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Theorem muladd11 7863
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 7681 . . . 4  |-  1  e.  CC
2 addcl 7713 . . . 4  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  +  A
)  e.  CC )
31, 2mpan 420 . . 3  |-  ( A  e.  CC  ->  (
1  +  A )  e.  CC )
4 adddi 7720 . . . 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 1288 . . 3  |-  ( ( ( 1  +  A
)  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  (
1  +  B ) )  =  ( ( ( 1  +  A
)  x.  1 )  +  ( ( 1  +  A )  x.  B ) ) )
63, 5sylan 281 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  (
1  +  B ) )  =  ( ( ( 1  +  A
)  x.  1 )  +  ( ( 1  +  A )  x.  B ) ) )
73mulid1d 7751 . . . 4  |-  ( A  e.  CC  ->  (
( 1  +  A
)  x.  1 )  =  ( 1  +  A ) )
87adantr 274 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  1 )  =  ( 1  +  A ) )
9 adddir 7725 . . . . 5  |-  ( ( 1  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( 1  +  A
)  x.  B )  =  ( ( 1  x.  B )  +  ( A  x.  B
) ) )
101, 9mp3an1 1287 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  B
)  =  ( ( 1  x.  B )  +  ( A  x.  B ) ) )
11 mulid2 7732 . . . . . 6  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
1211adantl 275 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  B
)  =  B )
1312oveq1d 5757 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  x.  B )  +  ( A  x.  B ) )  =  ( B  +  ( A  x.  B ) ) )
1410, 13eqtrd 2150 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  B
)  =  ( B  +  ( A  x.  B ) ) )
158, 14oveq12d 5760 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( 1  +  A )  x.  1 )  +  ( ( 1  +  A
)  x.  B ) )  =  ( ( 1  +  A )  +  ( B  +  ( A  x.  B
) ) ) )
166, 15eqtrd 2150 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 103    = wceq 1316    e. wcel 1465  (class class class)co 5742   CCcc 7586   1c1 7589    + caddc 7591    x. cmul 7593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-resscn 7680  ax-1cn 7681  ax-icn 7683  ax-addcl 7684  ax-mulcl 7686  ax-mulcom 7689  ax-mulass 7691  ax-distr 7692  ax-1rid 7695  ax-cnre 7699
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-iota 5058  df-fv 5101  df-ov 5745
This theorem is referenced by:  muladd11r  7886  bernneq  10380
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