Theorem mulcan1g

Description: A generalized form of the cancellation law for multiplication. (Contributed by Scott Fenton, 17-Jun-2013)

	Ref	Expression
Assertion	mulcan1g	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) = ( 𝐴 · 𝐶 ) ↔ ( 𝐴 = 0 ∨ 𝐵 = 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		mulcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 · 𝐵 ) ∈ ℂ )
2	1	3adant3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · 𝐵 ) ∈ ℂ )
3		mulcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · 𝐶 ) ∈ ℂ )
4	3	3adant2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · 𝐶 ) ∈ ℂ )
5	2 4	subeq0ad	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( ( 𝐴 · 𝐵 ) − ( 𝐴 · 𝐶 ) ) = 0 ↔ ( 𝐴 · 𝐵 ) = ( 𝐴 · 𝐶 ) ) )
6		simp1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → 𝐴 ∈ ℂ )
7		subcl	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 − 𝐶 ) ∈ ℂ )
8	7	3adant1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 − 𝐶 ) ∈ ℂ )
9	6 8	mul0ord	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · ( 𝐵 − 𝐶 ) ) = 0 ↔ ( 𝐴 = 0 ∨ ( 𝐵 − 𝐶 ) = 0 ) ) )
10		subdi	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 · 𝐵 ) − ( 𝐴 · 𝐶 ) ) )
11	10	eqeq1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · ( 𝐵 − 𝐶 ) ) = 0 ↔ ( ( 𝐴 · 𝐵 ) − ( 𝐴 · 𝐶 ) ) = 0 ) )
12		subeq0	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐵 − 𝐶 ) = 0 ↔ 𝐵 = 𝐶 ) )
13	12	3adant1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐵 − 𝐶 ) = 0 ↔ 𝐵 = 𝐶 ) )
14	13	orbi2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 = 0 ∨ ( 𝐵 − 𝐶 ) = 0 ) ↔ ( 𝐴 = 0 ∨ 𝐵 = 𝐶 ) ) )
15	9 11 14	3bitr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( ( 𝐴 · 𝐵 ) − ( 𝐴 · 𝐶 ) ) = 0 ↔ ( 𝐴 = 0 ∨ 𝐵 = 𝐶 ) ) )
16	5 15	bitr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) = ( 𝐴 · 𝐶 ) ↔ ( 𝐴 = 0 ∨ 𝐵 = 𝐶 ) ) )