Step |
Hyp |
Ref |
Expression |
1 |
|
imasdsf1o.u |
|- ( ph -> U = ( F "s R ) ) |
2 |
|
imasdsf1o.v |
|- ( ph -> V = ( Base ` R ) ) |
3 |
|
imasdsf1o.f |
|- ( ph -> F : V -1-1-onto-> B ) |
4 |
|
imasdsf1o.r |
|- ( ph -> R e. Z ) |
5 |
|
imasdsf1o.e |
|- E = ( ( dist ` R ) |` ( V X. V ) ) |
6 |
|
imasdsf1o.d |
|- D = ( dist ` U ) |
7 |
|
imasdsf1o.m |
|- ( ph -> E e. ( *Met ` V ) ) |
8 |
|
imasdsf1o.x |
|- ( ph -> X e. V ) |
9 |
|
imasdsf1o.y |
|- ( ph -> Y e. V ) |
10 |
|
eqid |
|- ( RR*s |`s ( RR* \ { -oo } ) ) = ( RR*s |`s ( RR* \ { -oo } ) ) |
11 |
|
eqid |
|- { h e. ( ( V X. V ) ^m ( 1 ... n ) ) | ( ( F ` ( 1st ` ( h ` 1 ) ) ) = ( F ` X ) /\ ( F ` ( 2nd ` ( h ` n ) ) ) = ( F ` Y ) /\ A. i e. ( 1 ... ( n - 1 ) ) ( F ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } = { h e. ( ( V X. V ) ^m ( 1 ... n ) ) | ( ( F ` ( 1st ` ( h ` 1 ) ) ) = ( F ` X ) /\ ( F ` ( 2nd ` ( h ` n ) ) ) = ( F ` Y ) /\ A. i e. ( 1 ... ( n - 1 ) ) ( F ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |
12 |
|
eqid |
|- U_ n e. NN ran ( g e. { h e. ( ( V X. V ) ^m ( 1 ... n ) ) | ( ( F ` ( 1st ` ( h ` 1 ) ) ) = ( F ` X ) /\ ( F ` ( 2nd ` ( h ` n ) ) ) = ( F ` Y ) /\ A. i e. ( 1 ... ( n - 1 ) ) ( F ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsum ( E o. g ) ) ) = U_ n e. NN ran ( g e. { h e. ( ( V X. V ) ^m ( 1 ... n ) ) | ( ( F ` ( 1st ` ( h ` 1 ) ) ) = ( F ` X ) /\ ( F ` ( 2nd ` ( h ` n ) ) ) = ( F ` Y ) /\ A. i e. ( 1 ... ( n - 1 ) ) ( F ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsum ( E o. g ) ) ) |
13 |
1 2 3 4 5 6 7 8 9 10 11 12
|
imasdsf1olem |
|- ( ph -> ( ( F ` X ) D ( F ` Y ) ) = ( X E Y ) ) |