Step |
Hyp |
Ref |
Expression |
1 |
|
dgrval.1 |
|- A = ( coeff ` F ) |
2 |
|
plyssc |
|- ( Poly ` S ) C_ ( Poly ` CC ) |
3 |
2
|
sseli |
|- ( F e. ( Poly ` S ) -> F e. ( Poly ` CC ) ) |
4 |
|
fveq2 |
|- ( f = F -> ( coeff ` f ) = ( coeff ` F ) ) |
5 |
4 1
|
eqtr4di |
|- ( f = F -> ( coeff ` f ) = A ) |
6 |
5
|
cnveqd |
|- ( f = F -> `' ( coeff ` f ) = `' A ) |
7 |
6
|
imaeq1d |
|- ( f = F -> ( `' ( coeff ` f ) " ( CC \ { 0 } ) ) = ( `' A " ( CC \ { 0 } ) ) ) |
8 |
7
|
supeq1d |
|- ( f = F -> sup ( ( `' ( coeff ` f ) " ( CC \ { 0 } ) ) , NN0 , < ) = sup ( ( `' A " ( CC \ { 0 } ) ) , NN0 , < ) ) |
9 |
|
df-dgr |
|- deg = ( f e. ( Poly ` CC ) |-> sup ( ( `' ( coeff ` f ) " ( CC \ { 0 } ) ) , NN0 , < ) ) |
10 |
|
nn0ssre |
|- NN0 C_ RR |
11 |
|
ltso |
|- < Or RR |
12 |
|
soss |
|- ( NN0 C_ RR -> ( < Or RR -> < Or NN0 ) ) |
13 |
10 11 12
|
mp2 |
|- < Or NN0 |
14 |
13
|
supex |
|- sup ( ( `' A " ( CC \ { 0 } ) ) , NN0 , < ) e. _V |
15 |
8 9 14
|
fvmpt |
|- ( F e. ( Poly ` CC ) -> ( deg ` F ) = sup ( ( `' A " ( CC \ { 0 } ) ) , NN0 , < ) ) |
16 |
3 15
|
syl |
|- ( F e. ( Poly ` S ) -> ( deg ` F ) = sup ( ( `' A " ( CC \ { 0 } ) ) , NN0 , < ) ) |