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Theorem muladd11 9027
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 8840 . . . 4  |-  1  e.  CC
2 addcl 8864 . . . 4  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  +  A
)  e.  CC )
31, 2mpan 651 . . 3  |-  ( A  e.  CC  ->  (
1  +  A )  e.  CC )
4 adddi 8871 . . . 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 1265 . . 3  |-  ( ( ( 1  +  A
)  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  (
1  +  B ) )  =  ( ( ( 1  +  A
)  x.  1 )  +  ( ( 1  +  A )  x.  B ) ) )
63, 5sylan 457 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  (
1  +  B ) )  =  ( ( ( 1  +  A
)  x.  1 )  +  ( ( 1  +  A )  x.  B ) ) )
73mulid1d 8897 . . . 4  |-  ( A  e.  CC  ->  (
( 1  +  A
)  x.  1 )  =  ( 1  +  A ) )
87adantr 451 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  1 )  =  ( 1  +  A ) )
9 adddir 8875 . . . . 5  |-  ( ( 1  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( 1  +  A
)  x.  B )  =  ( ( 1  x.  B )  +  ( A  x.  B
) ) )
101, 9mp3an1 1264 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  B
)  =  ( ( 1  x.  B )  +  ( A  x.  B ) ) )
11 mulid2 8881 . . . . . 6  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
1211adantl 452 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  B
)  =  B )
1312oveq1d 5915 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  x.  B )  +  ( A  x.  B ) )  =  ( B  +  ( A  x.  B ) ) )
1410, 13eqtrd 2348 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  B
)  =  ( B  +  ( A  x.  B ) ) )
158, 14oveq12d 5918 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( 1  +  A )  x.  1 )  +  ( ( 1  +  A
)  x.  B ) )  =  ( ( 1  +  A )  +  ( B  +  ( A  x.  B
) ) ) )
166, 15eqtrd 2348 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  (
1  +  B ) )  =  ( ( 1  +  A )  +  ( B  +  ( A  x.  B
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701  (class class class)co 5900   CCcc 8780   1c1 8783    + caddc 8785    x. cmul 8787
This theorem is referenced by:  bernneq  11274
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-mulcl 8844  ax-mulcom 8846  ax-mulass 8848  ax-distr 8849  ax-1rid 8852  ax-cnre 8855
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-iota 5256  df-fv 5300  df-ov 5903
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