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Metamath Proof Explorer

Theorem divdiv2

Description: Division by a fraction. (Contributed by NM, 27-Dec-2008)

Ref Expression
Assertion divdiv2 ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 / ( 𝐵 / 𝐶 ) ) = ( ( 𝐴 · 𝐶 ) / 𝐵 ) )

Proof

Step Hyp Ref Expression
1 ax-1cn ⊢ 1 ∈ ℂ
2 ax-1ne0 ⊢ 1 ≠ 0
3 1 2 pm3.2i ⊢ ( 1 ∈ ℂ ∧ 1 ≠ 0 )
4 divdivdiv ⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( 1 ∈ ℂ ∧ 1 ≠ 0 ) ) ∧ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) ) → ( ( 𝐴 / 1 ) / ( 𝐵 / 𝐶 ) ) = ( ( 𝐴 · 𝐶 ) / ( 1 · 𝐵 ) ) )
5 3 4 mpanl2 ⊢ ( ( 𝐴 ∈ ℂ ∧ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) ) → ( ( 𝐴 / 1 ) / ( 𝐵 / 𝐶 ) ) = ( ( 𝐴 · 𝐶 ) / ( 1 · 𝐵 ) ) )
6 5 3impb ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 1 ) / ( 𝐵 / 𝐶 ) ) = ( ( 𝐴 · 𝐶 ) / ( 1 · 𝐵 ) ) )
7 div1 ⊢ ( 𝐴 ∈ ℂ → ( 𝐴 / 1 ) = 𝐴 )
8 7 3ad2ant1 ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 / 1 ) = 𝐴 )
9 8 oveq1d ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 1 ) / ( 𝐵 / 𝐶 ) ) = ( 𝐴 / ( 𝐵 / 𝐶 ) ) )
10 mullid ⊢ ( 𝐵 ∈ ℂ → ( 1 · 𝐵 ) = 𝐵 )
11 10 ad2antrl ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( 1 · 𝐵 ) = 𝐵 )
12 11 3adant3 ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 1 · 𝐵 ) = 𝐵 )
13 12 oveq2d ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 · 𝐶 ) / ( 1 · 𝐵 ) ) = ( ( 𝐴 · 𝐶 ) / 𝐵 ) )
14 6 9 13 3eqtr3d ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 / ( 𝐵 / 𝐶 ) ) = ( ( 𝐴 · 𝐶 ) / 𝐵 ) )