Step |
Hyp |
Ref |
Expression |
1 |
|
mpfconst.b |
|- B = ( Base ` S ) |
2 |
|
mpfconst.q |
|- Q = ran ( ( I evalSub S ) ` R ) |
3 |
|
mpfconst.i |
|- ( ph -> I e. V ) |
4 |
|
mpfconst.s |
|- ( ph -> S e. CRing ) |
5 |
|
mpfconst.r |
|- ( ph -> R e. ( SubRing ` S ) ) |
6 |
|
mpfproj.j |
|- ( ph -> J e. I ) |
7 |
|
eqid |
|- ( ( I evalSub S ) ` R ) = ( ( I evalSub S ) ` R ) |
8 |
|
eqid |
|- ( I mVar ( S |`s R ) ) = ( I mVar ( S |`s R ) ) |
9 |
|
eqid |
|- ( S |`s R ) = ( S |`s R ) |
10 |
7 8 9 1 3 4 5 6
|
evlsvar |
|- ( ph -> ( ( ( I evalSub S ) ` R ) ` ( ( I mVar ( S |`s R ) ) ` J ) ) = ( f e. ( B ^m I ) |-> ( f ` J ) ) ) |
11 |
|
eqid |
|- ( I mPoly ( S |`s R ) ) = ( I mPoly ( S |`s R ) ) |
12 |
|
eqid |
|- ( S ^s ( B ^m I ) ) = ( S ^s ( B ^m I ) ) |
13 |
7 11 9 12 1
|
evlsrhm |
|- ( ( I e. V /\ S e. CRing /\ R e. ( SubRing ` S ) ) -> ( ( I evalSub S ) ` R ) e. ( ( I mPoly ( S |`s R ) ) RingHom ( S ^s ( B ^m I ) ) ) ) |
14 |
3 4 5 13
|
syl3anc |
|- ( ph -> ( ( I evalSub S ) ` R ) e. ( ( I mPoly ( S |`s R ) ) RingHom ( S ^s ( B ^m I ) ) ) ) |
15 |
|
eqid |
|- ( Base ` ( I mPoly ( S |`s R ) ) ) = ( Base ` ( I mPoly ( S |`s R ) ) ) |
16 |
|
eqid |
|- ( Base ` ( S ^s ( B ^m I ) ) ) = ( Base ` ( S ^s ( B ^m I ) ) ) |
17 |
15 16
|
rhmf |
|- ( ( ( I evalSub S ) ` R ) e. ( ( I mPoly ( S |`s R ) ) RingHom ( S ^s ( B ^m I ) ) ) -> ( ( I evalSub S ) ` R ) : ( Base ` ( I mPoly ( S |`s R ) ) ) --> ( Base ` ( S ^s ( B ^m I ) ) ) ) |
18 |
|
ffn |
|- ( ( ( I evalSub S ) ` R ) : ( Base ` ( I mPoly ( S |`s R ) ) ) --> ( Base ` ( S ^s ( B ^m I ) ) ) -> ( ( I evalSub S ) ` R ) Fn ( Base ` ( I mPoly ( S |`s R ) ) ) ) |
19 |
14 17 18
|
3syl |
|- ( ph -> ( ( I evalSub S ) ` R ) Fn ( Base ` ( I mPoly ( S |`s R ) ) ) ) |
20 |
9
|
subrgring |
|- ( R e. ( SubRing ` S ) -> ( S |`s R ) e. Ring ) |
21 |
5 20
|
syl |
|- ( ph -> ( S |`s R ) e. Ring ) |
22 |
11 8 15 3 21 6
|
mvrcl |
|- ( ph -> ( ( I mVar ( S |`s R ) ) ` J ) e. ( Base ` ( I mPoly ( S |`s R ) ) ) ) |
23 |
|
fnfvelrn |
|- ( ( ( ( I evalSub S ) ` R ) Fn ( Base ` ( I mPoly ( S |`s R ) ) ) /\ ( ( I mVar ( S |`s R ) ) ` J ) e. ( Base ` ( I mPoly ( S |`s R ) ) ) ) -> ( ( ( I evalSub S ) ` R ) ` ( ( I mVar ( S |`s R ) ) ` J ) ) e. ran ( ( I evalSub S ) ` R ) ) |
24 |
19 22 23
|
syl2anc |
|- ( ph -> ( ( ( I evalSub S ) ` R ) ` ( ( I mVar ( S |`s R ) ) ` J ) ) e. ran ( ( I evalSub S ) ` R ) ) |
25 |
24 2
|
eleqtrrdi |
|- ( ph -> ( ( ( I evalSub S ) ` R ) ` ( ( I mVar ( S |`s R ) ) ` J ) ) e. Q ) |
26 |
10 25
|
eqeltrrd |
|- ( ph -> ( f e. ( B ^m I ) |-> ( f ` J ) ) e. Q ) |