbothda 3597 . . . . . . . . 9  |-  ( X  e.  ( 0 [,] 1 )  ->  if ( X  =  0 ,  +oo ,  -u ( log `  X ) )  e.  RR* )
5330, 52eqeltrd 2359 . . . . . . . 8  |-  ( X  e.  ( 0 [,] 1 )  ->  ( F `  X )  e.  RR* )
5423, 53syl 15 . . . . . . 7  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  X )  e.  RR* )
55 neeq1 2456 . . . . . . . . . 10  |-  (  +oo  =  if ( X  =  0 ,  +oo ,  -u ( log `  X
) )  ->  (  +oo  =/=  -oo  <->  if ( X  =  0 ,  +oo ,  -u ( log `  X
) )  =/=  -oo ) )
56 neeq1 2456 . . . . . . . . . 10  |-  ( -u ( log `  X )  =  if ( X  =  0 ,  +oo , 
-u ( log `  X
) )  ->  ( -u ( log `  X
)  =/=  -oo  <->  if ( X  =  0 ,  +oo ,  -u ( log `  X
) )  =/=  -oo ) )
57 pnfnemnf 10461 . . . . . . . . . . 11  |-  +oo  =/=  -oo
5857a1i 10 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  X  =  0 )  ->  +oo  =/=  -oo )
5950renemnfd 8885 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 [,] 1 )  /\  -.  X  =  0
)  ->  -u ( log `  X )  =/=  -oo )
6055, 56, 58, 59ifbothda 3597 . . . . . . . . 9  |-  ( X  e.  ( 0 [,] 1 )  ->  if ( X  =  0 ,  +oo ,  -u ( log `  X ) )  =/=  -oo )
6130, 60eqnetrd 2466 . . . . . . . 8  |-  ( X  e.  ( 0 [,] 1 )  ->  ( F `  X )  =/=  -oo )
6223, 61syl 15 . . . . . . 7  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  X )  =/=  -oo )
63 xaddpnf1 10555 . . . . . . 7  |-  ( ( ( F `  X
)  e.  RR*  /\  ( F `  X )  =/=  -oo )  ->  (
( F `  X
) + e  +oo )  =  +oo )
6454, 62, 63syl2anc 642 . . . . . 6  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( ( F `
 X ) + e  +oo )  = 
+oo )
6522, 64syl5eq 2329 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( ( F `
 X ) + e ( F ` 
0 ) )  = 
+oo )
66 unitsscn 23282 . . . . . . . . 9  |-  ( 0 [,] 1 )  C_  CC
6766, 23sseldi 3180 . . . . . . . 8  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  X  e.  CC )
6867mul01d 9013 . . . . . . 7  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( X  x.  0 )  =  0 )
6968fveq2d 5531 . . . . . 6  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  0
) )  =  ( F `  0 ) )
7069, 21syl6eq 2333 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  0
) )  =  +oo )
7165, 70eqtr4d 2320 . . . 4  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( ( F `
 X ) + e ( F ` 
0 ) )  =  ( F `  ( X  x.  0 ) ) )
72 simpr 447 . . . . . 6  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  Y  =  0 )
7372fveq2d 5531 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  Y )  =  ( F `  0 ) )
7473oveq2d 5876 . . . 4  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( ( F `
 X ) + e ( F `  Y ) )  =  ( ( F `  X ) + e
( F `  0
) ) )
7572oveq2d 5876 . . . . 5  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( X  x.  Y )  =  ( X  x.  0 ) )
7675fveq2d 5531 . . . 4  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  Y
) )  =  ( F `  ( X  x.  0 ) ) )
7771, 74, 763eqtr4rd 2328 . . 3  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  =  0 )  ->  ( F `  ( X  x.  Y
) )  =  ( ( F `  X
) + e ( F `  Y ) ) )
787eleq2i 2349 . . . . . . 7  |-  ( X  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
X  e.  ( 0 [,] 1 ) )
79 elun 3318 . . . . . . 7  |-  ( X  e.  ( { 0 }  u.  ( 0 (,] 1 ) )  <-> 
( X  e.  {
0 }  \/  X  e.  ( 0 (,] 1
) ) )
8078, 79bitr3i 242 . . . . . 6  |-  ( X  e.  ( 0 [,] 1 )  <->  ( X  e.  { 0 }  \/  X  e.  ( 0 (,] 1 ) ) )
81 elsni 3666 . . . . . . 7  |-  ( X  e.  { 0 }  ->  X  =  0 )
8281orim1i 503 . . . . . 6  |-  ( ( X  e.  { 0 }  \/  X  e.  ( 0 (,] 1
) )  ->  ( X  =  0  \/  X  e.  ( 0 (,] 1 ) ) )
8380, 82sylbi 187 . . . . 5  |-  ( X  e.  ( 0 [,] 1 )  ->  ( X  =  0  \/  X  e.  ( 0 (,] 1 ) ) )
8483anim1i 551 . . . 4  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( X  =  0  \/  X  e.  ( 0 (,] 1
) )  /\  Y  e.  ( 0 (,] 1
) ) )
