Theorem adddir 8169

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 8163	. . 3 |- ( ( C e. CC /\ A e. CC /\ B e. CC ) -> ( C x. ( A + B ) ) = ( ( C x. A ) + ( C x. B ) ) )
2	1	3coml 1236	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C x. ( A + B ) ) = ( ( C x. A ) + ( C x. B ) ) )
3		addcl 8156	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
4		mulcom 8160	. . . 4 |- ( ( ( A + B ) e. CC /\ C e. CC ) -> ( ( A + B ) x. C ) = ( C x. ( A + B ) ) )
5	3, 4	sylan 283	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ C e. CC ) -> ( ( A + B ) x. C ) = ( C x. ( A + B ) ) )
6	5	3impa 1220	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) x. C ) = ( C x. ( A + B ) ) )
7		mulcom 8160	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
8	7	3adant2 1042	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
9		mulcom 8160	. . . 4 |- ( ( B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
10	9	3adant1 1041	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
11	8, 10	oveq12d 6035	. 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 2274	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 104   /\ w3a 1004   = wceq 1397   e. wcel 2202  (class class class)co 6017   CCcc 8029   + caddc 8034   x. cmul 8036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-addcl 8127  ax-mulcom 8132  ax-distr 8135

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6020

This theorem is referenced by:  mulrid  8175  adddiri  8189  adddird  8204  muladd11  8311  muladd  8562  demoivreALT  12334  dvds2ln  12384  dvds2add  12385  odd2np1lem  12432  cncrng  14582  sincosq1eq  15562  abssinper  15569