Metamath Proof Explorer


Theorem 0dgr

Description: A constant function has degree 0. (Contributed by Mario Carneiro, 24-Jul-2014)

Ref Expression
Assertion 0dgr
|- ( A e. CC -> ( deg ` ( CC X. { A } ) ) = 0 )

Proof

Step Hyp Ref Expression
1 ssid
 |-  CC C_ CC
2 plyconst
 |-  ( ( CC C_ CC /\ A e. CC ) -> ( CC X. { A } ) e. ( Poly ` CC ) )
3 1 2 mpan
 |-  ( A e. CC -> ( CC X. { A } ) e. ( Poly ` CC ) )
4 0nn0
 |-  0 e. NN0
5 4 a1i
 |-  ( A e. CC -> 0 e. NN0 )
6 simpl
 |-  ( ( A e. CC /\ k e. ( 0 ... 0 ) ) -> A e. CC )
7 fconstmpt
 |-  ( CC X. { A } ) = ( z e. CC |-> A )
8 0z
 |-  0 e. ZZ
9 exp0
 |-  ( z e. CC -> ( z ^ 0 ) = 1 )
10 9 oveq2d
 |-  ( z e. CC -> ( A x. ( z ^ 0 ) ) = ( A x. 1 ) )
11 mulid1
 |-  ( A e. CC -> ( A x. 1 ) = A )
12 10 11 sylan9eqr
 |-  ( ( A e. CC /\ z e. CC ) -> ( A x. ( z ^ 0 ) ) = A )
13 simpl
 |-  ( ( A e. CC /\ z e. CC ) -> A e. CC )
14 12 13 eqeltrd
 |-  ( ( A e. CC /\ z e. CC ) -> ( A x. ( z ^ 0 ) ) e. CC )
15 oveq2
 |-  ( k = 0 -> ( z ^ k ) = ( z ^ 0 ) )
16 15 oveq2d
 |-  ( k = 0 -> ( A x. ( z ^ k ) ) = ( A x. ( z ^ 0 ) ) )
17 16 fsum1
 |-  ( ( 0 e. ZZ /\ ( A x. ( z ^ 0 ) ) e. CC ) -> sum_ k e. ( 0 ... 0 ) ( A x. ( z ^ k ) ) = ( A x. ( z ^ 0 ) ) )
18 8 14 17 sylancr
 |-  ( ( A e. CC /\ z e. CC ) -> sum_ k e. ( 0 ... 0 ) ( A x. ( z ^ k ) ) = ( A x. ( z ^ 0 ) ) )
19 18 12 eqtrd
 |-  ( ( A e. CC /\ z e. CC ) -> sum_ k e. ( 0 ... 0 ) ( A x. ( z ^ k ) ) = A )
20 19 mpteq2dva
 |-  ( A e. CC -> ( z e. CC |-> sum_ k e. ( 0 ... 0 ) ( A x. ( z ^ k ) ) ) = ( z e. CC |-> A ) )
21 7 20 eqtr4id
 |-  ( A e. CC -> ( CC X. { A } ) = ( z e. CC |-> sum_ k e. ( 0 ... 0 ) ( A x. ( z ^ k ) ) ) )
22 3 5 6 21 dgrle
 |-  ( A e. CC -> ( deg ` ( CC X. { A } ) ) <_ 0 )
23 dgrcl
 |-  ( ( CC X. { A } ) e. ( Poly ` CC ) -> ( deg ` ( CC X. { A } ) ) e. NN0 )
24 nn0le0eq0
 |-  ( ( deg ` ( CC X. { A } ) ) e. NN0 -> ( ( deg ` ( CC X. { A } ) ) <_ 0 <-> ( deg ` ( CC X. { A } ) ) = 0 ) )
25 3 23 24 3syl
 |-  ( A e. CC -> ( ( deg ` ( CC X. { A } ) ) <_ 0 <-> ( deg ` ( CC X. { A } ) ) = 0 ) )
26 22 25 mpbid
 |-  ( A e. CC -> ( deg ` ( CC X. { A } ) ) = 0 )