Theorem adddir 7781

Description: Distributive law for complex numbers (right-distributivity). (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 7776	. . 3 |- ( ( C e. CC /\ A e. CC /\ B e. CC ) -> ( C x. ( A + B ) ) = ( ( C x. A ) + ( C x. B ) ) )
2	1	3coml 1189	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C x. ( A + B ) ) = ( ( C x. A ) + ( C x. B ) ) )
3		addcl 7769	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
4		mulcom 7773	. . . 4 |- ( ( ( A + B ) e. CC /\ C e. CC ) -> ( ( A + B ) x. C ) = ( C x. ( A + B ) ) )
5	3, 4	sylan 281	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ C e. CC ) -> ( ( A + B ) x. C ) = ( C x. ( A + B ) ) )
6	5	3impa 1177	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) x. C ) = ( C x. ( A + B ) ) )
7		mulcom 7773	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
8	7	3adant2 1001	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
9		mulcom 7773	. . . 4 |- ( ( B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
10	9	3adant1 1000	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
11	8, 10	oveq12d 5800	. 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 2183	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 103   /\ w3a 963   = wceq 1332   e. wcel 1481  (class class class)co 5782   CCcc 7642   + caddc 7647   x. cmul 7649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-addcl 7740  ax-mulcom 7745  ax-distr 7748

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139  df-ov 5785

This theorem is referenced by:  mulid1  7787  adddiri  7801  adddird  7815  muladd11  7919  muladd  8170  demoivreALT  11516  dvds2ln  11562  dvds2add  11563  odd2np1lem  11605  sincosq1eq  12968  abssinper  12975