Theorem adddirp1d 8206

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 8195	. . 3 |- ( ph -> 1 e. CC )
3		adddirp1d.b	. . 3 |- ( ph -> B e. CC )
4	1, 2, 3	adddird 8205	. 2 |- ( ph -> ( ( A + 1 ) x. B ) = ( ( A x. B ) + ( 1 x. B ) ) )
5	3	mulid2d 8198	. . 3 |- ( ph -> ( 1 x. B ) = B )
6	5	oveq2d 6034	. 2 |- ( ph -> ( ( A x. B ) + ( 1 x. B ) ) = ( ( A x. B ) + B ) )
7	4, 6	eqtrd 2264	1 |- ( ph -> ( ( A + 1 ) x. B ) = ( ( A x. B ) + B ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202  (class class class)co 6018   CCcc 8030   1c1 8033   + caddc 8035   x. cmul 8037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-resscn 8124  ax-1cn 8125  ax-icn 8127  ax-addcl 8128  ax-mulcl 8130  ax-mulcom 8133  ax-mulass 8135  ax-distr 8136  ax-1rid 8139  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6021

This theorem is referenced by:  modqvalp1  10606  hashxp  11091  fsumconst  12033  divalglemnqt  12499  pcexp  12900  mulgnnass  13762  cnfldmulg  14609  2lgsoddprmlem3d  15858