Theorem exidu1 10373

Description: Unicity of the left and right identity element of a magma when it exists. (Contributed by FL, 12-Dec-2009.)

Hypothesis

Ref	Expression
exidu1.1	|- X = ran G

Assertion

Ref	Expression
exidu1	|- (G e. (Magma i^i ExId ) -> E!u e. X A.x e. X ((uGx) = x /\ (xGu) = x))

Distinct variable groups:   u,G,x   u,X,x

Proof of Theorem exidu1

Step	Hyp	Ref	Expression
1		exidu1.1	. . . 4 |- X = ran G
2	1	isexid2 10372	. . 3 |- (G e. (Magma i^i ExId ) -> E.u e. X A.z e. X ((zGu) = z /\ (uGz) = z))
3		opreq2 4890	. . . . . . . . . . . 12 |- (z = x -> (uGz) = (uGx))
4		id 73	. . . . . . . . . . . 12 |- (z = x -> z = x)
5	3, 4	eqeq12d 1899	. . . . . . . . . . 11 |- (z = x -> ((uGz) = z <-> (uGx) = x))
6		opreq1 4889	. . . . . . . . . . . 12 |- (z = x -> (zGu) = (xGu))
7	6, 4	eqeq12d 1899	. . . . . . . . . . 11 |- (z = x -> ((zGu) = z <-> (xGu) = x))
8	5, 7	anbi12d 690	. . . . . . . . . 10 |- (z = x -> (((uGz) = z /\ (zGu) = z) <-> ((uGx) = x /\ (xGu) = x)))
9		ancom 482	. . . . . . . . . 10 |- (((zGu) = z /\ (uGz) = z) <-> ((uGz) = z /\ (zGu) = z))
10	8, 9	syl5bb 591	. . . . . . . . 9 |- (z = x -> (((zGu) = z /\ (uGz) = z) <-> ((uGx) = x /\ (xGu) = x)))
11	10	cbvralv 2280	. . . . . . . 8 |- (A.z e. X ((zGu) = z /\ (uGz) = z) <-> A.x e. X ((uGx) = x /\ (xGu) = x))
12	11	biimpi 168	. . . . . . 7 |- (A.z e. X ((zGu) = z /\ (uGz) = z) -> A.x e. X ((uGx) = x /\ (xGu) = x))
13	12	adantl 424	. . . . . 6 |- (((G e. (Magma i^i ExId ) /\ u e. X) /\ A.z e. X ((zGu) = z /\ (uGz) = z)) -> A.x e. X ((uGx) = x /\ (xGu) = x))
14		opreq1 4889	. . . . . . . . . . . . . . . . 17 |- (z = w -> (zGu) = (wGu))
15		id 73	. . . . . . . . . . . . . . . . 17 |- (z = w -> z = w)
16	14, 15	eqeq12d 1899	. . . . . . . . . . . . . . . 16 |- (z = w -> ((zGu) = z <-> (wGu) = w))
17		opreq2 4890	. . . . . . . . . . . . . . . . 17 |- (z = w -> (uGz) = (uGw))
18	17, 15	eqeq12d 1899	. . . . . . . . . . . . . . . 16 |- (z = w -> ((uGz) = z <-> (uGw) = w))
19	16, 18	anbi12d 690	. . . . . . . . . . . . . . 15 |- (z = w -> (((zGu) = z /\ (uGz) = z) <-> ((wGu) = w /\ (uGw) = w)))
20	19	rcla4va 2378	. . . . . . . . . . . . . 14 |- ((w e. X /\ A.z e. X ((zGu) = z /\ (uGz) = z)) -> ((wGu) = w /\ (uGw) = w))
21		eqtr3 1907	. . . . . . . . . . . . . 14 |- (((wGu) = w /\ (uGw) = w) -> (wGu) = (uGw))
22	20, 21	syl 12	. . . . . . . . . . . . 13 |- ((w e. X /\ A.z e. X ((zGu) = z /\ (uGz) = z)) -> (wGu) = (uGw))
23	22	expcom 403	. . . . . . . . . . . 12 |- (A.z e. X ((zGu) = z /\ (uGz) = z) -> (w e. X -> (wGu) = (uGw)))
24	23	adantl 424	. . . . . . . . . . 11 |- (((G e. (Magma i^i ExId ) /\ u e. X) /\ A.z e. X ((zGu) = z /\ (uGz) = z)) -> (w e. X -> (wGu) = (uGw)))