8521oveq1i 5870 . . . . . . . 8  |-  ( ( F `  0 ) + e ( F `
 Y ) )  =  (  +oo + e ( F `  Y ) )
86 simpr 447 . . . . . . . . . 10  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  ( 0 (,] 1 ) )
8717xrge0iifcv 23318 . . . . . . . . . . . 12  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  =  -u ( log `  Y
) )
88 0le0 9829 . . . . . . . . . . . . . . . . 17  |-  0  <_  0
89 ltpnf 10465 . . . . . . . . . . . . . . . . . 18  |-  ( 1  e.  RR  ->  1  <  +oo )
902, 89ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  1  <  +oo
91 iocssioo 23263 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 0  e.  RR*  /\ 
+oo  e.  RR* )  /\  ( 0  <_  0  /\  1  <  +oo )
)  ->  ( 0 (,] 1 )  C_  ( 0 (,)  +oo ) )
921, 18, 88, 90, 91mp4an 654 . . . . . . . . . . . . . . . 16  |-  ( 0 (,] 1 )  C_  ( 0 (,)  +oo )
93 ioorp 10729 . . . . . . . . . . . . . . . 16  |-  ( 0 (,)  +oo )  =  RR+
9492, 93sseqtri 3212 . . . . . . . . . . . . . . 15  |-  ( 0 (,] 1 )  C_  RR+
9594sseli 3178 . . . . . . . . . . . . . 14  |-  ( Y  e.  ( 0 (,] 1 )  ->  Y  e.  RR+ )
9695relogcld 19976 . . . . . . . . . . . . 13  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( log `  Y )  e.  RR )
9796renegcld 9212 . . . . . . . . . . . 12  |-  ( Y  e.  ( 0 (,] 1 )  ->  -u ( log `  Y )  e.  RR )
9887, 97eqeltrd 2359 . . . . . . . . . . 11  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  e.  RR )
9998rexrd 8883 . . . . . . . . . 10  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  e.  RR* )
10086, 99syl 15 . . . . . . . . 9  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  Y )  e.  RR* )
10198renemnfd 8885 . . . . . . . . . 10  |-  ( Y  e.  ( 0 (,] 1 )  ->  ( F `  Y )  =/=  -oo )
10286, 101syl 15 . . . . . . . . 9  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  Y )  =/=  -oo )
103 xaddpnf2 10556 . . . . . . . . 9  |-  ( ( ( F `  Y
)  e.  RR*  /\  ( F `  Y )  =/=  -oo )  ->  (  +oo + e ( F `
 Y ) )  =  +oo )
104100, 102, 103syl2anc 642 . . . . . . . 8  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  (  +oo + e ( F `  Y ) )  = 
+oo )
10585, 104syl5eq 2329 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 0 ) + e ( F `  Y ) )  = 
+oo )
106 rpssre 10366 . . . . . . . . . . . . 13  |-  RR+  C_  RR
10794, 106sstri 3190 . . . . . . . . . . . 12  |-  ( 0 (,] 1 )  C_  RR
108 ax-resscn 8796 . . . . . . . . . . . 12  |-  RR  C_  CC
109107, 108sstri 3190 . . . . . . . . . . 11  |-  ( 0 (,] 1 )  C_  CC
110109, 86sseldi 3180 . . . . . . . . . 10  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  CC )
111110mul02d 9012 . . . . . . . . 9  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( 0  x.  Y )  =  0 )
112111fveq2d 5531 . . . . . . . 8  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( 0  x.  Y
) )  =  ( F `  0 ) )
113112, 21syl6eq 2333 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( 0  x.  Y
) )  =  +oo )
114105, 113eqtr4d 2320 . . . . . 6  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 0 ) + e ( F `  Y ) )  =  ( F `  (
0  x.  Y ) ) )
115 simpl 443 . . . . . . . 8  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  =  0 )
116115fveq2d 5531 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  X )  =  ( F `  0 ) )
117116oveq1d 5875 . . . . . 6  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 X ) + e ( F `  Y ) )  =  ( ( F ` 
0 ) + e
( F `  Y
) ) )
118115oveq1d 5875 . . . . . . 7  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  =  ( 0  x.  Y ) )
119118fveq2d 5531 . . . . . 6  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  ( F `  ( 0  x.  Y ) ) )
120114, 117, 1193eqtr4rd 2328 . . . . 5  |-  ( ( X  =  0  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  ( ( F `  X
) + e ( F `  Y ) ) )
121 simpl 443 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  ( 0 (,] 1 ) )
