Theorem ringid 9469

Description: The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007.)

Hypotheses

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

Assertion

Ref	Expression
ringid	|- ((R e. Ring /\ A e. X) -> E.u e. X ((AHu) = A /\ (uHA) = A))

Distinct variable groups:   u,A   u,G   u,H   u,X

Proof of Theorem ringid

Step	Hyp	Ref	Expression
1		opreq1 4889	. . . . . . 7 |- (x = A -> (xHu) = (AHu))
2		id 73	. . . . . . 7 |- (x = A -> x = A)
3	1, 2	eqeq12d 1899	. . . . . 6 |- (x = A -> ((xHu) = x <-> (AHu) = A))
4		opreq2 4890	. . . . . . 7 |- (x = A -> (uHx) = (uHA))
5	4, 2	eqeq12d 1899	. . . . . 6 |- (x = A -> ((uHx) = x <-> (uHA) = A))
6	3, 5	anbi12d 690	. . . . 5 |- (x = A -> (((xHu) = x /\ (uHx) = x) <-> ((AHu) = A /\ (uHA) = A)))
7	6	rexbidv 2124	. . . 4 |- (x = A -> (E.u e. X ((xHu) = x /\ (uHx) = x) <-> E.u e. X ((AHu) = A /\ (uHA) = A)))
8	7	imbi2d 674	. . 3 |- (x = A -> ((R e. Ring -> E.u e. X ((xHu) = x /\ (uHx) = x)) <-> (R e. Ring -> E.u e. X ((AHu) = A /\ (uHA) = A))))
9		ringid.1	. . . . . . . 8 |- G = (1st` R)
10		ringid.2	. . . . . . . 8 |- H = (2nd` R)
11		ringid.3	. . . . . . . 8 |- X = ran G
12	9, 10, 11	ringi 9466	. . . . . . 7 |- (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))))
13	12	simprd 352	. . . . . 6 |- (R e. Ring -> (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)))
14	13	simprd 352	. . . . 5 |- (R e. Ring -> E.u e. X A.x e. X ((xHu) = x /\ (uHx) = x))
15		r19.12 2204	. . . . 5 |- (E.u e. X A.x e. X ((xHu) = x /\ (uHx) = x) -> A.x e. X E.u e. X ((xHu) = x /\ (uHx) = x))
16		ra4 2155	. . . . 5 |- (A.x e. X E.u e. X ((xHu) = x /\ (uHx) = x) -> (x e. X -> E.u e. X ((xHu) = x /\ (uHx) = x)))
17	14, 15, 16	3syl 24	. . . 4 |- (R e. Ring -> (x e. X -> E.u e. X ((xHu) = x /\ (uHx) = x)))
18	17	com12 14	. . 3 |- (x e. X -> (R e. Ring -> E.u e. X ((xHu) = x /\ (uHx) = x)))
19	8, 18	vtoclga 2352	. 2 |- (A e. X -> (R e. Ring -> E.u e. X ((AHu) = A /\ (uHA) = A)))
20	19	impcom 378	1 |- ((R e. Ring /\ A e. X) -> E.u e. X ((AHu) = A /\ (uHA) = A))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   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 is referenced by:  ring2 9474

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-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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