Theorem grpass 8047

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 3968	. . . . . 6 |- (x = A -> (xGy) = (AGy))
2	1	opreq1d 3975	. . . . 5 |- (x = A -> ((xGy)Gz) = ((AGy)Gz))
3		opreq1 3968	. . . . 5 |- (x = A -> (xG(yGz)) = (AG(yGz)))
4	2, 3	eqeq12d 1489	. . . 4 |- (x = A -> (((xGy)Gz) = (xG(yGz)) <-> ((AGy)Gz) = (AG(yGz))))
5		opreq2 3969	. . . . . 6 |- (y = B -> (AGy) = (AGB))
6	5	opreq1d 3975	. . . . 5 |- (y = B -> ((AGy)Gz) = ((AGB)Gz))
7		opreq1 3968	. . . . . 6 |- (y = B -> (yGz) = (BGz))
8	7	opreq2d 3976	. . . . 5 |- (y = B -> (AG(yGz)) = (AG(BGz)))
9	6, 8	eqeq12d 1489	. . . 4 |- (y = B -> (((AGy)Gz) = (AG(yGz)) <-> ((AGB)Gz) = (AG(BGz))))
10		opreq2 3969	. . . . 5 |- (z = C -> ((AGB)Gz) = ((AGB)GC))
11		opreq2 3969	. . . . . 6 |- (z = C -> (BGz) = (BGC))
12	11	opreq2d 3976	. . . . 5 |- (z = C -> (AG(BGz)) = (AG(BGC)))
13	10, 12	eqeq12d 1489	. . . 4 |- (z = C -> (((AGB)Gz) = (AG(BGz)) <-> ((AGB)GC) = (AG(BGC))))
14	4, 9, 13	rcla43v 1882	. . 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 8041	. . . . 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 592	. . . 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 795	. . 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 775   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646   X. cxp 3168  ran crn 3171  -->wf 3178  (class class class)co 3963  Grpcgr 8033

This theorem is referenced by:  grpidinvlem1 8048  grpidinvlem2 8049  grpidinvlem4 8051  grprcan 8063  grpinvid1 8072  grpinvid2 8073  grplcan 8075  grpasscan1 8077  grpinvop 8080  grpmuldivass 8088  grpnpcan 8091  grppnpcan2 8092  abl23 8104  abl4 8105  issubgi 8122  ghgrpilem4 8136  ringaass 8154  vcaass 8180  vcm 8190  nvass 8241  cayleylem2 10410

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-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-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-fo 3196  df-fv 3198  df-opr 3965  df-grp 8037

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