Theorem muladd11 8459

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 8272	. . . 4 |- 1 e. CC
2		addcl 8304	. . . 4 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 + A ) e. CC )
3	1, 2	mpan 428	. . 3 |- ( A e. CC -> ( 1 + A ) e. CC )
4		adddi 8311	. . . 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 1366	. . 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 8343	. . . 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 8317	. . . . 5 |- ( ( 1 e. CC /\ A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. B ) = ( ( 1 x. B ) + ( A x. B ) ) )
10	1, 9	mp3an1 1365	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. B ) = ( ( 1 x. B ) + ( A x. B ) ) )
11		mullid 8324	. . . . . 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 6100	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 x. B ) + ( A x. B ) ) = ( B + ( A x. B ) ) )
14	10, 13	eqtrd 2271	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. B ) = ( B + ( A x. B ) ) )
15	8, 14	oveq12d 6103	. 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 2271	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( 1 + A ) + ( B + ( A x. B ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209  (class class class)co 6085   CCcc 8177   1c1 8180   + caddc 8182   x. cmul 8184

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-resscn 8271  ax-1cn 8272  ax-icn 8274  ax-addcl 8275  ax-mulcl 8277  ax-mulcom 8280  ax-mulass 8282  ax-distr 8283  ax-1rid 8286  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088

This theorem is used by:  muladd11r  8482  bernneq  11098