Step |
Hyp |
Ref |
Expression |
1 |
|
simp2 |
|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> R : Q --> ( 0 [,] +oo ) ) |
2 |
|
simp1 |
|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> Q e. V ) |
3 |
|
fex |
|- ( ( R : Q --> ( 0 [,] +oo ) /\ Q e. V ) -> R e. _V ) |
4 |
1 2 3
|
syl2anc |
|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> R e. _V ) |
5 |
|
omsval |
|- ( R e. _V -> ( toOMeas ` R ) = ( a e. ~P U. dom R |-> inf ( ran ( x e. { z e. ~P dom R | ( a C_ U. z /\ z ~<_ _om ) } |-> sum* y e. x ( R ` y ) ) , ( 0 [,] +oo ) , < ) ) ) |
6 |
4 5
|
syl |
|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> ( toOMeas ` R ) = ( a e. ~P U. dom R |-> inf ( ran ( x e. { z e. ~P dom R | ( a C_ U. z /\ z ~<_ _om ) } |-> sum* y e. x ( R ` y ) ) , ( 0 [,] +oo ) , < ) ) ) |
7 |
|
simpr |
|- ( ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) /\ a = A ) -> a = A ) |
8 |
7
|
sseq1d |
|- ( ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) /\ a = A ) -> ( a C_ U. z <-> A C_ U. z ) ) |
9 |
8
|
anbi1d |
|- ( ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) /\ a = A ) -> ( ( a C_ U. z /\ z ~<_ _om ) <-> ( A C_ U. z /\ z ~<_ _om ) ) ) |
10 |
9
|
rabbidv |
|- ( ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) /\ a = A ) -> { z e. ~P dom R | ( a C_ U. z /\ z ~<_ _om ) } = { z e. ~P dom R | ( A C_ U. z /\ z ~<_ _om ) } ) |
11 |
10
|
mpteq1d |
|- ( ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) /\ a = A ) -> ( x e. { z e. ~P dom R | ( a C_ U. z /\ z ~<_ _om ) } |-> sum* y e. x ( R ` y ) ) = ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ _om ) } |-> sum* y e. x ( R ` y ) ) ) |
12 |
11
|
rneqd |
|- ( ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) /\ a = A ) -> ran ( x e. { z e. ~P dom R | ( a C_ U. z /\ z ~<_ _om ) } |-> sum* y e. x ( R ` y ) ) = ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ _om ) } |-> sum* y e. x ( R ` y ) ) ) |
13 |
12
|
infeq1d |
|- ( ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) /\ a = A ) -> inf ( ran ( x e. { z e. ~P dom R | ( a C_ U. z /\ z ~<_ _om ) } |-> sum* y e. x ( R ` y ) ) , ( 0 [,] +oo ) , < ) = inf ( ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ _om ) } |-> sum* y e. x ( R ` y ) ) , ( 0 [,] +oo ) , < ) ) |
14 |
|
simp3 |
|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> A C_ U. Q ) |
15 |
|
fdm |
|- ( R : Q --> ( 0 [,] +oo ) -> dom R = Q ) |
16 |
15
|
3ad2ant2 |
|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> dom R = Q ) |
17 |
16
|
unieqd |
|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> U. dom R = U. Q ) |
18 |
14 17
|
sseqtrrd |
|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> A C_ U. dom R ) |
19 |
2
|
uniexd |
|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> U. Q e. _V ) |
20 |
|
ssexg |
|- ( ( A C_ U. Q /\ U. Q e. _V ) -> A e. _V ) |
21 |
14 19 20
|
syl2anc |
|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> A e. _V ) |
22 |
|
elpwg |
|- ( A e. _V -> ( A e. ~P U. dom R <-> A C_ U. dom R ) ) |
23 |
21 22
|
syl |
|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> ( A e. ~P U. dom R <-> A C_ U. dom R ) ) |
24 |
18 23
|
mpbird |
|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> A e. ~P U. dom R ) |
25 |
|
xrltso |
|- < Or RR* |
26 |
|
iccssxr |
|- ( 0 [,] +oo ) C_ RR* |
27 |
|
soss |
|- ( ( 0 [,] +oo ) C_ RR* -> ( < Or RR* -> < Or ( 0 [,] +oo ) ) ) |
28 |
26 27
|
ax-mp |
|- ( < Or RR* -> < Or ( 0 [,] +oo ) ) |
29 |
25 28
|
mp1i |
|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> < Or ( 0 [,] +oo ) ) |
30 |
29
|
infexd |
|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> inf ( ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ _om ) } |-> sum* y e. x ( R ` y ) ) , ( 0 [,] +oo ) , < ) e. _V ) |
31 |
6 13 24 30
|
fvmptd |
|- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> ( ( toOMeas ` R ) ` A ) = inf ( ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ _om ) } |-> sum* y e. x ( R ` y ) ) , ( 0 [,] +oo ) , < ) ) |