12294, 121sseldi 3180 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  RR+ )
123 relogcl 19934 . . . . . . . . 9  |-  ( X  e.  RR+  ->  ( log `  X )  e.  RR )
124122, 123syl 15 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  X
)  e.  RR )
125124renegcld 9212 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( log `  X
)  e.  RR )
126 simpr 447 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  ( 0 (,] 1 ) )
12794, 126sseldi 3180 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  RR+ )
128 relogcl 19934 . . . . . . . . 9  |-  ( Y  e.  RR+  ->  ( log `  Y )  e.  RR )
129127, 128syl 15 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  Y
)  e.  RR )
130129renegcld 9212 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( log `  Y
)  e.  RR )
131 rexadd 10561 . . . . . . 7  |-  ( (
-u ( log `  X
)  e.  RR  /\  -u ( log `  Y
)  e.  RR )  ->  ( -u ( log `  X ) + e -u ( log `  Y ) )  =  ( -u ( log `  X )  +  -u ( log `  Y ) ) )
132125, 130, 131syl2anc 642 . . . . . 6  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( -u ( log `  X ) + e -u ( log `  Y ) )  =  ( -u ( log `  X )  +  -u ( log `  Y ) ) )
13317xrge0iifcv 23318 . . . . . . . 8  |-  ( X  e.  ( 0 (,] 1 )  ->  ( F `  X )  =  -u ( log `  X
) )
134133adantr 451 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  X )  =  -u ( log `  X ) )
13587adantl 452 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  Y )  =  -u ( log `  Y ) )
136134, 135oveq12d 5878 . . . . . 6  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( ( F `
 X ) + e ( F `  Y ) )  =  ( -u ( log `  X ) + e -u ( log `  Y
) ) )
137122rpred 10392 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  RR )
138127rpred 10392 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  RR )
139137, 138remulcld 8865 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  e.  RR )
140122rpgt0d 10395 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  0  <  X
)
141127rpgt0d 10395 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  0  <  Y
)
142137, 138, 140, 141mulgt0d 8973 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  0  <  ( X  x.  Y )
)
143 iocssicc 23261 . . . . . . . . . . . 12  |-  ( 0 (,] 1 )  C_  ( 0 [,] 1
)
144143, 121sseldi 3180 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  X  e.  ( 0 [,] 1 ) )
145143, 126sseldi 3180 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  Y  e.  ( 0 [,] 1 ) )
146 iimulcl 18437 . . . . . . . . . . 11  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  e.  ( 0 [,] 1 ) )  ->  ( X  x.  Y )  e.  ( 0 [,] 1 ) )
147144, 145, 146syl2anc 642 . . . . . . . . . 10  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  e.  ( 0 [,] 1 ) )
148 elicc2 10717 . . . . . . . . . . . . 13  |-  ( ( 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 ) ) )
14934, 2, 148mp2an 653 . . . . . . . . . . . 12  |-  ( ( X  x.  Y )  e.  ( 0 [,] 1 )  <->  ( ( X  x.  Y )  e.  RR  /\  0  <_ 
( X  x.  Y
)  /\  ( X  x.  Y )  <_  1
) )
150149biimpi 186 . . . . . . . . . . 11  |-  ( ( X  x.  Y )  e.  ( 0 [,] 1 )  ->  (
( X  x.  Y
)  e.  RR  /\  0  <_  ( X  x.  Y )  /\  ( X  x.  Y )  <_  1 ) )
151150simp3d 969 . . . . . . . . . 10  |-  ( ( X  x.  Y )  e.  ( 0 [,] 1 )  ->  ( X  x.  Y )  <_  1 )
152147, 151syl 15 . . . . . . . . 9  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  <_  1
)
153 elioc2 10715 . . . . . . . . . 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
) ) )
1541, 2, 153mp2an 653 . . . . . . . . 9  |-  ( ( X  x.  Y )  e.  ( 0 (,] 1 )  <->  ( ( X  x.  Y )  e.  RR  /\  0  < 
( X  x.  Y
)  /\  ( X  x.  Y )  <_  1
) )
155139, 142, 152, 154syl3anbrc 1136 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( X  x.  Y )  e.  ( 0 (,] 1 ) )
