Theorem addsub

Description: Law for addition and subtraction. (Contributed by NM, 19-Aug-2001) (Proof shortened by Andrew Salmon, 22-Oct-2011)

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

Proof

Step	Hyp	Ref	Expression
1		addcom	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
2	1	oveq1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) − 𝐶 ) = ( ( 𝐵 + 𝐴 ) − 𝐶 ) )
3	2	3adant3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) − 𝐶 ) = ( ( 𝐵 + 𝐴 ) − 𝐶 ) )
4		addsubass	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐵 + 𝐴 ) − 𝐶 ) = ( 𝐵 + ( 𝐴 − 𝐶 ) ) )
5	4	3com12	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐵 + 𝐴 ) − 𝐶 ) = ( 𝐵 + ( 𝐴 − 𝐶 ) ) )
6		subcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 − 𝐶 ) ∈ ℂ )
7		addcom	⊢ ( ( 𝐵 ∈ ℂ ∧ ( 𝐴 − 𝐶 ) ∈ ℂ ) → ( 𝐵 + ( 𝐴 − 𝐶 ) ) = ( ( 𝐴 − 𝐶 ) + 𝐵 ) )
8	6 7	sylan2	⊢ ( ( 𝐵 ∈ ℂ ∧ ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) ) → ( 𝐵 + ( 𝐴 − 𝐶 ) ) = ( ( 𝐴 − 𝐶 ) + 𝐵 ) )
9	8	3impb	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 + ( 𝐴 − 𝐶 ) ) = ( ( 𝐴 − 𝐶 ) + 𝐵 ) )
10	9	3com12	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 + ( 𝐴 − 𝐶 ) ) = ( ( 𝐴 − 𝐶 ) + 𝐵 ) )
11	3 5 10	3eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) − 𝐶 ) = ( ( 𝐴 − 𝐶 ) + 𝐵 ) )