Step |
Hyp |
Ref |
Expression |
1 |
|
imasbas.u |
|- ( ph -> U = ( F "s R ) ) |
2 |
|
imasbas.v |
|- ( ph -> V = ( Base ` R ) ) |
3 |
|
imasbas.f |
|- ( ph -> F : V -onto-> B ) |
4 |
|
imasbas.r |
|- ( ph -> R e. Z ) |
5 |
|
imasds.e |
|- E = ( dist ` R ) |
6 |
|
imasds.d |
|- D = ( dist ` U ) |
7 |
|
imasdsval.x |
|- ( ph -> X e. B ) |
8 |
|
imasdsval.y |
|- ( ph -> Y e. B ) |
9 |
|
imasdsval.s |
|- S = { h e. ( ( V X. V ) ^m ( 1 ... n ) ) | ( ( F ` ( 1st ` ( h ` 1 ) ) ) = X /\ ( F ` ( 2nd ` ( h ` n ) ) ) = Y /\ A. i e. ( 1 ... ( n - 1 ) ) ( F ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |
10 |
|
imasds.u |
|- T = ( E |` ( V X. V ) ) |
11 |
1 2 3 4 5 6 7 8 9
|
imasdsval |
|- ( ph -> ( X D Y ) = inf ( U_ n e. NN ran ( g e. S |-> ( RR*s gsum ( E o. g ) ) ) , RR* , < ) ) |
12 |
10
|
coeq1i |
|- ( T o. g ) = ( ( E |` ( V X. V ) ) o. g ) |
13 |
9
|
ssrab3 |
|- S C_ ( ( V X. V ) ^m ( 1 ... n ) ) |
14 |
13
|
sseli |
|- ( g e. S -> g e. ( ( V X. V ) ^m ( 1 ... n ) ) ) |
15 |
|
elmapi |
|- ( g e. ( ( V X. V ) ^m ( 1 ... n ) ) -> g : ( 1 ... n ) --> ( V X. V ) ) |
16 |
|
frn |
|- ( g : ( 1 ... n ) --> ( V X. V ) -> ran g C_ ( V X. V ) ) |
17 |
|
cores |
|- ( ran g C_ ( V X. V ) -> ( ( E |` ( V X. V ) ) o. g ) = ( E o. g ) ) |
18 |
14 15 16 17
|
4syl |
|- ( g e. S -> ( ( E |` ( V X. V ) ) o. g ) = ( E o. g ) ) |
19 |
12 18
|
eqtrid |
|- ( g e. S -> ( T o. g ) = ( E o. g ) ) |
20 |
19
|
oveq2d |
|- ( g e. S -> ( RR*s gsum ( T o. g ) ) = ( RR*s gsum ( E o. g ) ) ) |
21 |
20
|
mpteq2ia |
|- ( g e. S |-> ( RR*s gsum ( T o. g ) ) ) = ( g e. S |-> ( RR*s gsum ( E o. g ) ) ) |
22 |
21
|
rneqi |
|- ran ( g e. S |-> ( RR*s gsum ( T o. g ) ) ) = ran ( g e. S |-> ( RR*s gsum ( E o. g ) ) ) |
23 |
22
|
a1i |
|- ( n e. NN -> ran ( g e. S |-> ( RR*s gsum ( T o. g ) ) ) = ran ( g e. S |-> ( RR*s gsum ( E o. g ) ) ) ) |
24 |
23
|
iuneq2i |
|- U_ n e. NN ran ( g e. S |-> ( RR*s gsum ( T o. g ) ) ) = U_ n e. NN ran ( g e. S |-> ( RR*s gsum ( E o. g ) ) ) |
25 |
24
|
infeq1i |
|- inf ( U_ n e. NN ran ( g e. S |-> ( RR*s gsum ( T o. g ) ) ) , RR* , < ) = inf ( U_ n e. NN ran ( g e. S |-> ( RR*s gsum ( E o. g ) ) ) , RR* , < ) |
26 |
11 25
|
eqtr4di |
|- ( ph -> ( X D Y ) = inf ( U_ n e. NN ran ( g e. S |-> ( RR*s gsum ( T o. g ) ) ) , RR* , < ) ) |