| Step |
Hyp |
Ref |
Expression |
| 1 |
|
evls1gsumadd.q |
|- Q = ( S evalSub1 R ) |
| 2 |
|
evls1gsumadd.k |
|- K = ( Base ` S ) |
| 3 |
|
evls1gsumadd.w |
|- W = ( Poly1 ` U ) |
| 4 |
|
evls1gsumadd.0 |
|- .0. = ( 0g ` W ) |
| 5 |
|
evls1gsumadd.u |
|- U = ( S |`s R ) |
| 6 |
|
evls1gsumadd.p |
|- P = ( S ^s K ) |
| 7 |
|
evls1gsumadd.b |
|- B = ( Base ` W ) |
| 8 |
|
evls1gsumadd.s |
|- ( ph -> S e. CRing ) |
| 9 |
|
evls1gsumadd.r |
|- ( ph -> R e. ( SubRing ` S ) ) |
| 10 |
|
evls1gsumadd.y |
|- ( ( ph /\ x e. N ) -> Y e. B ) |
| 11 |
|
evls1gsumadd.n |
|- ( ph -> N C_ NN0 ) |
| 12 |
|
evls1gsumadd.f |
|- ( ph -> ( x e. N |-> Y ) finSupp .0. ) |
| 13 |
5
|
subrgring |
|- ( R e. ( SubRing ` S ) -> U e. Ring ) |
| 14 |
3
|
ply1ring |
|- ( U e. Ring -> W e. Ring ) |
| 15 |
|
ringcmn |
|- ( W e. Ring -> W e. CMnd ) |
| 16 |
9 13 14 15
|
4syl |
|- ( ph -> W e. CMnd ) |
| 17 |
|
crngring |
|- ( S e. CRing -> S e. Ring ) |
| 18 |
8 17
|
syl |
|- ( ph -> S e. Ring ) |
| 19 |
2
|
fvexi |
|- K e. _V |
| 20 |
18 19
|
jctir |
|- ( ph -> ( S e. Ring /\ K e. _V ) ) |
| 21 |
6
|
pwsring |
|- ( ( S e. Ring /\ K e. _V ) -> P e. Ring ) |
| 22 |
|
ringmnd |
|- ( P e. Ring -> P e. Mnd ) |
| 23 |
20 21 22
|
3syl |
|- ( ph -> P e. Mnd ) |
| 24 |
|
nn0ex |
|- NN0 e. _V |
| 25 |
24
|
a1i |
|- ( ph -> NN0 e. _V ) |
| 26 |
25 11
|
ssexd |
|- ( ph -> N e. _V ) |
| 27 |
1 2 6 5 3
|
evls1rhm |
|- ( ( S e. CRing /\ R e. ( SubRing ` S ) ) -> Q e. ( W RingHom P ) ) |
| 28 |
8 9 27
|
syl2anc |
|- ( ph -> Q e. ( W RingHom P ) ) |
| 29 |
|
rhmghm |
|- ( Q e. ( W RingHom P ) -> Q e. ( W GrpHom P ) ) |
| 30 |
|
ghmmhm |
|- ( Q e. ( W GrpHom P ) -> Q e. ( W MndHom P ) ) |
| 31 |
28 29 30
|
3syl |
|- ( ph -> Q e. ( W MndHom P ) ) |
| 32 |
7 4 16 23 26 31 10 12
|
gsummptmhm |
|- ( ph -> ( P gsum ( x e. N |-> ( Q ` Y ) ) ) = ( Q ` ( W gsum ( x e. N |-> Y ) ) ) ) |
| 33 |
32
|
eqcomd |
|- ( ph -> ( Q ` ( W gsum ( x e. N |-> Y ) ) ) = ( P gsum ( x e. N |-> ( Q ` Y ) ) ) ) |