25	24	imp 377	. . . . . . . . . 10 |- ((((G e. (Magma i^i ExId ) /\ u e. X) /\ A.z e. X ((zGu) = z /\ (uGz) = z)) /\ w e. X) -> (wGu) = (uGw))
26	25	adantr 425	. . . . . . . . 9 |- (((((G e. (Magma i^i ExId ) /\ u e. X) /\ A.z e. X ((zGu) = z /\ (uGz) = z)) /\ w e. X) /\ A.x e. X ((wGx) = x /\ (xGw) = x)) -> (wGu) = (uGw))
27		opreq2 4890	. . . . . . . . . . . . . . . . 17 |- (x = u -> (wGx) = (wGu))
28		id 73	. . . . . . . . . . . . . . . . 17 |- (x = u -> x = u)
29	27, 28	eqeq12d 1899	. . . . . . . . . . . . . . . 16 |- (x = u -> ((wGx) = x <-> (wGu) = u))
30		opreq1 4889	. . . . . . . . . . . . . . . . 17 |- (x = u -> (xGw) = (uGw))
31	30, 28	eqeq12d 1899	. . . . . . . . . . . . . . . 16 |- (x = u -> ((xGw) = x <-> (uGw) = u))
32	29, 31	anbi12d 690	. . . . . . . . . . . . . . 15 |- (x = u -> (((wGx) = x /\ (xGw) = x) <-> ((wGu) = u /\ (uGw) = u)))
33	32	rcla4va 2378	. . . . . . . . . . . . . 14 |- ((u e. X /\ A.x e. X ((wGx) = x /\ (xGw) = x)) -> ((wGu) = u /\ (uGw) = u))
34	33	simplld 348	. . . . . . . . . . . . 13 |- ((u e. X /\ A.x e. X ((wGx) = x /\ (xGw) = x)) -> (wGu) = u)
35	34	ex 402	. . . . . . . . . . . 12 |- (u e. X -> (A.x e. X ((wGx) = x /\ (xGw) = x) -> (wGu) = u))
36	35	adantl 424	. . . . . . . . . . 11 |- ((G e. (Magma i^i ExId ) /\ u e. X) -> (A.x e. X ((wGx) = x /\ (xGw) = x) -> (wGu) = u))
37	36	ad2antrr 440	. . . . . . . . . 10 |- ((((G e. (Magma i^i ExId ) /\ u e. X) /\ A.z e. X ((zGu) = z /\ (uGz) = z)) /\ w e. X) -> (A.x e. X ((wGx) = x /\ (xGw) = x) -> (wGu) = u))
38	37	imp 377	. . . . . . . . 9 |- (((((G e. (Magma i^i ExId ) /\ u e. X) /\ A.z e. X ((zGu) = z /\ (uGz) = z)) /\ w e. X) /\ A.x e. X ((wGx) = x /\ (xGw) = x)) -> (wGu) = u)
39	18	rcla4va 2378	. . . . . . . . . . . . . 14 |- ((w e. X /\ A.z e. X (uGz) = z) -> (uGw) = w)
40	39	ex 402	. . . . . . . . . . . . 13 |- (w e. X -> (A.z e. X (uGz) = z -> (uGw) = w))
41		simpr 350	. . . . . . . . . . . . . 14 |- (((zGu) = z /\ (uGz) = z) -> (uGz) = z)
42	41	ralimi 2168	. . . . . . . . . . . . 13 |- (A.z e. X ((zGu) = z /\ (uGz) = z) -> A.z e. X (uGz) = z)
43	40, 42	syl5com 63	. . . . . . . . . . . 12 |- (A.z e. X ((zGu) = z /\ (uGz) = z) -> (w e. X -> (uGw) = w))
44	43	adantl 424	. . . . . . . . . . 11 |- (((G e. (Magma i^i ExId ) /\ u e. X) /\ A.z e. X ((zGu) = z /\ (uGz) = z)) -> (w e. X -> (uGw) = w))
45	44	imp 377	. . . . . . . . . 10 |- ((((G e. (Magma i^i ExId ) /\ u e. X) /\ A.z e. X ((zGu) = z /\ (uGz) = z)) /\ w e. X) -> (uGw) = w)
46	45	adantr 425	. . . . . . . . 9 |- (((((G e. (Magma i^i ExId ) /\ u e. X) /\ A.z e. X ((zGu) = z /\ (uGz) = z)) /\ w e. X) /\ A.x e. X ((wGx) = x /\ (xGw) = x)) -> (uGw) = w)
