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Theorem muladd11r 7638
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 7504 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  1  e.  CC )
31, 2addcomd 7633 . . 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 7633 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  +  1 )  =  ( 1  +  B ) )
63, 5oveq12d 5670 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  + 
1 )  x.  ( B  +  1 ) )  =  ( ( 1  +  A )  x.  ( 1  +  B ) ) )
7 muladd11 7615 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  (
1  +  B ) )  =  ( ( 1  +  A )  +  ( B  +  ( A  x.  B
) ) ) )
8 mulcl 7469 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
94, 8addcld 7507 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( B  +  ( A  x.  B ) )  e.  CC )
102, 1, 9addassd 7510 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  +  ( B  +  ( A  x.  B ) ) )  =  ( 1  +  ( A  +  ( B  +  ( A  x.  B )
) ) ) )
111, 9addcld 7507 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( B  +  ( A  x.  B ) ) )  e.  CC )
122, 11addcomd 7633 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 1  +  ( A  +  ( B  +  ( A  x.  B ) ) ) )  =  ( ( A  +  ( B  +  ( A  x.  B ) ) )  +  1 ) )
131, 4, 8addassd 7510 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  ( A  x.  B ) )  =  ( A  +  ( B  +  ( A  x.  B
) ) ) )
14 addcl 7467 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
1514, 8addcomd 7633 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  ( A  x.  B ) )  =  ( ( A  x.  B )  +  ( A  +  B ) ) )
1613, 15eqtr3d 2122 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( B  +  ( A  x.  B ) ) )  =  ( ( A  x.  B )  +  ( A  +  B ) ) )
1716oveq1d 5667 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  ( B  +  ( A  x.  B )
) )  +  1 )  =  ( ( ( A  x.  B
)  +  ( A  +  B ) )  +  1 ) )
1810, 12, 173eqtrd 2124 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  +  ( B  +  ( A  x.  B ) ) )  =  ( ( ( A  x.  B
)  +  ( A  +  B ) )  +  1 ) )
196, 7, 183eqtrd 2124 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 1289    e. wcel 1438  (class class class)co 5652   CCcc 7348   1c1 7351    + caddc 7353    x. cmul 7355
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-resscn 7437  ax-1cn 7438  ax-icn 7440  ax-addcl 7441  ax-mulcl 7443  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-1rid 7452  ax-cnre 7456
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-iota 4980  df-fv 5023  df-ov 5655
This theorem is referenced by: (None)
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