Theorem muladd11 8108

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 7922	. . . 4 |- 1 e. CC
2		addcl 7954	. . . 4 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 + A ) e. CC )
3	1, 2	mpan 424	. . 3 |- ( A e. CC -> ( 1 + A ) e. CC )
4		adddi 7961	. . . 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 1336	. . 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 283	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( ( 1 + A ) x. 1 ) + ( ( 1 + A ) x. B ) ) )
7	3	mulridd 7992	. . . 4 |- ( A e. CC -> ( ( 1 + A ) x. 1 ) = ( 1 + A ) )
8	7	adantr 276	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. 1 ) = ( 1 + A ) )
9		adddir 7966	. . . . 5 |- ( ( 1 e. CC /\ A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. B ) = ( ( 1 x. B ) + ( A x. B ) ) )
10	1, 9	mp3an1 1335	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. B ) = ( ( 1 x. B ) + ( A x. B ) ) )
11		mullid 7973	. . . . . 6 |- ( B e. CC -> ( 1 x. B ) = B )
12	11	adantl 277	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( 1 x. B ) = B )
13	12	oveq1d 5906	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 x. B ) + ( A x. B ) ) = ( B + ( A x. B ) ) )
14	10, 13	eqtrd 2222	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. B ) = ( B + ( A x. B ) ) )
15	8, 14	oveq12d 5909	. 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 2222	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( 1 + A ) + ( B + ( A x. B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160  (class class class)co 5891   CCcc 7827   1c1 7830   + caddc 7832   x. cmul 7834

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 2171  ax-resscn 7921  ax-1cn 7922  ax-icn 7924  ax-addcl 7925  ax-mulcl 7927  ax-mulcom 7930  ax-mulass 7932  ax-distr 7933  ax-1rid 7936  ax-cnre 7940

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5193  df-fv 5239  df-ov 5894

This theorem is referenced by:  muladd11r  8131  bernneq  10659