15617xrge0iifcv 23318 . . . . . . . 8  |-  ( ( X  x.  Y )  e.  ( 0 (,] 1 )  ->  ( F `  ( X  x.  Y ) )  = 
-u ( log `  ( X  x.  Y )
) )
157155, 156syl 15 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  -u ( log `  ( X  x.  Y ) ) )
158 relogmul 19947 . . . . . . . . 9  |-  ( ( X  e.  RR+  /\  Y  e.  RR+ )  ->  ( log `  ( X  x.  Y ) )  =  ( ( log `  X
)  +  ( log `  Y ) ) )
159122, 127, 158syl2anc 642 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  ( X  x.  Y )
)  =  ( ( log `  X )  +  ( log `  Y
) ) )
160159negeqd 9048 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( log `  ( X  x.  Y )
)  =  -u (
( log `  X
)  +  ( log `  Y ) ) )
161124recnd 8863 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  X
)  e.  CC )
162129recnd 8863 . . . . . . . 8  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( log `  Y
)  e.  CC )
163 negdi 9106 . . . . . . . 8  |-  ( ( ( log `  X
)  e.  CC  /\  ( log `  Y )  e.  CC )  ->  -u ( ( log `  X
)  +  ( log `  Y ) )  =  ( -u ( log `  X )  +  -u ( log `  Y ) ) )
164161, 162, 163syl2anc 642 . . . . . . 7  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  -u ( ( log `  X )  +  ( log `  Y ) )  =  ( -u ( log `  X )  +  -u ( log `  Y
) ) )
165157, 160, 1643eqtrd 2321 . . . . . 6  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  (
-u ( log `  X
)  +  -u ( log `  Y ) ) )
166132, 136, 1653eqtr4rd 2328 . . . . 5  |-  ( ( X  e.  ( 0 (,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  ( ( F `  X
) + e ( F `  Y ) ) )
167120, 166jaoian 759 . . . 4  |-  ( ( ( X  =  0  \/  X  e.  ( 0 (,] 1 ) )  /\  Y  e.  ( 0 (,] 1
) )  ->  ( F `  ( X  x.  Y ) )  =  ( ( F `  X ) + e
( F `  Y
) ) )
16884, 167syl 15 . . 3  |-  ( ( X  e.  ( 0 [,] 1 )  /\  Y  e.  ( 0 (,] 1 ) )  ->  ( F `  ( X  x.  Y
) )  =  ( ( F `  X
) + e ( F `  Y ) ) )
16977, 168jaodan 760 . 2  |-  ( ( X  e.  ( 0 [,] 1 )  /\  ( Y  =  0  \/  Y  e.  (
0 (,] 1 ) ) )  ->  ( F `  ( X  x.  Y ) )  =  ( ( F `  X ) + e
( F `  Y
) ) )
17014, 169syl 15 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448    u. cun 3152    C_ wss 3154   ifcif 3567   {csn 3642   class class class wbr 4025    e. cmpt 4079   ` cfv 5257  (class class class)co 5860   CCcc 8737   RRcr 8738   0cc0 8739   1c1 8740    + caddc 8742    x. cmul 8744    +oocpnf 8866    -oocmnf 8867   RR*cxr 8868    < clt 8869    <_ cle 8870   -ucneg 9040   RR+crp 10356   + ecxad 10452   (,)cioo 10658   (,]cioc 10659   [,]cicc 10661   ↾t crest 13327  ordTopcordt 13400   logclog 19914
This theorem is referenced by:  xrge0iifmhm  23323  xrge0pluscn  23324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ioo 10662  df-ioc 10663  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-fl 10927  df-mod 10976  df-seq 11049  df-exp 11107  df-fac 11291  df-bc 11318  df-hash 11340  df-shft 11564  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-limsup 11947  df-clim 11964  df-rlim 11965  df-sum 12161  df-ef 12351  df-sin 12353  df-cos 12354  df-pi 12356  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-rest 13329  df-topn 13330  df-topgen 13346  df-pt 13347  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-qtop 13412  df-imas 13413  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-submnd 14418  df-mulg 14494  df-cntz 14795  df-cmn 15093  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-cnfld 16380  df-top 16638  df-bases 16640  df-topon 16641  df-topsp 16642  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-lp 16870  df-perf 16871  df-cn 16959  df-cnp 16960  df-haus 17045  df-tx 17259  df-hmeo 17448  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-xms 17887  df-ms 17888  df-tms 17889  df-cncf 18384  df-limc 19218  df-dv 19219  df-log 19916
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