Step |
Hyp |
Ref |
Expression |
1 |
|
polfval.o |
|- ._|_ = ( oc ` K ) |
2 |
|
polfval.a |
|- A = ( Atoms ` K ) |
3 |
|
polfval.m |
|- M = ( pmap ` K ) |
4 |
|
polfval.p |
|- P = ( _|_P ` K ) |
5 |
|
elex |
|- ( K e. B -> K e. _V ) |
6 |
|
fveq2 |
|- ( h = K -> ( Atoms ` h ) = ( Atoms ` K ) ) |
7 |
6 2
|
eqtr4di |
|- ( h = K -> ( Atoms ` h ) = A ) |
8 |
7
|
pweqd |
|- ( h = K -> ~P ( Atoms ` h ) = ~P A ) |
9 |
|
fveq2 |
|- ( h = K -> ( pmap ` h ) = ( pmap ` K ) ) |
10 |
9 3
|
eqtr4di |
|- ( h = K -> ( pmap ` h ) = M ) |
11 |
|
fveq2 |
|- ( h = K -> ( oc ` h ) = ( oc ` K ) ) |
12 |
11 1
|
eqtr4di |
|- ( h = K -> ( oc ` h ) = ._|_ ) |
13 |
12
|
fveq1d |
|- ( h = K -> ( ( oc ` h ) ` p ) = ( ._|_ ` p ) ) |
14 |
10 13
|
fveq12d |
|- ( h = K -> ( ( pmap ` h ) ` ( ( oc ` h ) ` p ) ) = ( M ` ( ._|_ ` p ) ) ) |
15 |
14
|
adantr |
|- ( ( h = K /\ p e. m ) -> ( ( pmap ` h ) ` ( ( oc ` h ) ` p ) ) = ( M ` ( ._|_ ` p ) ) ) |
16 |
15
|
iineq2dv |
|- ( h = K -> |^|_ p e. m ( ( pmap ` h ) ` ( ( oc ` h ) ` p ) ) = |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) |
17 |
7 16
|
ineq12d |
|- ( h = K -> ( ( Atoms ` h ) i^i |^|_ p e. m ( ( pmap ` h ) ` ( ( oc ` h ) ` p ) ) ) = ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) |
18 |
8 17
|
mpteq12dv |
|- ( h = K -> ( m e. ~P ( Atoms ` h ) |-> ( ( Atoms ` h ) i^i |^|_ p e. m ( ( pmap ` h ) ` ( ( oc ` h ) ` p ) ) ) ) = ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) ) |
19 |
|
df-polarityN |
|- _|_P = ( h e. _V |-> ( m e. ~P ( Atoms ` h ) |-> ( ( Atoms ` h ) i^i |^|_ p e. m ( ( pmap ` h ) ` ( ( oc ` h ) ` p ) ) ) ) ) |
20 |
2
|
fvexi |
|- A e. _V |
21 |
20
|
pwex |
|- ~P A e. _V |
22 |
21
|
mptex |
|- ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) e. _V |
23 |
18 19 22
|
fvmpt |
|- ( K e. _V -> ( _|_P ` K ) = ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) ) |
24 |
4 23
|
syl5eq |
|- ( K e. _V -> P = ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) ) |
25 |
5 24
|
syl |
|- ( K e. B -> P = ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) ) |