Theorem add12

Description: Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004)

	Ref	Expression
Assertion	add12	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 + ( 𝐵 + 𝐶 ) ) = ( 𝐵 + ( 𝐴 + 𝐶 ) ) )

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		addass	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) + 𝐶 ) = ( 𝐴 + ( 𝐵 + 𝐶 ) ) )
5		addass	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐵 + 𝐴 ) + 𝐶 ) = ( 𝐵 + ( 𝐴 + 𝐶 ) ) )
6	5	3com12	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐵 + 𝐴 ) + 𝐶 ) = ( 𝐵 + ( 𝐴 + 𝐶 ) ) )
7	3 4 6	3eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 + ( 𝐵 + 𝐶 ) ) = ( 𝐵 + ( 𝐴 + 𝐶 ) ) )