Metamath Proof Explorer

Theorem imasdsfn

Description: The distance function is a function on the base set. (Contributed by Mario Carneiro, 20-Aug-2015) (Proof shortened by AV, 6-Oct-2020)

Ref Expression
Hypotheses imasbas.u 
|- ( ph -> U = ( F "s R ) )
imasbas.v 
|- ( ph -> V = ( Base ` R ) )
imasbas.f 
|- ( ph -> F : V -onto-> B )
imasbas.r 
|- ( ph -> R e. Z )
imasds.e 
|- E = ( dist ` R )
imasds.d 
|- D = ( dist ` U )
Assertion imasdsfn 
|- ( ph -> D Fn ( B X. B ) )

Proof

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 eqid 
 |-  ( 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* , < ) ) = ( 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* , < ) )
8 xrltso 
 |-  < Or RR*
9 8 infex 
 |-  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* , < ) e. _V
10 7 9 fnmpoi 
 |-  ( 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* , < ) ) Fn ( B X. B )
11 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* , < ) ) )
12 11 fneq1d 
 |-  ( ph -> ( D Fn ( B X. B ) <-> ( 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* , < ) ) Fn ( B X. B ) ) )
13 10 12 mpbiri 
 |-  ( ph -> D Fn ( B X. B ) )