Theorem adddir 7018

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 7013	. . 3 |- ( ( C e. CC /\ A e. CC /\ B e. CC ) -> ( C x. ( A + B ) ) = ( ( C x. A ) + ( C x. B ) ) )
2	1	3coml 1111	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C x. ( A + B ) ) = ( ( C x. A ) + ( C x. B ) ) )
3		addcl 7006	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
4		mulcom 7010	. . . 4 |- ( ( ( A + B ) e. CC /\ C e. CC ) -> ( ( A + B ) x. C ) = ( C x. ( A + B ) ) )
5	3, 4	sylan 267	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ C e. CC ) -> ( ( A + B ) x. C ) = ( C x. ( A + B ) ) )
6	5	3impa 1099	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) x. C ) = ( C x. ( A + B ) ) )
7		mulcom 7010	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
8	7	3adant2 923	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
9		mulcom 7010	. . . 4 |- ( ( B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
10	9	3adant1 922	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
11	8, 10	oveq12d 5530	. 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 2082	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 97   /\ w3a 885   = wceq 1243   e. wcel 1393  (class class class)co 5512   CCcc 6887   + caddc 6892   x. cmul 6894

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-addcl 6980  ax-mulcom 6985  ax-distr 6988

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by:  mulid1  7024  adddiri  7038  adddird  7052  muladd11  7146  muladd  7381