Theorem dmdcan

Description: Cancellation law for division and multiplication. (Contributed by Scott Fenton, 7-Jun-2013) (Proof shortened by Fan Zheng, 3-Jul-2016)

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

Proof

Step	Hyp	Ref	Expression
1		simp1l	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → 𝐴 ∈ ℂ )
2		simp3	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → 𝐶 ∈ ℂ )
3		simp1r	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → 𝐴 ≠ 0 )
4		divcl	⊢ ( ( 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 𝐶 / 𝐴 ) ∈ ℂ )
5	2 1 3 4	syl3anc	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( 𝐶 / 𝐴 ) ∈ ℂ )
6		simp2l	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → 𝐵 ∈ ℂ )
7		simp2r	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → 𝐵 ≠ 0 )
8		div23	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐶 / 𝐴 ) ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 · ( 𝐶 / 𝐴 ) ) / 𝐵 ) = ( ( 𝐴 / 𝐵 ) · ( 𝐶 / 𝐴 ) ) )
9	1 5 6 7 8	syl112anc	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · ( 𝐶 / 𝐴 ) ) / 𝐵 ) = ( ( 𝐴 / 𝐵 ) · ( 𝐶 / 𝐴 ) ) )
10		divcan2	⊢ ( ( 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 𝐴 · ( 𝐶 / 𝐴 ) ) = 𝐶 )
11	2 1 3 10	syl3anc	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · ( 𝐶 / 𝐴 ) ) = 𝐶 )
12	11	oveq1d	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · ( 𝐶 / 𝐴 ) ) / 𝐵 ) = ( 𝐶 / 𝐵 ) )
13	9 12	eqtr3d	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 / 𝐵 ) · ( 𝐶 / 𝐴 ) ) = ( 𝐶 / 𝐵 ) )