Axiom ax-distr

Description: Distributive law for complex numbers (left-distributivity). Axiom 11 of 22 for real and complex numbers, justified by Theorem axdistr . Proofs should normally use adddi instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994)

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

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		cC	⊢ 𝐶
6	5 1	wcel	⊢ 𝐶 ∈ ℂ
7	2 4 6	w3a	⊢ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ )
8		cmul	⊢ ·
9		caddc	⊢ +
10	3 5 9	co	⊢ ( 𝐵 + 𝐶 )
11	0 10 8	co	⊢ ( 𝐴 · ( 𝐵 + 𝐶 ) )
12	0 3 8	co	⊢ ( 𝐴 · 𝐵 )
13	0 5 8	co	⊢ ( 𝐴 · 𝐶 )
14	12 13 9	co	⊢ ( ( 𝐴 · 𝐵 ) + ( 𝐴 · 𝐶 ) )
15	11 14	wceq	⊢ ( 𝐴 · ( 𝐵 + 𝐶 ) ) = ( ( 𝐴 · 𝐵 ) + ( 𝐴 · 𝐶 ) )
16	7 15	wi	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · ( 𝐵 + 𝐶 ) ) = ( ( 𝐴 · 𝐵 ) + ( 𝐴 · 𝐶 ) ) )