Theorem muladd11 8205

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 8018	. . . 4 |- 1 e. CC
2		addcl 8050	. . . 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 8057	. . . 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 1338	. . 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 8089	. . . 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 8063	. . . . 5 |- ( ( 1 e. CC /\ A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. B ) = ( ( 1 x. B ) + ( A x. B ) ) )
10	1, 9	mp3an1 1337	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. B ) = ( ( 1 x. B ) + ( A x. B ) ) )
11		mullid 8070	. . . . . 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 5959	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 x. B ) + ( A x. B ) ) = ( B + ( A x. B ) ) )
14	10, 13	eqtrd 2238	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. B ) = ( B + ( A x. B ) ) )
15	8, 14	oveq12d 5962	. 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 2238	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 1373   e. wcel 2176  (class class class)co 5944   CCcc 7923   1c1 7926   + caddc 7928   x. cmul 7930

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-resscn 8017  ax-1cn 8018  ax-icn 8020  ax-addcl 8021  ax-mulcl 8023  ax-mulcom 8026  ax-mulass 8028  ax-distr 8029  ax-1rid 8032  ax-cnre 8036

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947

This theorem is referenced by:  muladd11r  8228  bernneq  10805