Theorem muladdi 8428

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 8403	. 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 2164  (class class class)co 5918   CCcc 7870   + caddc 7875   x. cmul 7877

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-addcl 7968  ax-mulcl 7970  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-distr 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921

This theorem is referenced by: (None)