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Metamath Proof Explorer

Theorem mul31

Description: Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013)

Ref Expression
Assertion mul31 ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( ( 𝐶 · 𝐵 ) · 𝐴 ) )

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 mulcl ⊢ ( ( 𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐶 · 𝐵 ) ∈ ℂ )
6 5 ancoms ⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐶 · 𝐵 ) ∈ ℂ )
7 6 3adant1 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐶 · 𝐵 ) ∈ ℂ )
8 simp1 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → 𝐴 ∈ ℂ )
9 7 8 mulcomd ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐶 · 𝐵 ) · 𝐴 ) = ( 𝐴 · ( 𝐶 · 𝐵 ) ) )
10 3 4 9 3eqtr4d ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( ( 𝐶 · 𝐵 ) · 𝐴 ) )