Theorem npncan3

Description: Cancellation law for subtraction. (Contributed by Scott Fenton, 23-Jun-2013) (Proof shortened by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	npncan3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) + ( 𝐶 − 𝐴 ) ) = ( 𝐶 − 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		simp1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → 𝐴 ∈ ℂ )
2		subcl	⊢ ( ( 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 𝐶 − 𝐴 ) ∈ ℂ )
3	2	ancoms	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐶 − 𝐴 ) ∈ ℂ )
4	3	3adant2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐶 − 𝐴 ) ∈ ℂ )
5		simp2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → 𝐵 ∈ ℂ )
6		addsub	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐶 − 𝐴 ) ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 + ( 𝐶 − 𝐴 ) ) − 𝐵 ) = ( ( 𝐴 − 𝐵 ) + ( 𝐶 − 𝐴 ) ) )
7	1 4 5 6	syl3anc	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + ( 𝐶 − 𝐴 ) ) − 𝐵 ) = ( ( 𝐴 − 𝐵 ) + ( 𝐶 − 𝐴 ) ) )
8		pncan3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 + ( 𝐶 − 𝐴 ) ) = 𝐶 )
9	8	3adant2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 + ( 𝐶 − 𝐴 ) ) = 𝐶 )
10	9	oveq1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + ( 𝐶 − 𝐴 ) ) − 𝐵 ) = ( 𝐶 − 𝐵 ) )
11	7 10	eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) + ( 𝐶 − 𝐴 ) ) = ( 𝐶 − 𝐵 ) )