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Metamath Proof Explorer

Theorem add32r

Description: Commutative/associative law that swaps the last two terms in a triple sum, rearranging the parentheses. (Contributed by Paul Chapman, 18-May-2007)

		Ref	Expression
	Assertion	add32r	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 + ( 𝐵 + 𝐶 ) ) = ( ( 𝐴 + 𝐶 ) + 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		addass	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) + 𝐶 ) = ( 𝐴 + ( 𝐵 + 𝐶 ) ) )
2		add32	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) + 𝐶 ) = ( ( 𝐴 + 𝐶 ) + 𝐵 ) )
3	1 2	eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 + ( 𝐵 + 𝐶 ) ) = ( ( 𝐴 + 𝐶 ) + 𝐵 ) )