Theorem muladdi 8435

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 8410	. 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 427	1 |- ( ( A + B ) x. ( C + D ) ) = ( ( ( A x. C ) + ( D x. B ) ) + ( ( A x. D ) + ( C x. B ) ) )

Syntax hints:   = wceq 1364   e. wcel 2167  (class class class)co 5922   CCcc 7877   + caddc 7882   x. cmul 7884

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-addcl 7975  ax-mulcl 7977  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-distr 7983

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925

This theorem is referenced by:  karatsuba  12599