Theorem adddii 8189

Description: Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.)

Hypotheses

Ref	Expression
axi.1	⊢ 𝐴 ∈ ℂ
axi.2	⊢ 𝐵 ∈ ℂ
axi.3	⊢ 𝐶 ∈ ℂ

Assertion

Ref	Expression
adddii	⊢ (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))

Proof of Theorem adddii

Step	Hyp	Ref	Expression
1		axi.1	. 2 ⊢ 𝐴 ∈ ℂ
2		axi.2	. 2 ⊢ 𝐵 ∈ ℂ
3		axi.3	. 2 ⊢ 𝐶 ∈ ℂ
4		adddi 8164	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
5	1, 2, 3, 4	mp3an 1373	1 ⊢ (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))

Syntax hints:   = wceq 1397   ∈ wcel 2202  (class class class)co 6018  ℂcc 8030   + caddc 8035   · cmul 8037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-distr 8136

This theorem depends on definitions:  df-bi 117  df-3an 1006

This theorem is referenced by:  3t3e9  9301  numltc  9636  numsucc  9650  numma  9654  decmul10add  9679  4t3lem  9707  9t11e99  9740  decbin2  9751  binom2i  10911  3dec  10977  3dvds2dec  12432  decsplit  13007