Theorem subrecd

Description: Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jan-2017)

	Ref	Expression
Hypotheses	subrecd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	subrecd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	subrecd.3	⊢ ( 𝜑 → 𝐴 ≠ 0 )
	subrecd.4	⊢ ( 𝜑 → 𝐵 ≠ 0 )
Assertion	subrecd	⊢ ( 𝜑 → ( ( 1 / 𝐴 ) − ( 1 / 𝐵 ) ) = ( ( 𝐵 − 𝐴 ) / ( 𝐴 · 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		subrecd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		subrecd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		subrecd.3	⊢ ( 𝜑 → 𝐴 ≠ 0 )
4		subrecd.4	⊢ ( 𝜑 → 𝐵 ≠ 0 )
5		subrec	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 1 / 𝐴 ) − ( 1 / 𝐵 ) ) = ( ( 𝐵 − 𝐴 ) / ( 𝐴 · 𝐵 ) ) )
6	1 3 2 4 5	syl22anc	⊢ ( 𝜑 → ( ( 1 / 𝐴 ) − ( 1 / 𝐵 ) ) = ( ( 𝐵 − 𝐴 ) / ( 𝐴 · 𝐵 ) ) )