Theorem isass 10363

Description: The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.)

Hypothesis

Ref	Expression
isass.1	|- X = dom dom G

Assertion

Ref	Expression
isass	|- (G e. A -> (G e. Ass <-> A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz))))

Distinct variable groups:   x,G,y,z   x,X,y,z

Proof of Theorem isass

Step	Hyp	Ref	Expression
1		dmeq 4157	. . . . . . . . . 10 |- (g = G -> dom g = dom G)
2	1	dmeqd 4159	. . . . . . . . 9 |- (g = G -> dom dom g = dom dom G)
3	2	eleq2d 1964	. . . . . . . 8 |- (g = G -> (x e. dom dom g <-> x e. dom dom G))
4	2	eleq2d 1964	. . . . . . . 8 |- (g = G -> (y e. dom dom g <-> y e. dom dom G))
5	2	eleq2d 1964	. . . . . . . 8 |- (g = G -> (z e. dom dom g <-> z e. dom dom G))
6	3, 4, 5	3anbi123d 1168	. . . . . . 7 |- (g = G -> ((x e. dom dom g /\ y e. dom dom g /\ z e. dom dom g) <-> (x e. dom dom G /\ y e. dom dom G /\ z e. dom dom G)))
7		opreq 4888	. . . . . . . . . 10 |- (g = G -> (xgy) = (xGy))
8	7	opreq1d 4897	. . . . . . . . 9 |- (g = G -> ((xgy)gz) = ((xGy)gz))
9		opreq 4888	. . . . . . . . 9 |- (g = G -> ((xGy)gz) = ((xGy)Gz))
10	8, 9	eqtrd 1925	. . . . . . . 8 |- (g = G -> ((xgy)gz) = ((xGy)Gz))
11		opreq 4888	. . . . . . . . . 10 |- (g = G -> (ygz) = (yGz))
12	11	opreq2d 4898	. . . . . . . . 9 |- (g = G -> (xg(ygz)) = (xg(yGz)))
13		opreq 4888	. . . . . . . . 9 |- (g = G -> (xg(yGz)) = (xG(yGz)))
14	12, 13	eqtrd 1925	. . . . . . . 8 |- (g = G -> (xg(ygz)) = (xG(yGz)))
15	10, 14	eqeq12d 1899	. . . . . . 7 |- (g = G -> (((xgy)gz) = (xg(ygz)) <-> ((xGy)Gz) = (xG(yGz))))
16	6, 15	imbi12d 688	. . . . . 6 |- (g = G -> (((x e. dom dom g /\ y e. dom dom g /\ z e. dom dom g) -> ((xgy)gz) = (xg(ygz))) <-> ((x e. dom dom G /\ y e. dom dom G /\ z e. dom dom G) -> ((xGy)Gz) = (xG(yGz)))))
17	16	albidv 1656	. . . . 5 |- (g = G -> (A.z((x e. dom dom g /\ y e. dom dom g /\ z e. dom dom g) -> ((xgy)gz) = (xg(ygz))) <-> A.z((x e. dom dom G /\ y e. dom dom G /\ z e. dom dom G) -> ((xGy)Gz) = (xG(yGz)))))
18	17	2albidv 1658	. . . 4 |- (g = G -> (A.xA.yA.z((x e. dom dom g /\ y e. dom dom g /\ z e. dom dom g) -> ((xgy)gz) = (xg(ygz))) <-> A.xA.yA.z((x e. dom dom G /\ y e. dom dom G /\ z e. dom dom G) -> ((xGy)Gz) = (xG(yGz)))))
19		r3al 2151	. . . 4 |- (A.x e. dom dom gA.y e. dom dom gA.z e. dom dom g((xgy)gz) = (xg(ygz)) <-> A.xA.yA.z((x e. dom dom g /\ y e. dom dom g /\ z e. dom dom g) -> ((xgy)gz) = (xg(ygz))))
20		r3al 2151	. . . 4 |- (A.x e. dom dom GA.y e. dom dom GA.z e. dom dom G((xGy)Gz) = (xG(yGz)) <-> A.xA.yA.z((x e. dom dom G /\ y e. dom dom G /\ z e. dom dom G) -> ((xGy)Gz) = (xG(yGz))))
21	18, 19, 20	3bitr4g 614	. . 3 |- (g = G -> (A.x e. dom dom gA.y e. dom dom gA.z e. dom dom g((xgy)gz) = (xg(ygz)) <-> A.x e. dom dom GA.y e. dom dom GA.z e. dom dom G((xGy)Gz) = (xG(yGz))))
22		isass.1	. . . . . . 7 |- X = dom dom G
23	22	eqcomi 1888	. . . . . 6 |- dom dom G = X
24	23	a1i 8	. . . . 5 |- (g = G -> dom dom G = X)
25	24	raleqdv 2269	. . . 4 |- (g = G -> (A.x e. dom dom GA.y e. dom dom GA.z e. dom dom G((xGy)Gz) = (xG(yGz)) <-> A.x e. X A.y e. dom dom GA.z e. dom dom G((xGy)Gz) = (xG(yGz))))
26	24	raleqdv 2269	. . . . 5 |- (g = G -> (A.y e. dom dom GA.z e. dom dom G((xGy)Gz) = (xG(yGz)) <-> A.y e. X A.z e. dom dom G((xGy)Gz) = (xG(yGz))))
27	26	ralbidv 2123	. . . 4 |- (g = G -> (A.x e. X A.y e. dom dom GA.z e. dom dom G((xGy)Gz) = (xG(yGz)) <-> A.x e. X A.y e. X A.z e. dom dom G((xGy)Gz) = (xG(yGz))))
28	24	raleqdv 2269	. . . . 5 |- (g = G -> (A.z e. dom dom G((xGy)Gz) = (xG(yGz)) <-> A.z e. X ((xGy)Gz) = (xG(yGz))))
29	28	2ralbidv 2140	. . . 4 |- (g = G -> (A.x e. X A.y e. X A.z e. dom dom G((xGy)Gz) = (xG(yGz)) <-> A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz))))
30	25, 27, 29	3bitrd 603	. . 3 |- (g = G -> (A.x e. dom dom GA.y e. dom dom GA.z e. dom dom G((xGy)Gz) = (xG(yGz)) <-> A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz))))
31	21, 30	bitrd 587	. 2 |- (g = G -> (A.x e. dom dom gA.y e. dom dom gA.z e. dom dom g((xgy)gz) = (xg(ygz)) <-> A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz))))
32		df-ass 10360	. 2 |- Ass = {g | A.x e. dom dom gA.y e. dom dom gA.z e. dom dom g((xgy)gz) = (xg(ygz))}
33	31, 32	elab2g 2406	1 |- (G e. A -> (G e. Ass <-> A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz))))

Syntax hints:   -> wi 3   <-> wb 163   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105  dom cdm 3986  (class class class)co 4884  Asscass 10359

This theorem is referenced by:  issmgrp 10381

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-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

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-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-ass 10360

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