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Theorem adddirp1d 7417
Description: Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
adddirp1d.a  |-  ( ph  ->  A  e.  CC )
adddirp1d.b  |-  ( ph  ->  B  e.  CC )
Assertion
Ref Expression
adddirp1d  |-  ( ph  ->  ( ( A  + 
1 )  x.  B
)  =  ( ( A  x.  B )  +  B ) )

Proof of Theorem adddirp1d
StepHypRef Expression
1 adddirp1d.a . . 3  |-  ( ph  ->  A  e.  CC )
2 1cnd 7407 . . 3  |-  ( ph  ->  1  e.  CC )
3 adddirp1d.b . . 3  |-  ( ph  ->  B  e.  CC )
41, 2, 3adddird 7416 . 2  |-  ( ph  ->  ( ( A  + 
1 )  x.  B
)  =  ( ( A  x.  B )  +  ( 1  x.  B ) ) )
53mulid2d 7409 . . 3  |-  ( ph  ->  ( 1  x.  B
)  =  B )
65oveq2d 5607 . 2  |-  ( ph  ->  ( ( A  x.  B )  +  ( 1  x.  B ) )  =  ( ( A  x.  B )  +  B ) )
74, 6eqtrd 2115 1  |-  ( ph  ->  ( ( A  + 
1 )  x.  B
)  =  ( ( A  x.  B )  +  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434  (class class class)co 5591   CCcc 7251   1c1 7254    + caddc 7256    x. cmul 7258
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 7340  ax-1cn 7341  ax-icn 7343  ax-addcl 7344  ax-mulcl 7346  ax-mulcom 7349  ax-mulass 7351  ax-distr 7352  ax-1rid 7355  ax-cnre 7359
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 2614  df-un 2988  df-in 2990  df-ss 2997  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-iota 4934  df-fv 4977  df-ov 5594
This theorem is referenced by:  modqvalp1  9639  hashxp  10069  divalglemnqt  10700
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