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Metamath Proof Explorer

Theorem div11i

Description: One-to-one relationship for division. (Contributed by NM, 20-Aug-2001)

Ref Expression
Hypotheses divclz.1 ⊢ 𝐴 ∈ ℂ
divclz.2 ⊢ 𝐵 ∈ ℂ
divmulz.3 ⊢ 𝐶 ∈ ℂ
divass.4 ⊢ 𝐶 ≠ 0
Assertion div11i ( ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐶 ) ↔ 𝐴 = 𝐵 )

Proof

Step Hyp Ref Expression
1 divclz.1 ⊢ 𝐴 ∈ ℂ
2 divclz.2 ⊢ 𝐵 ∈ ℂ
3 divmulz.3 ⊢ 𝐶 ∈ ℂ
4 divass.4 ⊢ 𝐶 ≠ 0
5 3 4 pm3.2i ⊢ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 )
6 div11 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐶 ) ↔ 𝐴 = 𝐵 ) )
7 1 2 5 6 mp3an ⊢ ( ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐶 ) ↔ 𝐴 = 𝐵 )