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Metamath Proof Explorer

Axiom ax-mulrcl

Description: Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by Theorem axmulrcl . Proofs should normally use remulcl instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994)

		Ref	Expression
	Assertion	ax-mulrcl	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 · 𝐵 ) ∈ ℝ )

Detailed syntax breakdown

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