Theorem coe1term

Description: The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014)

	Ref	Expression
Hypothesis	coe1term.1	|- F = ( z e. CC |-> ( A x. ( z ^ N ) ) )
Assertion	coe1term	|- ( ( A e. CC /\ N e. NN0 /\ M e. NN0 ) -> ( ( coeff ` F ) ` M ) = if ( M = N , A , 0 ) )

Proof

Step	Hyp	Ref	Expression
1		coe1term.1	|- F = ( z e. CC |-> ( A x. ( z ^ N ) ) )
2	1	coe1termlem	|- ( ( A e. CC /\ N e. NN0 ) -> ( ( coeff ` F ) = ( n e. NN0 |-> if ( n = N , A , 0 ) ) /\ ( A =/= 0 -> ( deg ` F ) = N ) ) )
3	2	simpld	|- ( ( A e. CC /\ N e. NN0 ) -> ( coeff ` F ) = ( n e. NN0 |-> if ( n = N , A , 0 ) ) )
4	3	fveq1d	|- ( ( A e. CC /\ N e. NN0 ) -> ( ( coeff ` F ) ` M ) = ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` M ) )
5	4	3adant3	|- ( ( A e. CC /\ N e. NN0 /\ M e. NN0 ) -> ( ( coeff ` F ) ` M ) = ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` M ) )
6		eqid	|- ( n e. NN0 |-> if ( n = N , A , 0 ) ) = ( n e. NN0 |-> if ( n = N , A , 0 ) )
7		eqeq1	|- ( n = M -> ( n = N <-> M = N ) )
8	7	ifbid	|- ( n = M -> if ( n = N , A , 0 ) = if ( M = N , A , 0 ) )
9		simp3	|- ( ( A e. CC /\ N e. NN0 /\ M e. NN0 ) -> M e. NN0 )
10		simp1	|- ( ( A e. CC /\ N e. NN0 /\ M e. NN0 ) -> A e. CC )
11		0cn	|- 0 e. CC
12		ifcl	|- ( ( A e. CC /\ 0 e. CC ) -> if ( M = N , A , 0 ) e. CC )
13	10 11 12	sylancl	|- ( ( A e. CC /\ N e. NN0 /\ M e. NN0 ) -> if ( M = N , A , 0 ) e. CC )
14	6 8 9 13	fvmptd3	|- ( ( A e. CC /\ N e. NN0 /\ M e. NN0 ) -> ( ( n e. NN0 |-> if ( n = N , A , 0 ) ) ` M ) = if ( M = N , A , 0 ) )
15	5 14	eqtrd	|- ( ( A e. CC /\ N e. NN0 /\ M e. NN0 ) -> ( ( coeff ` F ) ` M ) = if ( M = N , A , 0 ) )