Theorem eft0val

Description: The value of the first term of the series expansion of the exponential function is 1. (Contributed by Paul Chapman, 21-Aug-2007) (Revised by Mario Carneiro, 29-Apr-2014)

	Ref	Expression
Assertion	eft0val	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 ↑ 0 ) / ( ! ‘ 0 ) ) = 1 )

Proof

Step	Hyp	Ref	Expression
1		exp0	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 0 ) = 1 )
2	1	oveq1d	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 ↑ 0 ) / ( ! ‘ 0 ) ) = ( 1 / ( ! ‘ 0 ) ) )
3		fac0	⊢ ( ! ‘ 0 ) = 1
4	3	oveq2i	⊢ ( 1 / ( ! ‘ 0 ) ) = ( 1 / 1 )
5		1div1e1	⊢ ( 1 / 1 ) = 1
6	4 5	eqtri	⊢ ( 1 / ( ! ‘ 0 ) ) = 1
7	2 6	eqtrdi	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 ↑ 0 ) / ( ! ‘ 0 ) ) = 1 )