Theorem muladd11 7241

Description: A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)

Assertion

Ref	Expression
muladd11	|- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( 1 + A ) + ( B + ( A x. B ) ) ) )

Proof of Theorem muladd11

Step	Hyp	Ref	Expression
1		ax-1cn 7069	. . . 4 |- 1 e. CC
2		addcl 7098	. . . 4 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 + A ) e. CC )
3	1, 2	mpan 414	. . 3 |- ( A e. CC -> ( 1 + A ) e. CC )
4		adddi 7105	. . . 4 |- ( ( ( 1 + A ) e. CC /\ 1 e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( ( 1 + A ) x. 1 ) + ( ( 1 + A ) x. B ) ) )
5	1, 4	mp3an2 1256	. . 3 |- ( ( ( 1 + A ) e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( ( 1 + A ) x. 1 ) + ( ( 1 + A ) x. B ) ) )
6	3, 5	sylan 277	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( ( 1 + A ) x. 1 ) + ( ( 1 + A ) x. B ) ) )
7	3	mulid1d 7136	. . . 4 |- ( A e. CC -> ( ( 1 + A ) x. 1 ) = ( 1 + A ) )
8	7	adantr 270	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. 1 ) = ( 1 + A ) )
9		adddir 7110	. . . . 5 |- ( ( 1 e. CC /\ A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. B ) = ( ( 1 x. B ) + ( A x. B ) ) )
10	1, 9	mp3an1 1255	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. B ) = ( ( 1 x. B ) + ( A x. B ) ) )
11		mulid2 7117	. . . . . 6 |- ( B e. CC -> ( 1 x. B ) = B )
12	11	adantl 271	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( 1 x. B ) = B )
13	12	oveq1d 5547	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 x. B ) + ( A x. B ) ) = ( B + ( A x. B ) ) )
14	10, 13	eqtrd 2113	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. B ) = ( B + ( A x. B ) ) )
15	8, 14	oveq12d 5550	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( 1 + A ) x. 1 ) + ( ( 1 + A ) x. B ) ) = ( ( 1 + A ) + ( B + ( A x. B ) ) ) )
16	6, 15	eqtrd 2113	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( 1 + A ) + ( B + ( A x. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433  (class class class)co 5532   CCcc 6979   1c1 6982   + caddc 6984   x. cmul 6986

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-resscn 7068  ax-1cn 7069  ax-icn 7071  ax-addcl 7072  ax-mulcl 7074  ax-mulcom 7077  ax-mulass 7079  ax-distr 7080  ax-1rid 7083  ax-cnre 7087

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by:  muladd11r  7264  bernneq  9593