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Theorem adddir 10367

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 10361	. . 3 ⊢ ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 · (𝐴 + 𝐵)) = ((𝐶 · 𝐴) + (𝐶 · 𝐵)))
2	1	3coml 1118	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐶 · (𝐴 + 𝐵)) = ((𝐶 · 𝐴) + (𝐶 · 𝐵)))
3		addcl 10354	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
4		mulcom 10358	. . 3 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐶) = (𝐶 · (𝐴 + 𝐵)))
5	3, 4	stoic3 1820	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐶) = (𝐶 · (𝐴 + 𝐵)))
6		mulcom 10358	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · 𝐶) = (𝐶 · 𝐴))
7	6	3adant2 1122	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · 𝐶) = (𝐶 · 𝐴))
8		mulcom 10358	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 · 𝐶) = (𝐶 · 𝐵))
9	8	3adant1 1121	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 · 𝐶) = (𝐶 · 𝐵))
10	7, 9	oveq12d 6940	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐶) + (𝐵 · 𝐶)) = ((𝐶 · 𝐴) + (𝐶 · 𝐵)))
11	2, 5, 10	3eqtr4d 2824	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 (class class class)co 6922 ℂcc 10270 + caddc 10275 · cmul 10277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-addcl 10332 ax-mulcom 10336 ax-distr 10339

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-iota 6099 df-fv 6143 df-ov 6925

This theorem is referenced by: mulid1 10374 adddiri 10390 adddird 10402 muladd11 10546 00id 10551 cnegex2 10558 muladd 10807 ser1const 13175 hashxplem 13534 demoivreALT 15333 dvds2ln 15421 dvds2add 15422 odd2np1lem 15468 cncrng 20163 icccvx 23157 cnlmod 23347 sincosq1eq 24702 abssinper 24708 sineq0 24711 bposlem9 25469 cncvcOLD 28010 ipasslem1 28258 ipasslem11 28267 cdj3i 29872 mblfinlem3 34076 expgrowth 39494 fmtnofac2lem 42505 2zrngALT 42967