Theorem adddirp1d 7145

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

Step	Hyp	Ref	Expression
1		adddirp1d.a	. . 3 |- ( ph -> A e. CC )
2		1cnd 7135	. . 3 |- ( ph -> 1 e. CC )
3		adddirp1d.b	. . 3 |- ( ph -> B e. CC )
4	1, 2, 3	adddird 7144	. 2 |- ( ph -> ( ( A + 1 ) x. B ) = ( ( A x. B ) + ( 1 x. B ) ) )
5	3	mulid2d 7137	. . 3 |- ( ph -> ( 1 x. B ) = B )
6	5	oveq2d 5548	. 2 |- ( ph -> ( ( A x. B ) + ( 1 x. B ) ) = ( ( A x. B ) + B ) )
7	4, 6	eqtrd 2113	1 |- ( ph -> ( ( A + 1 ) x. B ) = ( ( A x. B ) + B ) )

Syntax hints:   -> wi 4   = wceq 1284   e. wcel 1433  (class class class)co 5532   CCcc 6979   1c1 6982   + caddc 6984   x. cmul 6986

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-resscn 7068  ax-1cn 7069  ax-icn 7071  ax-addcl 7072  ax-mulcl 7074  ax-mulcom 7077  ax-mulass 7079  ax-distr 7080  ax-1rid 7083  ax-cnre 7087

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by:  modqvalp1  9345  divalglemnqt  10320