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Metamath Proof Explorer

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 ) ) )