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Metamath Proof Explorer

Theorem mul32

Description: Commutative/associative law. (Contributed by NM, 8-Oct-1999)

Ref Expression
Assertion mul32 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) · 𝐵 ) )

Proof

Step Hyp Ref Expression
1 mulcom ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
2 1 oveq2d ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · ( 𝐵 · 𝐶 ) ) = ( 𝐴 · ( 𝐶 · 𝐵 ) ) )
3 2 3adant1 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · ( 𝐵 · 𝐶 ) ) = ( 𝐴 · ( 𝐶 · 𝐵 ) ) )
4 mulass ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐵 · 𝐶 ) ) )
5 mulass ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐶 ) · 𝐵 ) = ( 𝐴 · ( 𝐶 · 𝐵 ) ) )
6 5 3com23 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐶 ) · 𝐵 ) = ( 𝐴 · ( 𝐶 · 𝐵 ) ) )
7 3 4 6 3eqtr4d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) · 𝐵 ) )