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Metamath Proof Explorer

Theorem recdiv

Description: The reciprocal of a ratio. (Contributed by NM, 3-Aug-2004)

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

Proof

Step Hyp Ref Expression
1 1div1e1 ⊢ ( 1 / 1 ) = 1
2 1 oveq1i ⊢ ( ( 1 / 1 ) / ( 𝐴 / 𝐵 ) ) = ( 1 / ( 𝐴 / 𝐵 ) )
3 ax-1cn ⊢ 1 ∈ ℂ
4 ax-1ne0 ⊢ 1 ≠ 0
5 3 4 pm3.2i ⊢ ( 1 ∈ ℂ ∧ 1 ≠ 0 )
6 divdivdiv ⊢ ( ( ( 1 ∈ ℂ ∧ ( 1 ∈ ℂ ∧ 1 ≠ 0 ) ) ∧ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) ) → ( ( 1 / 1 ) / ( 𝐴 / 𝐵 ) ) = ( ( 1 · 𝐵 ) / ( 1 · 𝐴 ) ) )
7 3 5 6 mpanl12 ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 1 / 1 ) / ( 𝐴 / 𝐵 ) ) = ( ( 1 · 𝐵 ) / ( 1 · 𝐴 ) ) )
8 2 7 eqtr3id ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( 1 / ( 𝐴 / 𝐵 ) ) = ( ( 1 · 𝐵 ) / ( 1 · 𝐴 ) ) )
9 mullid ⊢ ( 𝐵 ∈ ℂ → ( 1 · 𝐵 ) = 𝐵 )
10 mullid ⊢ ( 𝐴 ∈ ℂ → ( 1 · 𝐴 ) = 𝐴 )
11 9 10 oveqan12rd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 1 · 𝐵 ) / ( 1 · 𝐴 ) ) = ( 𝐵 / 𝐴 ) )
12 11 ad2ant2r ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 1 · 𝐵 ) / ( 1 · 𝐴 ) ) = ( 𝐵 / 𝐴 ) )
13 8 12 eqtrd ⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( 1 / ( 𝐴 / 𝐵 ) ) = ( 𝐵 / 𝐴 ) )