Theorem bccn1

Description: Generalized binomial coefficient: C choose 1 . (Contributed by Steve Rodriguez, 22-Apr-2020)

	Ref	Expression
Hypothesis	bccn0.c	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion	bccn1	⊢ ( 𝜑 → ( 𝐶 C𝑐 1 ) = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		bccn0.c	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
2		0nn0	⊢ 0 ∈ ℕ0
3	2	a1i	⊢ ( 𝜑 → 0 ∈ ℕ0 )
4	1 3	bccp1k	⊢ ( 𝜑 → ( 𝐶 C𝑐 ( 0 + 1 ) ) = ( ( 𝐶 C𝑐 0 ) · ( ( 𝐶 − 0 ) / ( 0 + 1 ) ) ) )
5		0p1e1	⊢ ( 0 + 1 ) = 1
6	5	oveq2i	⊢ ( 𝐶 C𝑐 ( 0 + 1 ) ) = ( 𝐶 C𝑐 1 )
7	6	a1i	⊢ ( 𝜑 → ( 𝐶 C𝑐 ( 0 + 1 ) ) = ( 𝐶 C𝑐 1 ) )
8	1	bccn0	⊢ ( 𝜑 → ( 𝐶 C𝑐 0 ) = 1 )
9	1	subid1d	⊢ ( 𝜑 → ( 𝐶 − 0 ) = 𝐶 )
10	5	a1i	⊢ ( 𝜑 → ( 0 + 1 ) = 1 )
11	9 10	oveq12d	⊢ ( 𝜑 → ( ( 𝐶 − 0 ) / ( 0 + 1 ) ) = ( 𝐶 / 1 ) )
12	1	div1d	⊢ ( 𝜑 → ( 𝐶 / 1 ) = 𝐶 )
13	11 12	eqtrd	⊢ ( 𝜑 → ( ( 𝐶 − 0 ) / ( 0 + 1 ) ) = 𝐶 )
14	8 13	oveq12d	⊢ ( 𝜑 → ( ( 𝐶 C𝑐 0 ) · ( ( 𝐶 − 0 ) / ( 0 + 1 ) ) ) = ( 1 · 𝐶 ) )
15	4 7 14	3eqtr3d	⊢ ( 𝜑 → ( 𝐶 C𝑐 1 ) = ( 1 · 𝐶 ) )
16	1	mullidd	⊢ ( 𝜑 → ( 1 · 𝐶 ) = 𝐶 )
17	15 16	eqtrd	⊢ ( 𝜑 → ( 𝐶 C𝑐 1 ) = 𝐶 )