Metamath Proof Explorer


Axiom ax-mulass

Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass . Proofs should normally use mulass instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994)

Ref Expression
Assertion ax-mulass ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐵 · 𝐶 ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA 𝐴
1 cc
2 0 1 wcel 𝐴 ∈ ℂ
3 cB 𝐵
4 3 1 wcel 𝐵 ∈ ℂ
5 cC 𝐶
6 5 1 wcel 𝐶 ∈ ℂ
7 2 4 6 w3a ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ )
8 cmul ·
9 0 3 8 co ( 𝐴 · 𝐵 )
10 9 5 8 co ( ( 𝐴 · 𝐵 ) · 𝐶 )
11 3 5 8 co ( 𝐵 · 𝐶 )
12 0 11 8 co ( 𝐴 · ( 𝐵 · 𝐶 ) )
13 10 12 wceq ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐵 · 𝐶 ) )
14 7 13 wi ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) · 𝐶 ) = ( 𝐴 · ( 𝐵 · 𝐶 ) ) )