Theorem divsub1dir

Description: Distribution of division over subtraction by 1. (Contributed by AV, 6-Jun-2020)

	Ref	Expression
Assertion	divsub1dir	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) − 1 ) = ( ( 𝐴 − 𝐵 ) / 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		3simpc	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) )
2		divid	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 / 𝐵 ) = 1 )
3	1 2	syl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 / 𝐵 ) = 1 )
4	3	eqcomd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → 1 = ( 𝐵 / 𝐵 ) )
5	4	oveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) − 1 ) = ( ( 𝐴 / 𝐵 ) − ( 𝐵 / 𝐵 ) ) )
6		divsubdir	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 − 𝐵 ) / 𝐵 ) = ( ( 𝐴 / 𝐵 ) − ( 𝐵 / 𝐵 ) ) )
7	1 6	syld3an3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 − 𝐵 ) / 𝐵 ) = ( ( 𝐴 / 𝐵 ) − ( 𝐵 / 𝐵 ) ) )
8	5 7	eqtr4d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) − 1 ) = ( ( 𝐴 − 𝐵 ) / 𝐵 ) )