Theorem seq1i

Description: Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 30-Apr-2014)

	Ref	Expression
Hypotheses	seq1i.1	⊢ 𝑀 ∈ ℤ
	seq1i.2	⊢ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) = 𝐴 )
Assertion	seq1i	⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑀 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		seq1i.1	⊢ 𝑀 ∈ ℤ
2		seq1i.2	⊢ ( 𝜑 → ( 𝐹 ‘ 𝑀 ) = 𝐴 )
3		seq1	⊢ ( 𝑀 ∈ ℤ → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑀 ) = ( 𝐹 ‘ 𝑀 ) )
4	1 3	ax-mp	⊢ ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑀 ) = ( 𝐹 ‘ 𝑀 )
5	4 2	eqtrid	⊢ ( 𝜑 → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑀 ) = 𝐴 )