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Metamath Proof Explorer

Theorem divdiv32i

Description: Swap denominators in a division. (Contributed by NM, 15-Sep-1999)

Ref Expression
Hypotheses divclz.1 ⊢ A ∈ ℂ
divclz.2 ⊢ B ∈ ℂ
divmulz.3 ⊢ C ∈ ℂ
divmul.4 ⊢ B ≠ 0
divdiv23.5 ⊢ C ≠ 0
Assertion divdiv32i ⊢ A B C = A C B

Proof

Step Hyp Ref Expression
1 divclz.1 ⊢ A ∈ ℂ
2 divclz.2 ⊢ B ∈ ℂ
3 divmulz.3 ⊢ C ∈ ℂ
4 divmul.4 ⊢ B ≠ 0
5 divdiv23.5 ⊢ C ≠ 0
6 1 2 3 divdiv23zi ⊢ B ≠ 0 ∧ C ≠ 0 → A B C = A C B
7 4 5 6 mp2an ⊢ A B C = A C B