Theorem subrec

Description: Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jul-2015)

	Ref	Expression
Assertion	subrec	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 1 / 𝐴 ) − ( 1 / 𝐵 ) ) = ( ( 𝐵 − 𝐴 ) / ( 𝐴 · 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		simpll	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → 𝐴 ∈ ℂ )
2		simprl	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → 𝐵 ∈ ℂ )
3		simplr	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → 𝐴 ≠ 0 )
4		simprr	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → 𝐵 ≠ 0 )
5	1 2 3 4	subrecd	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 1 / 𝐴 ) − ( 1 / 𝐵 ) ) = ( ( 𝐵 − 𝐴 ) / ( 𝐴 · 𝐵 ) ) )