Step |
Hyp |
Ref |
Expression |
1 |
|
prdsbasmpt2.y |
|- Y = ( S Xs_ ( x e. I |-> R ) ) |
2 |
|
prdsbasmpt2.b |
|- B = ( Base ` Y ) |
3 |
|
prdsbasmpt2.s |
|- ( ph -> S e. V ) |
4 |
|
prdsbasmpt2.i |
|- ( ph -> I e. W ) |
5 |
|
prdsbasmpt2.r |
|- ( ph -> A. x e. I R e. X ) |
6 |
|
prdsdsval2.f |
|- ( ph -> F e. B ) |
7 |
|
prdsdsval2.g |
|- ( ph -> G e. B ) |
8 |
|
prdsdsval3.k |
|- K = ( Base ` R ) |
9 |
|
prdsdsval3.e |
|- E = ( ( dist ` R ) |` ( K X. K ) ) |
10 |
|
prdsdsval3.d |
|- D = ( dist ` Y ) |
11 |
|
eqid |
|- ( dist ` R ) = ( dist ` R ) |
12 |
1 2 3 4 5 6 7 11 10
|
prdsdsval2 |
|- ( ph -> ( F D G ) = sup ( ( ran ( x e. I |-> ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) u. { 0 } ) , RR* , < ) ) |
13 |
|
eqidd |
|- ( ph -> I = I ) |
14 |
1 2 3 4 5 8 6
|
prdsbascl |
|- ( ph -> A. x e. I ( F ` x ) e. K ) |
15 |
1 2 3 4 5 8 7
|
prdsbascl |
|- ( ph -> A. x e. I ( G ` x ) e. K ) |
16 |
9
|
oveqi |
|- ( ( F ` x ) E ( G ` x ) ) = ( ( F ` x ) ( ( dist ` R ) |` ( K X. K ) ) ( G ` x ) ) |
17 |
|
ovres |
|- ( ( ( F ` x ) e. K /\ ( G ` x ) e. K ) -> ( ( F ` x ) ( ( dist ` R ) |` ( K X. K ) ) ( G ` x ) ) = ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) |
18 |
16 17
|
eqtrid |
|- ( ( ( F ` x ) e. K /\ ( G ` x ) e. K ) -> ( ( F ` x ) E ( G ` x ) ) = ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) |
19 |
18
|
ex |
|- ( ( F ` x ) e. K -> ( ( G ` x ) e. K -> ( ( F ` x ) E ( G ` x ) ) = ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) ) |
20 |
19
|
ral2imi |
|- ( A. x e. I ( F ` x ) e. K -> ( A. x e. I ( G ` x ) e. K -> A. x e. I ( ( F ` x ) E ( G ` x ) ) = ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) ) |
21 |
14 15 20
|
sylc |
|- ( ph -> A. x e. I ( ( F ` x ) E ( G ` x ) ) = ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) |
22 |
|
mpteq12 |
|- ( ( I = I /\ A. x e. I ( ( F ` x ) E ( G ` x ) ) = ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) -> ( x e. I |-> ( ( F ` x ) E ( G ` x ) ) ) = ( x e. I |-> ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) ) |
23 |
13 21 22
|
syl2anc |
|- ( ph -> ( x e. I |-> ( ( F ` x ) E ( G ` x ) ) ) = ( x e. I |-> ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) ) |
24 |
23
|
rneqd |
|- ( ph -> ran ( x e. I |-> ( ( F ` x ) E ( G ` x ) ) ) = ran ( x e. I |-> ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) ) |
25 |
24
|
uneq1d |
|- ( ph -> ( ran ( x e. I |-> ( ( F ` x ) E ( G ` x ) ) ) u. { 0 } ) = ( ran ( x e. I |-> ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) u. { 0 } ) ) |
26 |
25
|
supeq1d |
|- ( ph -> sup ( ( ran ( x e. I |-> ( ( F ` x ) E ( G ` x ) ) ) u. { 0 } ) , RR* , < ) = sup ( ( ran ( x e. I |-> ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) u. { 0 } ) , RR* , < ) ) |
27 |
12 26
|
eqtr4d |
|- ( ph -> ( F D G ) = sup ( ( ran ( x e. I |-> ( ( F ` x ) E ( G ` x ) ) ) u. { 0 } ) , RR* , < ) ) |