Theorem ringdiNEW 17147

Description: Distributive law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.)

Hypotheses

Ref	Expression
ringdi.1NEW	|- B = (base` R)
ringdi.2NEW	|- P = (+g` R)
ringdi.3NEW	|- T = (.r` R)

Assertion

Ref	Expression
ringdiNEW	|- ((R e. RingNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> (XT(YPZ)) = ((XTY)P(XTZ)))

Proof of Theorem ringdiNEW

Step	Hyp	Ref	Expression
1		ringdi.1NEW	. . 3 |- B = (base` R)
2		ringdi.2NEW	. . 3 |- P = (+g` R)
3		ringdi.3NEW	. . 3 |- T = (.r` R)
4	1, 2, 3	ringiNEW 17143	. 2 |- ((R e. RingNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> ((XTY) e. B /\ (((XTY)TZ) = (XT(YTZ)) /\ (XT(YPZ)) = ((XTY)P(XTZ)) /\ ((XPY)TZ) = ((XTZ)P(YTZ)))))
5		simpr2 883	. 2 |- (((XTY) e. B /\ (((XTY)TZ) = (XT(YTZ)) /\ (XT(YPZ)) = ((XTY)P(XTZ)) /\ ((XPY)TZ) = ((XTZ)P(YTZ)))) -> (XT(YPZ)) = ((XTY)P(XTZ)))
6	4, 5	syl 12	1 |- ((R e. RingNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> (XT(YPZ)) = ((XTY)P(XTZ)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  ` cfv 3998  (class class class)co 4884  basecbs 16758  +_gcplusg 17080  ._rcmulr 17085  RingNEWcrg 17086

This theorem is referenced by:  ringrzNEW 17157

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-tru 1262  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-struct 16708  df-ringNEW 17094

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