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Theorem muladd11 9784
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 9580 . . . 4  |-  1  e.  CC
2 addcl 9604 . . . 4  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  +  A
)  e.  CC )
31, 2mpan 668 . . 3  |-  ( A  e.  CC  ->  (
1  +  A )  e.  CC )
4 adddi 9611 . . . 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 1314 . . 3  |-  ( ( ( 1  +  A
)  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  (
1  +  B ) )  =  ( ( ( 1  +  A
)  x.  1 )  +  ( ( 1  +  A )  x.  B ) ) )
63, 5sylan 469 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  (
1  +  B ) )  =  ( ( ( 1  +  A
)  x.  1 )  +  ( ( 1  +  A )  x.  B ) ) )
73mulid1d 9643 . . . 4  |-  ( A  e.  CC  ->  (
( 1  +  A
)  x.  1 )  =  ( 1  +  A ) )
87adantr 463 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  1 )  =  ( 1  +  A ) )
9 adddir 9617 . . . . 5  |-  ( ( 1  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  (
( 1  +  A
)  x.  B )  =  ( ( 1  x.  B )  +  ( A  x.  B
) ) )
101, 9mp3an1 1313 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  B
)  =  ( ( 1  x.  B )  +  ( A  x.  B ) ) )
11 mulid2 9624 . . . . . 6  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
1211adantl 464 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  x.  B
)  =  B )
1312oveq1d 6293 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  x.  B )  +  ( A  x.  B ) )  =  ( B  +  ( A  x.  B ) ) )
1410, 13eqtrd 2443 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  B
)  =  ( B  +  ( A  x.  B ) ) )
158, 14oveq12d 6296 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( 1  +  A )  x.  1 )  +  ( ( 1  +  A
)  x.  B ) )  =  ( ( 1  +  A )  +  ( B  +  ( A  x.  B
) ) ) )
166, 15eqtrd 2443 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 367    = wceq 1405    e. wcel 1842  (class class class)co 6278   CCcc 9520   1c1 9523    + caddc 9525    x. cmul 9527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-mulcl 9584  ax-mulcom 9586  ax-mulass 9588  ax-distr 9589  ax-1rid 9592  ax-cnre 9595
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-iota 5533  df-fv 5577  df-ov 6281
This theorem is referenced by:  bernneq  12336
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