Theorem cscval

Description: Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014)

	Ref	Expression
Assertion	cscval	⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) ≠ 0 ) → ( csc ‘ 𝐴 ) = ( 1 / ( sin ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		fveq2	⊢ ( 𝑦 = 𝐴 → ( sin ‘ 𝑦 ) = ( sin ‘ 𝐴 ) )
2	1	neeq1d	⊢ ( 𝑦 = 𝐴 → ( ( sin ‘ 𝑦 ) ≠ 0 ↔ ( sin ‘ 𝐴 ) ≠ 0 ) )
3	2	elrab	⊢ ( 𝐴 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ↔ ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) ≠ 0 ) )
4		fveq2	⊢ ( 𝑥 = 𝐴 → ( sin ‘ 𝑥 ) = ( sin ‘ 𝐴 ) )
5	4	oveq2d	⊢ ( 𝑥 = 𝐴 → ( 1 / ( sin ‘ 𝑥 ) ) = ( 1 / ( sin ‘ 𝐴 ) ) )
6		df-csc	⊢ csc = ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ↦ ( 1 / ( sin ‘ 𝑥 ) ) )
7		ovex	⊢ ( 1 / ( sin ‘ 𝐴 ) ) ∈ V
8	5 6 7	fvmpt	⊢ ( 𝐴 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( csc ‘ 𝐴 ) = ( 1 / ( sin ‘ 𝐴 ) ) )
9	3 8	sylbir	⊢ ( ( 𝐴 ∈ ℂ ∧ ( sin ‘ 𝐴 ) ≠ 0 ) → ( csc ‘ 𝐴 ) = ( 1 / ( sin ‘ 𝐴 ) ) )