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Theorem muladd11r 7401
Description: A simple product of sums expansion. (Contributed by AV, 30-Jul-2021.)
Assertion
Ref Expression
muladd11r  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  + 
1 )  x.  ( B  +  1 ) )  =  ( ( ( A  x.  B
)  +  ( A  +  B ) )  +  1 ) )

Proof of Theorem muladd11r
StepHypRef Expression
1 simpl 107 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
2 1cnd 7267 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  1  e.  CC )
31, 2addcomd 7396 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  1 )  =  ( 1  +  A ) )
4 simpr 108 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
54, 2addcomd 7396 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  +  1 )  =  ( 1  +  B ) )
63, 5oveq12d 5582 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  + 
1 )  x.  ( B  +  1 ) )  =  ( ( 1  +  A )  x.  ( 1  +  B ) ) )
7 muladd11 7378 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  (
1  +  B ) )  =  ( ( 1  +  A )  +  ( B  +  ( A  x.  B
) ) ) )
8 mulcl 7232 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
94, 8addcld 7270 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  +  ( A  x.  B ) )  e.  CC )
102, 1, 9addassd 7273 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  +  ( B  +  ( A  x.  B ) ) )  =  ( 1  +  ( A  +  ( B  +  ( A  x.  B )
) ) ) )
111, 9addcld 7270 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( B  +  ( A  x.  B ) ) )  e.  CC )
122, 11addcomd 7396 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  +  ( A  +  ( B  +  ( A  x.  B ) ) ) )  =  ( ( A  +  ( B  +  ( A  x.  B ) ) )  +  1 ) )
131, 4, 8addassd 7273 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  ( A  x.  B ) )  =  ( A  +  ( B  +  ( A  x.  B
) ) ) )
14 addcl 7230 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
1514, 8addcomd 7396 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  ( A  x.  B ) )  =  ( ( A  x.  B )  +  ( A  +  B ) ) )
1613, 15eqtr3d 2117 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( B  +  ( A  x.  B ) ) )  =  ( ( A  x.  B )  +  ( A  +  B ) ) )
1716oveq1d 5579 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  ( B  +  ( A  x.  B )
) )  +  1 )  =  ( ( ( A  x.  B
)  +  ( A  +  B ) )  +  1 ) )
1810, 12, 173eqtrd 2119 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  +  ( B  +  ( A  x.  B ) ) )  =  ( ( ( A  x.  B
)  +  ( A  +  B ) )  +  1 ) )
196, 7, 183eqtrd 2119 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  + 
1 )  x.  ( B  +  1 ) )  =  ( ( ( A  x.  B
)  +  ( A  +  B ) )  +  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434  (class class class)co 5564   CCcc 7111   1c1 7114    + caddc 7116    x. cmul 7118
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-resscn 7200  ax-1cn 7201  ax-icn 7203  ax-addcl 7204  ax-mulcl 7206  ax-addcom 7208  ax-mulcom 7209  ax-addass 7210  ax-mulass 7211  ax-distr 7212  ax-1rid 7215  ax-cnre 7219
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-iota 4917  df-fv 4960  df-ov 5567
This theorem is referenced by: (None)
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