Theorem muladdi 8328

Description: Product of two sums. (Contributed by NM, 17-May-1999.)

Hypotheses

Ref	Expression
mulm1.1	|- A e. CC
mulneg.2	|- B e. CC
subdi.3	|- C e. CC
muladdi.4	|- D e. CC

Assertion

Ref	Expression
muladdi	|- ( ( A + B ) x. ( C + D ) ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) )

Proof of Theorem muladdi

Step	Hyp	Ref	Expression
1		mulm1.1	. 2 |- A e. CC
2		mulneg.2	. 2 |- B e. CC
3		subdi.3	. 2 |- C e. CC
4		muladdi.4	. 2 |- D e. CC
5		muladd 8303	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) x. ( C + D ) ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) ) )
6	1, 2, 3, 4, 5	mp4an 425	1 |- ( ( A + B ) x. ( C + D ) ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) )

Syntax hints:   = wceq 1348   e. wcel 2141  (class class class)co 5853   CCcc 7772   + caddc 7777   x. cmul 7779

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-addcl 7870  ax-mulcl 7872  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-distr 7878

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by: (None)