Step |
Hyp |
Ref |
Expression |
1 |
|
ismhp.h |
|- H = ( I mHomP R ) |
2 |
|
ismhp.p |
|- P = ( I mPoly R ) |
3 |
|
ismhp.b |
|- B = ( Base ` P ) |
4 |
|
ismhp.0 |
|- .0. = ( 0g ` R ) |
5 |
|
ismhp.d |
|- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } |
6 |
|
ismhp.n |
|- ( ph -> N e. NN0 ) |
7 |
|
reldmmhp |
|- Rel dom mHomP |
8 |
|
id |
|- ( X e. ( H ` N ) -> X e. ( H ` N ) ) |
9 |
7 1 8
|
elfvov1 |
|- ( X e. ( H ` N ) -> I e. _V ) |
10 |
7 1 8
|
elfvov2 |
|- ( X e. ( H ` N ) -> R e. _V ) |
11 |
9 10
|
jca |
|- ( X e. ( H ` N ) -> ( I e. _V /\ R e. _V ) ) |
12 |
11
|
anim2i |
|- ( ( ph /\ X e. ( H ` N ) ) -> ( ph /\ ( I e. _V /\ R e. _V ) ) ) |
13 |
|
reldmmpl |
|- Rel dom mPoly |
14 |
13 2 3
|
elbasov |
|- ( X e. B -> ( I e. _V /\ R e. _V ) ) |
15 |
14
|
adantr |
|- ( ( X e. B /\ ( X supp .0. ) C_ { g e. D | ( ( CCfld |`s NN0 ) gsum g ) = N } ) -> ( I e. _V /\ R e. _V ) ) |
16 |
15
|
anim2i |
|- ( ( ph /\ ( X e. B /\ ( X supp .0. ) C_ { g e. D | ( ( CCfld |`s NN0 ) gsum g ) = N } ) ) -> ( ph /\ ( I e. _V /\ R e. _V ) ) ) |
17 |
|
simprl |
|- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> I e. _V ) |
18 |
|
simprr |
|- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> R e. _V ) |
19 |
6
|
adantr |
|- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> N e. NN0 ) |
20 |
1 2 3 4 5 17 18 19
|
mhpval |
|- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> ( H ` N ) = { f e. B | ( f supp .0. ) C_ { g e. D | ( ( CCfld |`s NN0 ) gsum g ) = N } } ) |
21 |
20
|
eleq2d |
|- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> ( X e. ( H ` N ) <-> X e. { f e. B | ( f supp .0. ) C_ { g e. D | ( ( CCfld |`s NN0 ) gsum g ) = N } } ) ) |
22 |
|
oveq1 |
|- ( f = X -> ( f supp .0. ) = ( X supp .0. ) ) |
23 |
22
|
sseq1d |
|- ( f = X -> ( ( f supp .0. ) C_ { g e. D | ( ( CCfld |`s NN0 ) gsum g ) = N } <-> ( X supp .0. ) C_ { g e. D | ( ( CCfld |`s NN0 ) gsum g ) = N } ) ) |
24 |
23
|
elrab |
|- ( X e. { f e. B | ( f supp .0. ) C_ { g e. D | ( ( CCfld |`s NN0 ) gsum g ) = N } } <-> ( X e. B /\ ( X supp .0. ) C_ { g e. D | ( ( CCfld |`s NN0 ) gsum g ) = N } ) ) |
25 |
21 24
|
bitrdi |
|- ( ( ph /\ ( I e. _V /\ R e. _V ) ) -> ( X e. ( H ` N ) <-> ( X e. B /\ ( X supp .0. ) C_ { g e. D | ( ( CCfld |`s NN0 ) gsum g ) = N } ) ) ) |
26 |
12 16 25
|
pm5.21nd |
|- ( ph -> ( X e. ( H ` N ) <-> ( X e. B /\ ( X supp .0. ) C_ { g e. D | ( ( CCfld |`s NN0 ) gsum g ) = N } ) ) ) |