Theorem ringass 9473

Description: Associative 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
ringass	|- ((R e. Ring /\ (A e. X /\ B e. X /\ C e. X)) -> ((AHB)HC) = (AH(BHC)))

Proof of Theorem ringass

Step	Hyp	Ref	Expression
1		opreq1 4889	. . . . . 6 |- (x = A -> (xHy) = (AHy))
2	1	opreq1d 4897	. . . . 5 |- (x = A -> ((xHy)Hz) = ((AHy)Hz))
3		opreq1 4889	. . . . 5 |- (x = A -> (xH(yHz)) = (AH(yHz)))
4	2, 3	eqeq12d 1899	. . . 4 |- (x = A -> (((xHy)Hz) = (xH(yHz)) <-> ((AHy)Hz) = (AH(yHz))))
5		opreq2 4890	. . . . . 6 |- (y = B -> (AHy) = (AHB))
6	5	opreq1d 4897	. . . . 5 |- (y = B -> ((AHy)Hz) = ((AHB)Hz))
7		opreq1 4889	. . . . . 6 |- (y = B -> (yHz) = (BHz))
8	7	opreq2d 4898	. . . . 5 |- (y = B -> (AH(yHz)) = (AH(BHz)))
9	6, 8	eqeq12d 1899	. . . 4 |- (y = B -> (((AHy)Hz) = (AH(yHz)) <-> ((AHB)Hz) = (AH(BHz))))
10		opreq2 4890	. . . . 5 |- (z = C -> ((AHB)Hz) = ((AHB)HC))
11		opreq2 4890	. . . . . 6 |- (z = C -> (BHz) = (BHC))
12	11	opreq2d 4898	. . . . 5 |- (z = C -> (AH(BHz)) = (AH(BHC)))
13	10, 12	eqeq12d 1899	. . . 4 |- (z = C -> (((AHB)Hz) = (AH(BHz)) <-> ((AHB)HC) = (AH(BHC))))
14	4, 9, 13	rcla43v 2386	. . 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)) -> ((AHB)HC) = (AH(BHC))))
15		ringdi.1	. . . . . . 7 |- G = (1st` R)
16		ringdi.2	. . . . . . 7 |- H = (2nd` R)
17		ringdi.3	. . . . . . 7 |- X = ran G
18	15, 16, 17	ringi 9466	. . . . . 6 |- (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))))
19	18	simprd 352	. . . . 5 |- (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)))
20	19	simplld 348	. . . 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))))
21		simp1 876	. . . . . . 7 |- ((((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) -> ((xHy)Hz) = (xH(yHz)))
22	21	ralimi 2168	. . . . . 6 |- (A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) -> A.z e. X ((xHy)Hz) = (xH(yHz)))
23	22	ralimi 2168	. . . . 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 ((xHy)Hz) = (xH(yHz)))
24	23	ralimi 2168	. . . 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 ((xHy)Hz) = (xH(yHz)))
25	20, 24	syl 12	. . 3 |- (R e. Ring -> A.x e. X A.y e. X A.z e. X ((xHy)Hz) = (xH(yHz)))
26	14, 25	syl5 20	. 2 |- ((A e. X /\ B e. X /\ C e. X) -> (R e. Ring -> ((AHB)HC) = (AH(BHC))))
27	26	impcom 378	1 |- ((R e. Ring /\ (A e. X /\ B e. X /\ C e. X)) -> ((AHB)HC) = (AH(BHC)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   X. cxp 3984  ran crn 3987  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Abelcabl 9407  Ringcring 9463

This theorem is referenced by:  unmnd 10405  zerdivemp1 14785  ringneglmul 16104  ringnegrmul 16105  zerdivemp1x 16108  isdivrng2 16111  crngm23 16150  crngm4 16151  prnc 16215

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-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-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-f 4010  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-ring 9464

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