Theorem grpass 8044

Description: A group operation is associative.

Hypothesis

Ref	Expression
grpfo.1	|- X = ran G

Assertion

Ref	Expression
grpass	|- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)GC) = (AG(BGC)))

Proof of Theorem grpass

Step	Hyp	Ref	Expression
1		opreq1 3974	. . . . . 6 |- (x = A -> (xGy) = (AGy))
2	1	opreq1d 3981	. . . . 5 |- (x = A -> ((xGy)Gz) = ((AGy)Gz))
3		opreq1 3974	. . . . 5 |- (x = A -> (xG(yGz)) = (AG(yGz)))
4	2, 3	eqeq12d 1492	. . . 4 |- (x = A -> (((xGy)Gz) = (xG(yGz)) <-> ((AGy)Gz) = (AG(yGz))))
5		opreq2 3975	. . . . . 6 |- (y = B -> (AGy) = (AGB))
6	5	opreq1d 3981	. . . . 5 |- (y = B -> ((AGy)Gz) = ((AGB)Gz))
7		opreq1 3974	. . . . . 6 |- (y = B -> (yGz) = (BGz))
8	7	opreq2d 3982	. . . . 5 |- (y = B -> (AG(yGz)) = (AG(BGz)))
9	6, 8	eqeq12d 1492	. . . 4 |- (y = B -> (((AGy)Gz) = (AG(yGz)) <-> ((AGB)Gz) = (AG(BGz))))
10		opreq2 3975	. . . . 5 |- (z = C -> ((AGB)Gz) = ((AGB)GC))
11		opreq2 3975	. . . . . 6 |- (z = C -> (BGz) = (BGC))
12	11	opreq2d 3982	. . . . 5 |- (z = C -> (AG(BGz)) = (AG(BGC)))
13	10, 12	eqeq12d 1492	. . . 4 |- (z = C -> (((AGB)Gz) = (AG(BGz)) <-> ((AGB)GC) = (AG(BGC))))
14	4, 9, 13	rcla43v 1885	. . 3 |- ((A e. X /\ B e. X /\ C e. X) -> (A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz)) -> ((AGB)GC) = (AG(BGC))))
15		grpfo.1	. . . . . 6 |- X = ran G
16	15	isgrp 8038	. . . . 5 |- (G e. Grp -> (G e. Grp <-> (G:(X X. X)-->X /\ A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz)) /\ E.u e. X A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u))))
17	16	ibi 594	. . . 4 |- (G e. Grp -> (G:(X X. X)-->X /\ A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz)) /\ E.u e. X A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u)))
18	17	3simp2d 797	. . 3 |- (G e. Grp -> A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz)))
19	14, 18	syl5com 52	. 2 |- (G e. Grp -> ((A e. X /\ B e. X /\ C e. X) -> ((AGB)GC) = (AG(BGC))))
20	19	imp 350	1 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)GC) = (AG(BGC)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  E.wrex 1649   X. cxp 3174  ran crn 3177  -->wf 3184  (class class class)co 3969  Grpcgr 8030

This theorem is referenced by:  grpidinvlem1 8045  grpidinvlem2 8046  grpidinvlem4 8048  grprcan 8059  grpinvid1 8068  grpinvid2 8069  grplcan 8071  grpasscan1 8073  grpinvop 8076  grpmuldivass 8084  grpnpcan 8087  grppnpcan2 8088  abl23 8100  abl4 8101  issubgi 8118  ghgrpilem4 8132  ringaass 8150  vcaass 8176  vcm 8186  nvass 8237  cayleylem2 10405

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-grp 8034

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