Axiom ax-distr 10003

Description: Distributive law for complex numbers (left-distributivity). Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 9979. Proofs should normally use adddi 10025 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

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

Detailed syntax breakdown of Axiom ax-distr

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 9934	. . . 4 class ℂ
3	1, 2	wcel 1990	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 1990	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 1990	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1037	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		caddc 9939	. . . . 5 class +
10	4, 6, 9	co 6650	. . . 4 class (𝐵 + 𝐶)
11		cmul 9941	. . . 4 class ·
12	1, 10, 11	co 6650	. . 3 class (𝐴 · (𝐵 + 𝐶))
13	1, 4, 11	co 6650	. . . 4 class (𝐴 · 𝐵)
14	1, 6, 11	co 6650	. . . 4 class (𝐴 · 𝐶)
15	13, 14, 9	co 6650	. . 3 class ((𝐴 · 𝐵) + (𝐴 · 𝐶))
16	12, 15	wceq 1483	. 2 wff (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))
17	8, 16	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

This axiom is referenced by:  adddi  10025