This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem divmul

Description: Relationship between division and multiplication. (Contributed by NM, 2-Aug-2004) (Revised by Mario Carneiro, 17-Feb-2014)

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

Proof

Step Hyp Ref Expression
1 divval ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) → ( 𝐴 / 𝐶 ) = ( ℩ 𝑥 ∈ ℂ ( 𝐶 · 𝑥 ) = 𝐴 ) )
2 1 3expb ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 / 𝐶 ) = ( ℩ 𝑥 ∈ ℂ ( 𝐶 · 𝑥 ) = 𝐴 ) )
3 2 3adant2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( 𝐴 / 𝐶 ) = ( ℩ 𝑥 ∈ ℂ ( 𝐶 · 𝑥 ) = 𝐴 ) )
4 3 eqeq1d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐶 ) = 𝐵 ↔ ( ℩ 𝑥 ∈ ℂ ( 𝐶 · 𝑥 ) = 𝐴 ) = 𝐵 ) )
5 simp2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → 𝐵 ∈ ℂ )
6 receu ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) → ∃! 𝑥 ∈ ℂ ( 𝐶 · 𝑥 ) = 𝐴 )
7 6 3expb ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ∃! 𝑥 ∈ ℂ ( 𝐶 · 𝑥 ) = 𝐴 )
8 oveq2 ⊢ ( 𝑥 = 𝐵 → ( 𝐶 · 𝑥 ) = ( 𝐶 · 𝐵 ) )
9 8 eqeq1d ⊢ ( 𝑥 = 𝐵 → ( ( 𝐶 · 𝑥 ) = 𝐴 ↔ ( 𝐶 · 𝐵 ) = 𝐴 ) )
10 9 riota2 ⊢ ( ( 𝐵 ∈ ℂ ∧ ∃! 𝑥 ∈ ℂ ( 𝐶 · 𝑥 ) = 𝐴 ) → ( ( 𝐶 · 𝐵 ) = 𝐴 ↔ ( ℩ 𝑥 ∈ ℂ ( 𝐶 · 𝑥 ) = 𝐴 ) = 𝐵 ) )
11 5 7 10 3imp3i2an ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐶 · 𝐵 ) = 𝐴 ↔ ( ℩ 𝑥 ∈ ℂ ( 𝐶 · 𝑥 ) = 𝐴 ) = 𝐵 ) )
12 4 11 bitr4d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐶 ) = 𝐵 ↔ ( 𝐶 · 𝐵 ) = 𝐴 ) )