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Metamath Proof Explorer

Theorem dmmcand

Description: Cancellation law for division and multiplication. (Contributed by Glauco Siliprandi, 5-Apr-2020)

Ref Expression
Hypotheses dmmcand.a ⊢ φ → A ∈ ℂ
dmmcand.b ⊢ φ → B ∈ ℂ
dmmcand.c ⊢ φ → C ∈ ℂ
dmmcand.bn0 ⊢ φ → B ≠ 0
Assertion dmmcand ⊢ φ → A B ⁢ B ⁢ C = A ⁢ C

Proof

Step Hyp Ref Expression
1 dmmcand.a ⊢ φ → A ∈ ℂ
2 dmmcand.b ⊢ φ → B ∈ ℂ
3 dmmcand.c ⊢ φ → C ∈ ℂ
4 dmmcand.bn0 ⊢ φ → B ≠ 0
5 2 3 mulcld ⊢ φ → B ⁢ C ∈ ℂ
6 1 2 5 4 div32d ⊢ φ → A B ⁢ B ⁢ C = A ⁢ B ⁢ C B
7 3 2 4 divcan3d ⊢ φ → B ⁢ C B = C
8 7 oveq2d ⊢ φ → A ⁢ B ⁢ C B = A ⁢ C
9 eqidd ⊢ φ → A ⁢ C = A ⁢ C
10 6 8 9 3eqtrd ⊢ φ → A B ⁢ B ⁢ C = A ⁢ C