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 ) ) |
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 ) ) |