Step |
Hyp |
Ref |
Expression |
1 |
|
xrge0cmn |
|- ( RR*s |`s ( 0 [,] +oo ) ) e. CMnd |
2 |
|
cmnmnd |
|- ( ( RR*s |`s ( 0 [,] +oo ) ) e. CMnd -> ( RR*s |`s ( 0 [,] +oo ) ) e. Mnd ) |
3 |
1 2
|
ax-mp |
|- ( RR*s |`s ( 0 [,] +oo ) ) e. Mnd |
4 |
|
xrge0tps |
|- ( RR*s |`s ( 0 [,] +oo ) ) e. TopSp |
5 |
|
eqeq1 |
|- ( y = x -> ( y = 0 <-> x = 0 ) ) |
6 |
|
fveq2 |
|- ( y = x -> ( log ` y ) = ( log ` x ) ) |
7 |
6
|
negeqd |
|- ( y = x -> -u ( log ` y ) = -u ( log ` x ) ) |
8 |
5 7
|
ifbieq2d |
|- ( y = x -> if ( y = 0 , +oo , -u ( log ` y ) ) = if ( x = 0 , +oo , -u ( log ` x ) ) ) |
9 |
8
|
cbvmptv |
|- ( y e. ( 0 [,] 1 ) |-> if ( y = 0 , +oo , -u ( log ` y ) ) ) = ( x e. ( 0 [,] 1 ) |-> if ( x = 0 , +oo , -u ( log ` x ) ) ) |
10 |
|
eqid |
|- ( ( ordTop ` <_ ) |`t ( 0 [,] +oo ) ) = ( ( ordTop ` <_ ) |`t ( 0 [,] +oo ) ) |
11 |
|
eqid |
|- ( +e |` ( ( 0 [,] +oo ) X. ( 0 [,] +oo ) ) ) = ( +e |` ( ( 0 [,] +oo ) X. ( 0 [,] +oo ) ) ) |
12 |
9 10 11
|
xrge0pluscn |
|- ( +e |` ( ( 0 [,] +oo ) X. ( 0 [,] +oo ) ) ) e. ( ( ( ( ordTop ` <_ ) |`t ( 0 [,] +oo ) ) tX ( ( ordTop ` <_ ) |`t ( 0 [,] +oo ) ) ) Cn ( ( ordTop ` <_ ) |`t ( 0 [,] +oo ) ) ) |
13 |
|
xrsbas |
|- RR* = ( Base ` RR*s ) |
14 |
|
eqid |
|- ( RR*s |`s ( 0 [,] +oo ) ) = ( RR*s |`s ( 0 [,] +oo ) ) |
15 |
|
xrsadd |
|- +e = ( +g ` RR*s ) |
16 |
|
xaddf |
|- +e : ( RR* X. RR* ) --> RR* |
17 |
|
ffn |
|- ( +e : ( RR* X. RR* ) --> RR* -> +e Fn ( RR* X. RR* ) ) |
18 |
16 17
|
ax-mp |
|- +e Fn ( RR* X. RR* ) |
19 |
|
iccssxr |
|- ( 0 [,] +oo ) C_ RR* |
20 |
13 14 15 18 19
|
ressplusf |
|- ( +f ` ( RR*s |`s ( 0 [,] +oo ) ) ) = ( +e |` ( ( 0 [,] +oo ) X. ( 0 [,] +oo ) ) ) |
21 |
20
|
eqcomi |
|- ( +e |` ( ( 0 [,] +oo ) X. ( 0 [,] +oo ) ) ) = ( +f ` ( RR*s |`s ( 0 [,] +oo ) ) ) |
22 |
|
xrge0base |
|- ( 0 [,] +oo ) = ( Base ` ( RR*s |`s ( 0 [,] +oo ) ) ) |
23 |
|
ovex |
|- ( 0 [,] +oo ) e. _V |
24 |
|
xrstset |
|- ( ordTop ` <_ ) = ( TopSet ` RR*s ) |
25 |
14 24
|
resstset |
|- ( ( 0 [,] +oo ) e. _V -> ( ordTop ` <_ ) = ( TopSet ` ( RR*s |`s ( 0 [,] +oo ) ) ) ) |
26 |
23 25
|
ax-mp |
|- ( ordTop ` <_ ) = ( TopSet ` ( RR*s |`s ( 0 [,] +oo ) ) ) |
27 |
22 26
|
topnval |
|- ( ( ordTop ` <_ ) |`t ( 0 [,] +oo ) ) = ( TopOpen ` ( RR*s |`s ( 0 [,] +oo ) ) ) |
28 |
21 27
|
istmd |
|- ( ( RR*s |`s ( 0 [,] +oo ) ) e. TopMnd <-> ( ( RR*s |`s ( 0 [,] +oo ) ) e. Mnd /\ ( RR*s |`s ( 0 [,] +oo ) ) e. TopSp /\ ( +e |` ( ( 0 [,] +oo ) X. ( 0 [,] +oo ) ) ) e. ( ( ( ( ordTop ` <_ ) |`t ( 0 [,] +oo ) ) tX ( ( ordTop ` <_ ) |`t ( 0 [,] +oo ) ) ) Cn ( ( ordTop ` <_ ) |`t ( 0 [,] +oo ) ) ) ) ) |
29 |
3 4 12 28
|
mpbir3an |
|- ( RR*s |`s ( 0 [,] +oo ) ) e. TopMnd |