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Theorem adddir 11222

Description: Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004.)

**Assertion**
Ref	Expression
adddir	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

**Proof of Theorem adddir**
Step	Hyp	Ref	Expression
1		adddi 11214	. . 3 ⊢ ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 · (𝐴 + 𝐵)) = ((𝐶 · 𝐴) + (𝐶 · 𝐵)))
2	1	3coml 1145	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐶 · (𝐴 + 𝐵)) = ((𝐶 · 𝐴) + (𝐶 · 𝐵)))
3		addcl 11207	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
4		mulcom 11211	. . 3 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐶) = (𝐶 · (𝐴 + 𝐵)))
5	3, 4	stoic3 1809	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐶) = (𝐶 · (𝐴 + 𝐵)))
6		mulcom 11211	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · 𝐶) = (𝐶 · 𝐴))
7	6	3adant2 1149	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · 𝐶) = (𝐶 · 𝐴))
8		mulcom 11211	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 · 𝐶) = (𝐶 · 𝐵))
9	8	3adant1 1148	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 · 𝐶) = (𝐶 · 𝐵))
10	7, 9	oveq12d 7432	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐶) + (𝐵 · 𝐶)) = ((𝐶 · 𝐴) + (𝐶 · 𝐵)))
11	2, 5, 10	3eqtr4d 2805	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 (class class class)co 7414 ℂcc 11123 + caddc 11128 · cmul 11130

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-ext 2732 ax-addcl 11185 ax-mulcom 11189 ax-distr 11192

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2100 df-clab 2739 df-cleq 2752 df-clel 2835 df-rab 3413 df-v 3452 df-dif 3902 df-un 3904 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-iota 6489 df-fv 6541 df-ov 7417

This theorem is used by: mulrid 11231 adddiri 11247 adddird 11259 muladd11 11405 00id 11410 cnegex2 11417 muladd 11671 ser1const 14123 hashxplem 14499 demoivreALT 16290 dvds2ln 16380 dvds2add 16381 odd2np1lem 16431 cncrng 21607 icccvx 25179 cnlmod 25369 sincosq1eq 26751 abssinper 26759 sineq0 26762 bposlem9 27529 cncvcOLD 31065 ipasslem1 31313 ipasslem11 31322 cdj3i 32923 mblfinlem3 38409 expgrowth 45160 fmtnofac2lem 48472 2zrngALT 49170