Metamath Proof Explorer


Theorem divmul3

Description: Relationship between division and multiplication. (Contributed by NM, 13-Feb-2006)

Ref Expression
Assertion divmul3 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐶 ) = 𝐵𝐴 = ( 𝐵 · 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 divmul2 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐶 ) = 𝐵𝐴 = ( 𝐶 · 𝐵 ) ) )
2 mulcom ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
3 2 adantrr ( ( 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
4 3 3adant1 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
5 4 eqeq2d ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 = ( 𝐵 · 𝐶 ) ↔ 𝐴 = ( 𝐶 · 𝐵 ) ) )
6 1 5 bitr4d ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐶 ) = 𝐵𝐴 = ( 𝐵 · 𝐶 ) ) )