Theorem adddir 9088

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

Assertion

Ref	Expression
adddir	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) )

Proof of Theorem adddir

Step	Hyp	Ref	Expression
1		adddi 9084	. . 3 |- ( ( C e. CC /\ A e. CC /\ B e. CC ) -> ( C x. ( A + B ) ) = ( ( C x. A ) + ( C x. B ) ) )
2	1	3coml 1161	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C x. ( A + B ) ) = ( ( C x. A ) + ( C x. B ) ) )
3		addcl 9077	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
4		mulcom 9081	. . . 4 |- ( ( ( A + B ) e. CC /\ C e. CC ) -> ( ( A + B ) x. C ) = ( C x. ( A + B ) ) )
5	3, 4	sylan 459	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ C e. CC ) -> ( ( A + B ) x. C ) = ( C x. ( A + B ) ) )
6	5	3impa 1149	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) x. C ) = ( C x. ( A + B ) ) )
7		mulcom 9081	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
8	7	3adant2 977	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
9		mulcom 9081	. . . 4 |- ( ( B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
10	9	3adant1 976	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
11	8, 10	oveq12d 6102	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. C ) + ( B x. C ) ) = ( ( C x. A ) + ( C x. B ) ) )
12	2, 6, 11	3eqtr4d 2480	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) )

Syntax hints:   -> wi 4   /\ wa 360   /\ w3a 937   = wceq 1653   e. wcel 1726  (class class class)co 6084   CCcc 8993   + caddc 8998   x. cmul 9000

This theorem is referenced by:  mulid1  9093  adddiri  9106  adddird  9118  muladd11  9241  00id  9246  cnegex2  9253  muladd  9471  ser1const  11384  hashxplem  11701  demoivreALT  12807  dvds2ln  12885  dvds2add  12886  odd2np1lem  12912  cncrng  16727  icccvx  18980  sincosq1eq  20425  abssinper  20431  sineq0  20434  bposlem9  21081  cnrngo  21996  cncvc  22067  ipasslem1  22337  ipasslem11  22346  cdj3i  23949  mblfinlem3  26257  expgrowth  27543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-addcl 9055  ax-mulcom 9059  ax-distr 9062

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-ov 6087