Theorem dvn0

Description: Zero times iterated derivative. (Contributed by Stefan O'Rear, 15-Nov-2014) (Revised by Mario Carneiro, 11-Feb-2015)

	Ref	Expression
Assertion	dvn0	⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ) → ( ( 𝑆 D𝑛 𝐹 ) ‘ 0 ) = 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) = ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) )
2	1	dvnfval	⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ) → ( 𝑆 D𝑛 𝐹 ) = seq 0 ( ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) , ( ℕ0 × { 𝐹 } ) ) )
3	2	fveq1d	⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ) → ( ( 𝑆 D𝑛 𝐹 ) ‘ 0 ) = ( seq 0 ( ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) , ( ℕ0 × { 𝐹 } ) ) ‘ 0 ) )
4		0z	⊢ 0 ∈ ℤ
5		simpr	⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ) → 𝐹 ∈ ( ℂ ↑pm 𝑆 ) )
6		0nn0	⊢ 0 ∈ ℕ0
7		fvconst2g	⊢ ( ( 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ∧ 0 ∈ ℕ0 ) → ( ( ℕ0 × { 𝐹 } ) ‘ 0 ) = 𝐹 )
8	5 6 7	sylancl	⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ) → ( ( ℕ0 × { 𝐹 } ) ‘ 0 ) = 𝐹 )
9	4 8	seq1i	⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ) → ( seq 0 ( ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) , ( ℕ0 × { 𝐹 } ) ) ‘ 0 ) = 𝐹 )
10	3 9	eqtrd	⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ) → ( ( 𝑆 D𝑛 𝐹 ) ‘ 0 ) = 𝐹 )