Step |
Hyp |
Ref |
Expression |
1 |
|
prdsbasmpt.y |
|- Y = ( S Xs_ R ) |
2 |
|
prdsbasmpt.b |
|- B = ( Base ` Y ) |
3 |
|
prdsbasmpt.s |
|- ( ph -> S e. V ) |
4 |
|
prdsbasmpt.i |
|- ( ph -> I e. W ) |
5 |
|
prdsbasmpt.r |
|- ( ph -> R Fn I ) |
6 |
|
prdsplusgval.f |
|- ( ph -> F e. B ) |
7 |
|
prdsplusgval.g |
|- ( ph -> G e. B ) |
8 |
|
prdsdsval.d |
|- D = ( dist ` Y ) |
9 |
|
fnex |
|- ( ( R Fn I /\ I e. W ) -> R e. _V ) |
10 |
5 4 9
|
syl2anc |
|- ( ph -> R e. _V ) |
11 |
|
fndm |
|- ( R Fn I -> dom R = I ) |
12 |
5 11
|
syl |
|- ( ph -> dom R = I ) |
13 |
1 3 10 2 12 8
|
prdsds |
|- ( ph -> D = ( f e. B , g e. B |-> sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) ) |
14 |
|
fveq1 |
|- ( f = F -> ( f ` x ) = ( F ` x ) ) |
15 |
|
fveq1 |
|- ( g = G -> ( g ` x ) = ( G ` x ) ) |
16 |
14 15
|
oveqan12d |
|- ( ( f = F /\ g = G ) -> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) = ( ( F ` x ) ( dist ` ( R ` x ) ) ( G ` x ) ) ) |
17 |
16
|
adantl |
|- ( ( ph /\ ( f = F /\ g = G ) ) -> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) = ( ( F ` x ) ( dist ` ( R ` x ) ) ( G ` x ) ) ) |
18 |
17
|
mpteq2dv |
|- ( ( ph /\ ( f = F /\ g = G ) ) -> ( x e. I |-> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) = ( x e. I |-> ( ( F ` x ) ( dist ` ( R ` x ) ) ( G ` x ) ) ) ) |
19 |
18
|
rneqd |
|- ( ( ph /\ ( f = F /\ g = G ) ) -> ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) = ran ( x e. I |-> ( ( F ` x ) ( dist ` ( R ` x ) ) ( G ` x ) ) ) ) |
20 |
19
|
uneq1d |
|- ( ( ph /\ ( f = F /\ g = G ) ) -> ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) u. { 0 } ) = ( ran ( x e. I |-> ( ( F ` x ) ( dist ` ( R ` x ) ) ( G ` x ) ) ) u. { 0 } ) ) |
21 |
20
|
supeq1d |
|- ( ( ph /\ ( f = F /\ g = G ) ) -> sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) = sup ( ( ran ( x e. I |-> ( ( F ` x ) ( dist ` ( R ` x ) ) ( G ` x ) ) ) u. { 0 } ) , RR* , < ) ) |
22 |
|
xrltso |
|- < Or RR* |
23 |
22
|
supex |
|- sup ( ( ran ( x e. I |-> ( ( F ` x ) ( dist ` ( R ` x ) ) ( G ` x ) ) ) u. { 0 } ) , RR* , < ) e. _V |
24 |
23
|
a1i |
|- ( ph -> sup ( ( ran ( x e. I |-> ( ( F ` x ) ( dist ` ( R ` x ) ) ( G ` x ) ) ) u. { 0 } ) , RR* , < ) e. _V ) |
25 |
13 21 6 7 24
|
ovmpod |
|- ( ph -> ( F D G ) = sup ( ( ran ( x e. I |-> ( ( F ` x ) ( dist ` ( R ` x ) ) ( G ` x ) ) ) u. { 0 } ) , RR* , < ) ) |