| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ply1val.1 |
|- P = ( Poly1 ` R ) |
| 2 |
|
ply1val.2 |
|- S = ( PwSer1 ` R ) |
| 3 |
|
fveq2 |
|- ( r = R -> ( PwSer1 ` r ) = ( PwSer1 ` R ) ) |
| 4 |
3 2
|
eqtr4di |
|- ( r = R -> ( PwSer1 ` r ) = S ) |
| 5 |
|
oveq2 |
|- ( r = R -> ( 1o mPoly r ) = ( 1o mPoly R ) ) |
| 6 |
5
|
fveq2d |
|- ( r = R -> ( Base ` ( 1o mPoly r ) ) = ( Base ` ( 1o mPoly R ) ) ) |
| 7 |
4 6
|
oveq12d |
|- ( r = R -> ( ( PwSer1 ` r ) |`s ( Base ` ( 1o mPoly r ) ) ) = ( S |`s ( Base ` ( 1o mPoly R ) ) ) ) |
| 8 |
|
df-ply1 |
|- Poly1 = ( r e. _V |-> ( ( PwSer1 ` r ) |`s ( Base ` ( 1o mPoly r ) ) ) ) |
| 9 |
|
ovex |
|- ( S |`s ( Base ` ( 1o mPoly R ) ) ) e. _V |
| 10 |
7 8 9
|
fvmpt |
|- ( R e. _V -> ( Poly1 ` R ) = ( S |`s ( Base ` ( 1o mPoly R ) ) ) ) |
| 11 |
|
fvprc |
|- ( -. R e. _V -> ( Poly1 ` R ) = (/) ) |
| 12 |
|
ress0 |
|- ( (/) |`s ( Base ` ( 1o mPoly R ) ) ) = (/) |
| 13 |
11 12
|
eqtr4di |
|- ( -. R e. _V -> ( Poly1 ` R ) = ( (/) |`s ( Base ` ( 1o mPoly R ) ) ) ) |
| 14 |
|
fvprc |
|- ( -. R e. _V -> ( PwSer1 ` R ) = (/) ) |
| 15 |
2 14
|
eqtrid |
|- ( -. R e. _V -> S = (/) ) |
| 16 |
15
|
oveq1d |
|- ( -. R e. _V -> ( S |`s ( Base ` ( 1o mPoly R ) ) ) = ( (/) |`s ( Base ` ( 1o mPoly R ) ) ) ) |
| 17 |
13 16
|
eqtr4d |
|- ( -. R e. _V -> ( Poly1 ` R ) = ( S |`s ( Base ` ( 1o mPoly R ) ) ) ) |
| 18 |
10 17
|
pm2.61i |
|- ( Poly1 ` R ) = ( S |`s ( Base ` ( 1o mPoly R ) ) ) |
| 19 |
1 18
|
eqtri |
|- P = ( S |`s ( Base ` ( 1o mPoly R ) ) ) |