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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 )