Theorem ringass 8085

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 3953	. . . . . 6 |- (x = A -> (xHy) = (AHy))
2	1	opreq1d 3960	. . . . 5 |- (x = A -> ((xHy)Hz) = ((AHy)Hz))
3		opreq1 3953	. . . . 5 |- (x = A -> (xH(yHz)) = (AH(yHz)))
4	2, 3	eqeq12d 1481	. . . 4 |- (x = A -> (((xHy)Hz) = (xH(yHz)) <-> ((AHy)Hz) = (AH(yHz))))
5		opreq2 3954	. . . . . 6 |- (y = B -> (AHy) = (AHB))
6	5	opreq1d 3960	. . . . 5 |- (y = B -> ((AHy)Hz) = ((AHB)Hz))
7		opreq1 3953	. . . . . 6 |- (y = B -> (yHz) = (BHz))
8	7	opreq2d 3961	. . . . 5 |- (y = B -> (AH(yHz)) = (AH(BHz)))
9	6, 8	eqeq12d 1481	. . . 4 |- (y = B -> (((AHy)Hz) = (AH(yHz)) <-> ((AHB)Hz) = (AH(BHz))))
10		opreq2 3954	. . . . 5 |- (z = C -> ((AHB)Hz) = ((AHB)HC))
11		opreq2 3954	. . . . . 6 |- (z = C -> (BHz) = (BHC))
12	11	opreq2d 3961	. . . . 5 |- (z = C -> (AH(BHz)) = (AH(BHC)))
13	10, 12	eqeq12d 1481	. . . 4 |- (z = C -> (((AHB)Hz) = (AH(BHz)) <-> ((AHB)HC) = (AH(BHC))))
14	4, 9, 13	rcla43v 1873	. . 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 8079	. . . . . 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	pm3.27d 325	. . . . 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	pm3.26d 321	. . . 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		3simp1 786	. . . . . . 7 |- ((((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) -> ((xHy)Hz) = (xH(yHz)))
22	21	r19.20si 1698	. . . . . 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	r19.20si 1698	. . . . 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	r19.20si 1698	. . . 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 10	. . 3 |- (R e. Ring -> A.x e. X A.y e. X A.z e. X ((xHy)Hz) = (xH(yHz)))
26	14, 25	syl5 21	. 2 |- ((A e. X /\ B e. X /\ C e. X) -> (R e. Ring -> ((AHB)HC) = (AH(BHC))))
27	26	impcom 351	1 |- ((R e. Ring /\ (A e. X /\ B e. X /\ C e. X)) -> ((AHB)HC) = (AH(BHC)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  A.wral 1637  E.wrex 1638   X. cxp 3158  ran crn 3161  -->wf 3168  ` cfv 3172  (class class class)co 3948  1stc1st 4061  2ndc2nd 4062  Abelcabl 8035  Ringcring 8076

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-opr 3950  df-1st 4063  df-2nd 4064  df-ring 8077

Copyright terms: Public domain