47	26, 38, 46	3eqtr3d 1934	. . . . . . . 8 |- (((((G e. (Magma i^i ExId ) /\ u e. X) /\ A.z e. X ((zGu) = z /\ (uGz) = z)) /\ w e. X) /\ A.x e. X ((wGx) = x /\ (xGw) = x)) -> u = w)
48	47	ex 402	. . . . . . 7 |- ((((G e. (Magma i^i ExId ) /\ u e. X) /\ A.z e. X ((zGu) = z /\ (uGz) = z)) /\ w e. X) -> (A.x e. X ((wGx) = x /\ (xGw) = x) -> u = w))
49	48	r19.21aiva 2176	. . . . . 6 |- (((G e. (Magma i^i ExId ) /\ u e. X) /\ A.z e. X ((zGu) = z /\ (uGz) = z)) -> A.w e. X (A.x e. X ((wGx) = x /\ (xGw) = x) -> u = w))
50	13, 49	jca 310	. . . . 5 |- (((G e. (Magma i^i ExId ) /\ u e. X) /\ A.z e. X ((zGu) = z /\ (uGz) = z)) -> (A.x e. X ((uGx) = x /\ (xGu) = x) /\ A.w e. X (A.x e. X ((wGx) = x /\ (xGw) = x) -> u = w)))
51	50	ex 402	. . . 4 |- ((G e. (Magma i^i ExId ) /\ u e. X) -> (A.z e. X ((zGu) = z /\ (uGz) = z) -> (A.x e. X ((uGx) = x /\ (xGu) = x) /\ A.w e. X (A.x e. X ((wGx) = x /\ (xGw) = x) -> u = w))))
52	51	reximdva 2203	. . 3 |- (G e. (Magma i^i ExId ) -> (E.u e. X A.z e. X ((zGu) = z /\ (uGz) = z) -> E.u e. X (A.x e. X ((uGx) = x /\ (xGu) = x) /\ A.w e. X (A.x e. X ((wGx) = x /\ (xGw) = x) -> u = w))))
53	2, 52	mpd 29	. 2 |- (G e. (Magma i^i ExId ) -> E.u e. X (A.x e. X ((uGx) = x /\ (xGu) = x) /\ A.w e. X (A.x e. X ((wGx) = x /\ (xGw) = x) -> u = w)))
54		opreq1 4889	. . . . . 6 |- (u = w -> (uGx) = (wGx))
55	54	eqeq1d 1892	. . . . 5 |- (u = w -> ((uGx) = x <-> (wGx) = x))
56		opreq2 4890	. . . . . 6 |- (u = w -> (xGu) = (xGw))
57	56	eqeq1d 1892	. . . . 5 |- (u = w -> ((xGu) = x <-> (xGw) = x))
58	55, 57	anbi12d 690	. . . 4 |- (u = w -> (((uGx) = x /\ (xGu) = x) <-> ((wGx) = x /\ (xGw) = x)))
59	58	ralbidv 2123	. . 3 |- (u = w -> (A.x e. X ((uGx) = x /\ (xGu) = x) <-> A.x e. X ((wGx) = x /\ (xGw) = x)))
60	59	reu8 2448	. 2 |- (E!u e. X A.x e. X ((uGx) = x /\ (xGu) = x) <-> E.u e. X (A.x e. X ((uGx) = x /\ (xGu) = x) /\ A.w e. X (A.x e. X ((wGx) = x /\ (xGw) = x) -> u = w)))
61	53, 60	sylibr 217	1 |- (G e. (Magma i^i ExId ) -> E!u e. X A.x e. X ((uGx) = x /\ (xGu) = x))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  E!wreu 2107   i^i cin 2592  ran crn 3987  (class class class)co 4884   ExId cexid 10361  Magmacmagm 10365

This theorem is referenced by:  iorlid 10375  cmpidelt 10376  exidresid 16032

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-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-reu 2111  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-fo 4012  df-fv 4014  df-opr 4886  df-exid 10362  df-mgm 10366

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