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Theorem muladd11r 7925
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 108 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
2 1cnd 7789 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  1  e.  CC )
31, 2addcomd 7920 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  1 )  =  ( 1  +  A ) )
4 simpr 109 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
54, 2addcomd 7920 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  +  1 )  =  ( 1  +  B ) )
63, 5oveq12d 5792 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  + 
1 )  x.  ( B  +  1 ) )  =  ( ( 1  +  A )  x.  ( 1  +  B ) ) )
7 muladd11 7902 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  (
1  +  B ) )  =  ( ( 1  +  A )  +  ( B  +  ( A  x.  B
) ) ) )
8 mulcl 7754 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
94, 8addcld 7792 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  +  ( A  x.  B ) )  e.  CC )
102, 1, 9addassd 7795 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  +  ( B  +  ( A  x.  B ) ) )  =  ( 1  +  ( A  +  ( B  +  ( A  x.  B )
) ) ) )
111, 9addcld 7792 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( B  +  ( A  x.  B ) ) )  e.  CC )
122, 11addcomd 7920 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  +  ( A  +  ( B  +  ( A  x.  B ) ) ) )  =  ( ( A  +  ( B  +  ( A  x.  B ) ) )  +  1 ) )
131, 4, 8addassd 7795 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  ( A  x.  B ) )  =  ( A  +  ( B  +  ( A  x.  B
) ) ) )
14 addcl 7752 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
1514, 8addcomd 7920 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  ( A  x.  B ) )  =  ( ( A  x.  B )  +  ( A  +  B ) ) )
1613, 15eqtr3d 2174 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( B  +  ( A  x.  B ) ) )  =  ( ( A  x.  B )  +  ( A  +  B ) ) )
1716oveq1d 5789 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  ( B  +  ( A  x.  B )
) )  +  1 )  =  ( ( ( A  x.  B
)  +  ( A  +  B ) )  +  1 ) )
1810, 12, 173eqtrd 2176 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  +  ( B  +  ( A  x.  B ) ) )  =  ( ( ( A  x.  B
)  +  ( A  +  B ) )  +  1 ) )
196, 7, 183eqtrd 2176 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 103    = wceq 1331    e. wcel 1480  (class class class)co 5774   CCcc 7625   1c1 7628    + caddc 7630    x. cmul 7632
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-resscn 7719  ax-1cn 7720  ax-icn 7722  ax-addcl 7723  ax-mulcl 7725  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-1rid 7734  ax-cnre 7738
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777
This theorem is referenced by: (None)
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