Description: Cancellation law for multiplication. Theorem I.7 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | mulcand.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℂ ) | |
mulcand.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℂ ) | ||
mulcand.3 | ⊢ ( 𝜑 → 𝐶 ∈ ℂ ) | ||
mulcand.4 | ⊢ ( 𝜑 → 𝐶 ≠ 0 ) | ||
Assertion | mulcan2d | ⊢ ( 𝜑 → ( ( 𝐴 · 𝐶 ) = ( 𝐵 · 𝐶 ) ↔ 𝐴 = 𝐵 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulcand.1 | ⊢ ( 𝜑 → 𝐴 ∈ ℂ ) | |
2 | mulcand.2 | ⊢ ( 𝜑 → 𝐵 ∈ ℂ ) | |
3 | mulcand.3 | ⊢ ( 𝜑 → 𝐶 ∈ ℂ ) | |
4 | mulcand.4 | ⊢ ( 𝜑 → 𝐶 ≠ 0 ) | |
5 | 1 3 | mulcomd | ⊢ ( 𝜑 → ( 𝐴 · 𝐶 ) = ( 𝐶 · 𝐴 ) ) |
6 | 2 3 | mulcomd | ⊢ ( 𝜑 → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) ) |
7 | 5 6 | eqeq12d | ⊢ ( 𝜑 → ( ( 𝐴 · 𝐶 ) = ( 𝐵 · 𝐶 ) ↔ ( 𝐶 · 𝐴 ) = ( 𝐶 · 𝐵 ) ) ) |
8 | 1 2 3 4 | mulcand | ⊢ ( 𝜑 → ( ( 𝐶 · 𝐴 ) = ( 𝐶 · 𝐵 ) ↔ 𝐴 = 𝐵 ) ) |
9 | 7 8 | bitrd | ⊢ ( 𝜑 → ( ( 𝐴 · 𝐶 ) = ( 𝐵 · 𝐶 ) ↔ 𝐴 = 𝐵 ) ) |