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 |
1 2 3 4 5 6
|
imasds |
|- ( ph -> D = ( x e. B , y e. B |-> inf ( U_ n e. NN ran ( g e. { 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 ) ) ) ) ) } |-> ( RR*s gsum ( E o. g ) ) ) , RR* , < ) ) ) |
11 |
|
simplrl |
|- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ n e. NN ) -> x = X ) |
12 |
11
|
eqeq2d |
|- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ n e. NN ) -> ( ( F ` ( 1st ` ( h ` 1 ) ) ) = x <-> ( F ` ( 1st ` ( h ` 1 ) ) ) = X ) ) |
13 |
|
simplrr |
|- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ n e. NN ) -> y = Y ) |
14 |
13
|
eqeq2d |
|- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ n e. NN ) -> ( ( F ` ( 2nd ` ( h ` n ) ) ) = y <-> ( F ` ( 2nd ` ( h ` n ) ) ) = Y ) ) |
15 |
12 14
|
3anbi12d |
|- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ n e. NN ) -> ( ( ( 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 ) ) ) ) ) <-> ( ( 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 ) ) ) ) ) ) ) |
16 |
15
|
rabbidv |
|- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ n e. NN ) -> { 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 ) ) ) ) ) } = { 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 ) ) ) ) ) } ) |
17 |
16 9
|
eqtr4di |
|- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ n e. NN ) -> { 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 ) ) ) ) ) } = S ) |
18 |
17
|
mpteq1d |
|- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ n e. NN ) -> ( g e. { 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 ) ) ) ) ) } |-> ( RR*s gsum ( E o. g ) ) ) = ( g e. S |-> ( RR*s gsum ( E o. g ) ) ) ) |
19 |
18
|
rneqd |
|- ( ( ( ph /\ ( x = X /\ y = Y ) ) /\ n e. NN ) -> ran ( g e. { 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 ) ) ) ) ) } |-> ( RR*s gsum ( E o. g ) ) ) = ran ( g e. S |-> ( RR*s gsum ( E o. g ) ) ) ) |
20 |
19
|
iuneq2dv |
|- ( ( ph /\ ( x = X /\ y = Y ) ) -> U_ n e. NN ran ( g e. { 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 ) ) ) ) ) } |-> ( RR*s gsum ( E o. g ) ) ) = U_ n e. NN ran ( g e. S |-> ( RR*s gsum ( E o. g ) ) ) ) |
21 |
20
|
infeq1d |
|- ( ( ph /\ ( x = X /\ y = Y ) ) -> inf ( U_ n e. NN ran ( g e. { 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 ) ) ) ) ) } |-> ( RR*s gsum ( E o. g ) ) ) , RR* , < ) = inf ( U_ n e. NN ran ( g e. S |-> ( RR*s gsum ( E o. g ) ) ) , RR* , < ) ) |
22 |
|
xrltso |
|- < Or RR* |
23 |
22
|
infex |
|- inf ( U_ n e. NN ran ( g e. S |-> ( RR*s gsum ( E o. g ) ) ) , RR* , < ) e. _V |
24 |
23
|
a1i |
|- ( ph -> inf ( U_ n e. NN ran ( g e. S |-> ( RR*s gsum ( E o. g ) ) ) , RR* , < ) e. _V ) |
25 |
10 21 7 8 24
|
ovmpod |
|- ( ph -> ( X D Y ) = inf ( U_ n e. NN ran ( g e. S |-> ( RR*s gsum ( E o. g ) ) ) , RR* , < ) ) |