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Theorem muladd11 9642
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
StepHypRef Expression
1 ax-1cn 9443 . . . 4  |-  1  e.  CC
2 addcl 9467 . . . 4  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  +  A
)  e.  CC )
31, 2mpan 670 . . 3  |-  ( A  e.  CC  ->  (
1  +  A )  e.  CC )
4 adddi 9474 . . . 4  |-  ( ( ( 1  +  A
)  e.  CC  /\  1  e.  CC  /\  B  e.  CC )  ->  (
( 1  +  A
)  x.  ( 1  +  B ) )  =  ( ( ( 1  +  A )  x.  1 )  +  ( ( 1  +  A )  x.  B
) ) )
51, 4mp3an2 1303 . . 3  |-  ( ( ( 1  +  A
)  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  (
1  +  B ) )  =  ( ( ( 1  +  A
)  x.  1 )  +  ( ( 1  +  A )  x.  B ) ) )
63, 5sylan 471 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  (
1  +  B ) )  =  ( ( ( 1  +  A
)  x.  1 )  +  ( ( 1  +  A )  x.  B ) ) )
73mulid1d 9506 . . . 4  |-  ( A  e.  CC  ->  (
( 1  +  A
)  x.  1 )  =  ( 1  +  A ) )
87adantr 465 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  1 )  =  ( 1  +  A ) )
9 adddir 9480 . . . . 5  |-  ( ( 1  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( 1  +  A
)  x.  B )  =  ( ( 1  x.  B )  +  ( A  x.  B
) ) )
101, 9mp3an1 1302 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  B
)  =  ( ( 1  x.  B )  +  ( A  x.  B ) ) )
11 mulid2 9487 . . . . . 6  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
1211adantl 466 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  B
)  =  B )
1312oveq1d 6207 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  x.  B )  +  ( A  x.  B ) )  =  ( B  +  ( A  x.  B ) ) )
1410, 13eqtrd 2492 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  B
)  =  ( B  +  ( A  x.  B ) ) )
158, 14oveq12d 6210 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( 1  +  A )  x.  1 )  +  ( ( 1  +  A
)  x.  B ) )  =  ( ( 1  +  A )  +  ( B  +  ( A  x.  B
) ) ) )
166, 15eqtrd 2492 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  (
1  +  B ) )  =  ( ( 1  +  A )  +  ( B  +  ( A  x.  B
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758  (class class class)co 6192   CCcc 9383   1c1 9386    + caddc 9388    x. cmul 9390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-mulcl 9447  ax-mulcom 9449  ax-mulass 9451  ax-distr 9452  ax-1rid 9455  ax-cnre 9458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-iota 5481  df-fv 5526  df-ov 6195
This theorem is referenced by:  bernneq  12093
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