Theorem pnncan

Description: Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005) (Revised by Mario Carneiro, 27-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		simp1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → 𝐴 ∈ ℂ )
2		simp2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → 𝐵 ∈ ℂ )
3	1 2	addcld	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 + 𝐵 ) ∈ ℂ )
4		simp3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → 𝐶 ∈ ℂ )
5		subsub	⊢ ( ( ( 𝐴 + 𝐵 ) ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) − ( 𝐴 − 𝐶 ) ) = ( ( ( 𝐴 + 𝐵 ) − 𝐴 ) + 𝐶 ) )
6	3 1 4 5	syl3anc	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) − ( 𝐴 − 𝐶 ) ) = ( ( ( 𝐴 + 𝐵 ) − 𝐴 ) + 𝐶 ) )
7		pncan2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) − 𝐴 ) = 𝐵 )
8	7	3adant3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) − 𝐴 ) = 𝐵 )
9	8	oveq1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( ( 𝐴 + 𝐵 ) − 𝐴 ) + 𝐶 ) = ( 𝐵 + 𝐶 ) )
10	6 9	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) − ( 𝐴 − 𝐶 ) ) = ( 𝐵 + 𝐶 ) )