Theorem divrec

Description: Relationship between division and reciprocal. Theorem I.9 of Apostol p. 18. (Contributed by NM, 2-Aug-2004) (Revised by Mario Carneiro, 27-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		simp2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → 𝐵 ∈ ℂ )
2		simp1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → 𝐴 ∈ ℂ )
3		reccl	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 1 / 𝐵 ) ∈ ℂ )
4	3	3adant1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 1 / 𝐵 ) ∈ ℂ )
5	1 2 4	mul12d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 · ( 𝐴 · ( 1 / 𝐵 ) ) ) = ( 𝐴 · ( 𝐵 · ( 1 / 𝐵 ) ) ) )
6		recid	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 · ( 1 / 𝐵 ) ) = 1 )
7	6	3adant1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 · ( 1 / 𝐵 ) ) = 1 )
8	7	oveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 · ( 𝐵 · ( 1 / 𝐵 ) ) ) = ( 𝐴 · 1 ) )
9	2	mulid1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 · 1 ) = 𝐴 )
10	5 8 9	3eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 · ( 𝐴 · ( 1 / 𝐵 ) ) ) = 𝐴 )
11	2 4	mulcld	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 · ( 1 / 𝐵 ) ) ∈ ℂ )
12		3simpc	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) )
13		divmul	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐴 · ( 1 / 𝐵 ) ) ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 / 𝐵 ) = ( 𝐴 · ( 1 / 𝐵 ) ) ↔ ( 𝐵 · ( 𝐴 · ( 1 / 𝐵 ) ) ) = 𝐴 ) )
14	2 11 12 13	syl3anc	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) = ( 𝐴 · ( 1 / 𝐵 ) ) ↔ ( 𝐵 · ( 𝐴 · ( 1 / 𝐵 ) ) ) = 𝐴 ) )
15	10 14	mpbird	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 / 𝐵 ) = ( 𝐴 · ( 1 / 𝐵 ) ) )