Theorem ununr 14769

Description: The unit of a unital ring is unique.

Hypotheses

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

Assertion

Ref	Expression
ununr	|- (R e. Ring -> E!x e. X A.y e. X ((yHx) = y /\ (xHy) = y))

Distinct variable groups:   x,H,y   x,R,y   x,X,y

Proof of Theorem ununr

Step	Hyp	Ref	Expression
1		eqid 1884	. . . . . 6 |- (1st` R) = (1st` R)
2		ununr.1	. . . . . 6 |- H = (2nd` R)
3		ununr.2	. . . . . 6 |- X = ran (1st` R)
4	1, 2, 3	ringi 9466	. . . . 5 |- (R e. Ring -> (((1st` R) e. Abel /\ H:(X X. X)-->X) /\ (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(y(1st` R)z)) = ((xHy)(1st` R)(xHz)) /\ ((x(1st` R)y)Hz) = ((xHz)(1st` R)(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y))))
5	4	simprd 352	. . . 4 |- (R e. Ring -> (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(y(1st` R)z)) = ((xHy)(1st` R)(xHz)) /\ ((x(1st` R)y)Hz) = ((xHz)(1st` R)(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y)))
6	5	simprd 352	. . 3 |- (R e. Ring -> E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y))
7		opreq1 4889	. . . . . . . . . 10 |- (y = x -> (yHa) = (xHa))
8		id 73	. . . . . . . . . 10 |- (y = x -> y = x)
9	7, 8	eqeq12d 1899	. . . . . . . . 9 |- (y = x -> ((yHa) = y <-> (xHa) = x))
10		opreq2 4890	. . . . . . . . . 10 |- (y = x -> (aHy) = (aHx))
11	10, 8	eqeq12d 1899	. . . . . . . . 9 |- (y = x -> ((aHy) = y <-> (aHx) = x))
12	9, 11	anbi12d 690	. . . . . . . 8 |- (y = x -> (((yHa) = y /\ (aHy) = y) <-> ((xHa) = x /\ (aHx) = x)))
13	12	rcla4v 2376	. . . . . . 7 |- (x e. X -> (A.y e. X ((yHa) = y /\ (aHy) = y) -> ((xHa) = x /\ (aHx) = x)))
14		opreq1 4889	. . . . . . . . . . . 12 |- (y = a -> (yHx) = (aHx))
15		id 73	. . . . . . . . . . . 12 |- (y = a -> y = a)
16	14, 15	eqeq12d 1899	. . . . . . . . . . 11 |- (y = a -> ((yHx) = y <-> (aHx) = a))
17		opreq2 4890	. . . . . . . . . . . 12 |- (y = a -> (xHy) = (xHa))
18	17, 15	eqeq12d 1899	. . . . . . . . . . 11 |- (y = a -> ((xHy) = y <-> (xHa) = a))
19	16, 18	anbi12d 690	. . . . . . . . . 10 |- (y = a -> (((yHx) = y /\ (xHy) = y) <-> ((aHx) = a /\ (xHa) = a)))
20	19	rcla4v 2376	. . . . . . . . 9 |- (a e. X -> (A.y e. X ((yHx) = y /\ (xHy) = y) -> ((aHx) = a /\ (xHa) = a)))
21		eqeq1 1890	. . . . . . . . . . . . 13 |- ((aHx) = x -> ((aHx) = a <-> x = a))
22	21	biimpd 170	. . . . . . . . . . . 12 |- ((aHx) = x -> ((aHx) = a -> x = a))
23	22	adantl 424	. . . . . . . . . . 11 |- (((xHa) = x /\ (aHx) = x) -> ((aHx) = a -> x = a))
24	23	com12 14	. . . . . . . . . 10 |- ((aHx) = a -> (((xHa) = x /\ (aHx) = x) -> x = a))
25	24	adantr 425	. . . . . . . . 9 |- (((aHx) = a /\ (xHa) = a) -> (((xHa) = x /\ (aHx) = x) -> x = a))
26	20, 25	syl6 25	. . . . . . . 8 |- (a e. X -> (A.y e. X ((yHx) = y /\ (xHy) = y) -> (((xHa) = x /\ (aHx) = x) -> x = a)))
27	26	com13 37	. . . . . . 7 |- (((xHa) = x /\ (aHx) = x) -> (A.y e. X ((yHx) = y /\ (xHy) = y) -> (a e. X -> x = a)))
28	13, 27	syl6 25	. . . . . 6 |- (x e. X -> (A.y e. X ((yHa) = y /\ (aHy) = y) -> (A.y e. X ((yHx) = y /\ (xHy) = y) -> (a e. X -> x = a))))
29	28	com24 41	. . . . 5 |- (x e. X -> (a e. X -> (A.y e. X ((yHx) = y /\ (xHy) = y) -> (A.y e. X ((yHa) = y /\ (aHy) = y) -> x = a))))
30	29	imp4b 392	. . . 4 |- ((x e. X /\ a e. X) -> ((A.y e. X ((yHx) = y /\ (xHy) = y) /\ A.y e. X ((yHa) = y /\ (aHy) = y)) -> x = a))
31	30	rgen2a 2160	. . 3 |- A.x e. X A.a e. X ((A.y e. X ((yHx) = y /\ (xHy) = y) /\ A.y e. X ((yHa) = y /\ (aHy) = y)) -> x = a)
32	6, 31	jctir 317	. 2 |- (R e. Ring -> (E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y) /\ A.x e. X A.a e. X ((A.y e. X ((yHx) = y /\ (xHy) = y) /\ A.y e. X ((yHa) = y /\ (aHy) = y)) -> x = a)))
33		opreq2 4890	. . . . . 6 |- (x = a -> (yHx) = (yHa))
34	33	eqeq1d 1892	. . . . 5 |- (x = a -> ((yHx) = y <-> (yHa) = y))
35		opreq1 4889	. . . . . 6 |- (x = a -> (xHy) = (aHy))
36	35	eqeq1d 1892	. . . . 5 |- (x = a -> ((xHy) = y <-> (aHy) = y))
37	34, 36	anbi12d 690	. . . 4 |- (x = a -> (((yHx) = y /\ (xHy) = y) <-> ((yHa) = y /\ (aHy) = y)))
38	37	ralbidv 2123	. . 3 |- (x = a -> (A.y e. X ((yHx) = y /\ (xHy) = y) <-> A.y e. X ((yHa) = y /\ (aHy) = y)))
39	38	reu4 2446	. 2 |- (E!x e. X A.y e. X ((yHx) = y /\ (xHy) = y) <-> (E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y) /\ A.x e. X A.a e. X ((A.y e. X ((yHx) = y /\ (xHy) = y) /\ A.y e. X ((yHa) = y /\ (aHy) = y)) -> x = a)))
40	32, 39	sylibr 217	1 |- (R e. Ring -> E!x e. X A.y e. X ((yHx) = y /\ (xHy) = y))

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

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