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Theorem adddirp1d 7575
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 7565 . . 3  |-  ( ph  ->  1  e.  CC )
3 adddirp1d.b . . 3  |-  ( ph  ->  B  e.  CC )
41, 2, 3adddird 7574 . 2  |-  ( ph  ->  ( ( A  + 
1 )  x.  B
)  =  ( ( A  x.  B )  +  ( 1  x.  B ) ) )
53mulid2d 7567 . . 3  |-  ( ph  ->  ( 1  x.  B
)  =  B )
65oveq2d 5682 . 2  |-  ( ph  ->  ( ( A  x.  B )  +  ( 1  x.  B ) )  =  ( ( A  x.  B )  +  B ) )
74, 6eqtrd 2121 1  |-  ( ph  ->  ( ( A  + 
1 )  x.  B
)  =  ( ( A  x.  B )  +  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1290    e. wcel 1439  (class class class)co 5666   CCcc 7409   1c1 7412    + caddc 7414    x. cmul 7416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-resscn 7498  ax-1cn 7499  ax-icn 7501  ax-addcl 7502  ax-mulcl 7504  ax-mulcom 7507  ax-mulass 7509  ax-distr 7510  ax-1rid 7513  ax-cnre 7517
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-iota 4993  df-fv 5036  df-ov 5669
This theorem is referenced by:  modqvalp1  9811  hashxp  10295  fsumconst  10909  divalglemnqt  11259
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