Theorem addsubass

Description: Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003) (Revised by Mario Carneiro, 27-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		simp1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → 𝐴 ∈ ℂ )
2		subcl	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 − 𝐶 ) ∈ ℂ )
3	2	3adant1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 − 𝐶 ) ∈ ℂ )
4		simp3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → 𝐶 ∈ ℂ )
5	1 3 4	addassd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + ( 𝐵 − 𝐶 ) ) + 𝐶 ) = ( 𝐴 + ( ( 𝐵 − 𝐶 ) + 𝐶 ) ) )
6		npcan	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐵 − 𝐶 ) + 𝐶 ) = 𝐵 )
7	6	3adant1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐵 − 𝐶 ) + 𝐶 ) = 𝐵 )
8	7	oveq2d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 + ( ( 𝐵 − 𝐶 ) + 𝐶 ) ) = ( 𝐴 + 𝐵 ) )
9	5 8	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + ( 𝐵 − 𝐶 ) ) + 𝐶 ) = ( 𝐴 + 𝐵 ) )
10	9	oveq1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( ( 𝐴 + ( 𝐵 − 𝐶 ) ) + 𝐶 ) − 𝐶 ) = ( ( 𝐴 + 𝐵 ) − 𝐶 ) )
11	1 3	addcld	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 + ( 𝐵 − 𝐶 ) ) ∈ ℂ )
12		pncan	⊢ ( ( ( 𝐴 + ( 𝐵 − 𝐶 ) ) ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( ( 𝐴 + ( 𝐵 − 𝐶 ) ) + 𝐶 ) − 𝐶 ) = ( 𝐴 + ( 𝐵 − 𝐶 ) ) )
13	11 4 12	syl2anc	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( ( 𝐴 + ( 𝐵 − 𝐶 ) ) + 𝐶 ) − 𝐶 ) = ( 𝐴 + ( 𝐵 − 𝐶 ) ) )
14	10 13	eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) − 𝐶 ) = ( 𝐴 + ( 𝐵 − 𝐶 ) ) )