Theorem ringdir 8147

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

Hypotheses

Ref	Expression
ringdi.1	|- G = (1st` R)
ringdi.2	|- H = (2nd` R)
ringdi.3	|- X = ran G

Assertion

Ref	Expression
ringdir	|- ((R e. Ring /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)HC) = ((AHC)G(BHC)))

Proof of Theorem ringdir

Step	Hyp	Ref	Expression
1		opreq1 3968	. . . . . . 7 |- (x = A -> (xGy) = (AGy))
2	1	opreq1d 3975	. . . . . 6 |- (x = A -> ((xGy)Hz) = ((AGy)Hz))
3		opreq1 3968	. . . . . . 7 |- (x = A -> (xHz) = (AHz))
4	3	opreq1d 3975	. . . . . 6 |- (x = A -> ((xHz)G(yHz)) = ((AHz)G(yHz)))
5	2, 4	eqeq12d 1489	. . . . 5 |- (x = A -> (((xGy)Hz) = ((xHz)G(yHz)) <-> ((AGy)Hz) = ((AHz)G(yHz))))
6		opreq2 3969	. . . . . . 7 |- (y = B -> (AGy) = (AGB))
7	6	opreq1d 3975	. . . . . 6 |- (y = B -> ((AGy)Hz) = ((AGB)Hz))
8		opreq1 3968	. . . . . . 7 |- (y = B -> (yHz) = (BHz))
9	8	opreq2d 3976	. . . . . 6 |- (y = B -> ((AHz)G(yHz)) = ((AHz)G(BHz)))
10	7, 9	eqeq12d 1489	. . . . 5 |- (y = B -> (((AGy)Hz) = ((AHz)G(yHz)) <-> ((AGB)Hz) = ((AHz)G(BHz))))
11		opreq2 3969	. . . . . 6 |- (z = C -> ((AGB)Hz) = ((AGB)HC))
12		opreq2 3969	. . . . . . 7 |- (z = C -> (AHz) = (AHC))
13		opreq2 3969	. . . . . . 7 |- (z = C -> (BHz) = (BHC))
14	12, 13	opreq12d 3978	. . . . . 6 |- (z = C -> ((AHz)G(BHz)) = ((AHC)G(BHC)))
15	11, 14	eqeq12d 1489	. . . . 5 |- (z = C -> (((AGB)Hz) = ((AHz)G(BHz)) <-> ((AGB)HC) = ((AHC)G(BHC))))
16	5, 10, 15	rcla43v 1882	. . . 4 |- ((A e. X /\ B e. X /\ C e. X) -> (A.x e. X A.y e. X A.z e. X ((xGy)Hz) = ((xHz)G(yHz)) -> ((AGB)HC) = ((AHC)G(BHC))))
17		3simp3 790	. . . . . . 7 |- ((((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) -> ((xGy)Hz) = ((xHz)G(yHz)))
18	17	r19.20si 1706	. . . . . 6 |- (A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) -> A.z e. X ((xGy)Hz) = ((xHz)G(yHz)))
19	18	r19.20si 1706	. . . . 5 |- (A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) -> A.y e. X A.z e. X ((xGy)Hz) = ((xHz)G(yHz)))
20	19	r19.20si 1706	. . . 4 |- (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) -> A.x e. X A.y e. X A.z e. X ((xGy)Hz) = ((xHz)G(yHz)))
21	16, 20	syl5 21	. . 3 |- ((A e. X /\ B e. X /\ C e. X) -> (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) -> ((AGB)HC) = ((AHC)G(BHC))))
22		ringdi.1	. . . . . 6 |- G = (1st` R)
23		ringdi.2	. . . . . 6 |- H = (2nd` R)
24		ringdi.3	. . . . . 6 |- X = ran G
25	22, 23, 24	ringi 8142	. . . . 5 |- (R e. Ring -> ((G e. Abel /\ H:(X X. X)-->X) /\ (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y))))
26	25	pm3.27d 325	. . . 4 |- (R e. Ring -> (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y)))
27	26	pm3.26d 321	. . 3 |- (R e. Ring -> A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))))
28	21, 27	syl5 21	. 2 |- ((A e. X /\ B e. X /\ C e. X) -> (R e. Ring -> ((AGB)HC) = ((AHC)G(BHC))))
29	28	impcom 351	1 |- ((R e. Ring /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)HC) = ((AHC)G(BHC)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646   X. cxp 3168  ran crn 3171  -->wf 3178  ` cfv 3182  (class class class)co 3963  1stc1st 4077  2ndc2nd 4078  Abelcabl 8099  Ringcring 8139

This theorem is referenced by:  ring2 8149

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-1st 4079  df-2nd 4080  df-ring 8140

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