Theorem ringideu 9470

Description: The unit element of a ring is unique.

Hypotheses

Ref	Expression
ringid.1	|- G = (1st` R)
ringid.2	|- H = (2nd` R)
ringid.3	|- X = ran G

Assertion

Ref	Expression
ringideu	|- (R e. Ring -> E!u e. X A.x e. X ((xHu) = x /\ (uHx) = x))

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

Proof of Theorem ringideu

Step	Hyp	Ref	Expression
1		ringid.1	. . . . 5 |- G = (1st` R)
2		ringid.2	. . . . 5 |- H = (2nd` R)
3		ringid.3	. . . . 5 |- X = ran G
4	1, 2, 3	ringi 9466	. . . 4 |- (R e. Ring -> ((G e. Abel /\ H:(X X. X)-->X) /\ (A.u e. X A.x e. X A.y e. X (((uHx)Hy) = (uH(xHy)) /\ (uH(xGy)) = ((uHx)G(uHy)) /\ ((uGx)Hy) = ((uHy)G(xHy))) /\ E.u e. X A.x e. X ((xHu) = x /\ (uHx) = x))))
5		simprr 451	. . . 4 |- (((G e. Abel /\ H:(X X. X)-->X) /\ (A.u e. X A.x e. X A.y e. X (((uHx)Hy) = (uH(xHy)) /\ (uH(xGy)) = ((uHx)G(uHy)) /\ ((uGx)Hy) = ((uHy)G(xHy))) /\ E.u e. X A.x e. X ((xHu) = x /\ (uHx) = x))) -> E.u e. X A.x e. X ((xHu) = x /\ (uHx) = x))
6	4, 5	syl 12	. . 3 |- (R e. Ring -> E.u e. X A.x e. X ((xHu) = x /\ (uHx) = x))
7		opreq1 4889	. . . . . . . . . 10 |- (x = w -> (xHu) = (wHu))
8		id 73	. . . . . . . . . 10 |- (x = w -> x = w)
9	7, 8	eqeq12d 1899	. . . . . . . . 9 |- (x = w -> ((xHu) = x <-> (wHu) = w))
10	9	rcla4v 2376	. . . . . . . 8 |- (w e. X -> (A.x e. X (xHu) = x -> (wHu) = w))
11		simpl 346	. . . . . . . . 9 |- (((xHu) = x /\ (uHx) = x) -> (xHu) = x)
12	11	ralimi 2168	. . . . . . . 8 |- (A.x e. X ((xHu) = x /\ (uHx) = x) -> A.x e. X (xHu) = x)
13	10, 12	syl5 20	. . . . . . 7 |- (w e. X -> (A.x e. X ((xHu) = x /\ (uHx) = x) -> (wHu) = w))
14		opreq2 4890	. . . . . . . . . 10 |- (x = u -> (wHx) = (wHu))
15		id 73	. . . . . . . . . 10 |- (x = u -> x = u)
16	14, 15	eqeq12d 1899	. . . . . . . . 9 |- (x = u -> ((wHx) = x <-> (wHu) = u))
17	16	rcla4v 2376	. . . . . . . 8 |- (u e. X -> (A.x e. X (wHx) = x -> (wHu) = u))
18		simpr 350	. . . . . . . . 9 |- (((xHw) = x /\ (wHx) = x) -> (wHx) = x)
19	18	ralimi 2168	. . . . . . . 8 |- (A.x e. X ((xHw) = x /\ (wHx) = x) -> A.x e. X (wHx) = x)
20	17, 19	syl5 20	. . . . . . 7 |- (u e. X -> (A.x e. X ((xHw) = x /\ (wHx) = x) -> (wHu) = u))
21	13, 20	im2anan9 622	. . . . . 6 |- ((w e. X /\ u e. X) -> ((A.x e. X ((xHu) = x /\ (uHx) = x) /\ A.x e. X ((xHw) = x /\ (wHx) = x)) -> ((wHu) = w /\ (wHu) = u)))
22		eqtr2 1905	. . . . . . 7 |- (((wHu) = u /\ (wHu) = w) -> u = w)
23	22	ancoms 484	. . . . . 6 |- (((wHu) = w /\ (wHu) = u) -> u = w)
24	21, 23	syl6 25	. . . . 5 |- ((w e. X /\ u e. X) -> ((A.x e. X ((xHu) = x /\ (uHx) = x) /\ A.x e. X ((xHw) = x /\ (wHx) = x)) -> u = w))
25	24	ancoms 484	. . . 4 |- ((u e. X /\ w e. X) -> ((A.x e. X ((xHu) = x /\ (uHx) = x) /\ A.x e. X ((xHw) = x /\ (wHx) = x)) -> u = w))
26	25	rgen2a 2160	. . 3 |- A.u e. X A.w e. X ((A.x e. X ((xHu) = x /\ (uHx) = x) /\ A.x e. X ((xHw) = x /\ (wHx) = x)) -> u = w)
27	6, 26	jctir 317	. 2 |- (R e. Ring -> (E.u e. X A.x e. X ((xHu) = x /\ (uHx) = x) /\ A.u e. X A.w e. X ((A.x e. X ((xHu) = x /\ (uHx) = x) /\ A.x e. X ((xHw) = x /\ (wHx) = x)) -> u = w)))
28		opreq2 4890	. . . . . 6 |- (u = w -> (xHu) = (xHw))
29	28	eqeq1d 1892	. . . . 5 |- (u = w -> ((xHu) = x <-> (xHw) = x))
30		opreq1 4889	. . . . . 6 |- (u = w -> (uHx) = (wHx))
31	30	eqeq1d 1892	. . . . 5 |- (u = w -> ((uHx) = x <-> (wHx) = x))
32	29, 31	anbi12d 690	. . . 4 |- (u = w -> (((xHu) = x /\ (uHx) = x) <-> ((xHw) = x /\ (wHx) = x)))
33	32	ralbidv 2123	. . 3 |- (u = w -> (A.x e. X ((xHu) = x /\ (uHx) = x) <-> A.x e. X ((xHw) = x /\ (wHx) = x)))
34	33	reu4 2446	. 2 |- (E!u e. X A.x e. X ((xHu) = x /\ (uHx) = x) <-> (E.u e. X A.x e. X ((xHu) = x /\ (uHx) = x) /\ A.u e. X A.w e. X ((A.x e. X ((xHu) = x /\ (uHx) = x) /\ A.x e. X ((xHw) = x /\ (wHx) = x)) -> u = w)))
35	27, 34	sylibr 217	1 |- (R e. Ring -> E!u e. X A.x e. X ((xHu) = x /\ (uHx) = x))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  E!wreu 2107   X. cxp 3984  ran crn 3987  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Abelcabl 9407  Ringcring 9463

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-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-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-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-ring 9464

Copyright terms: Public domain