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Metamath Proof Explorer

Axiom ax-mulcom

Description: Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by Theorem axmulcom . Proofs should normally use mulcom instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994)

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

Detailed syntax breakdown

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