Theorem ringid 8141

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.x e. X ((AHx) = A /\ (xHA) = A))

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

Proof of Theorem ringid

Step	Hyp	Ref	Expression
1		opreq1 3974	. . . . . . 7 |- (y = A -> (yHx) = (AHx))
2		id 59	. . . . . . 7 |- (y = A -> y = A)
3	1, 2	eqeq12d 1492	. . . . . 6 |- (y = A -> ((yHx) = y <-> (AHx) = A))
4		opreq2 3975	. . . . . . 7 |- (y = A -> (xHy) = (xHA))
5	4, 2	eqeq12d 1492	. . . . . 6 |- (y = A -> ((xHy) = y <-> (xHA) = A))
6	3, 5	anbi12d 630	. . . . 5 |- (y = A -> (((yHx) = y /\ (xHy) = y) <-> ((AHx) = A /\ (xHA) = A)))
7	6	rexbidv 1667	. . . 4 |- (y = A -> (E.x e. X ((yHx) = y /\ (xHy) = y) <-> E.x e. X ((AHx) = A /\ (xHA) = A)))
8	7	imbi2d 614	. . 3 |- (y = A -> ((R e. Ring -> E.x e. X ((yHx) = y /\ (xHy) = y)) <-> (R e. Ring -> E.x e. X ((AHx) = A /\ (xHA) = 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 8138	. . . . . . 7 |- (R e. Ring -> ((G e. Abel /\ H:(X X. X)-->X) /\ (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y))))
13	12	pm3.27d 325	. . . . . 6 |- (R e. Ring -> (A.x e. X A.y e. X A.z e. X (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y)))
14	13	pm3.27d 325	. . . . 5 |- (R e. Ring -> E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y))
15		r19.12 1743	. . . . 5 |- (E.x e. X A.y e. X ((yHx) = y /\ (xHy) = y) -> A.y e. X E.x e. X ((yHx) = y /\ (xHy) = y))
16		ra4 1697	. . . . 5 |- (A.y e. X E.x e. X ((yHx) = y /\ (xHy) = y) -> (y e. X -> E.x e. X ((yHx) = y /\ (xHy) = y)))
17	14, 15, 16	3syl 20	. . . 4 |- (R e. Ring -> (y e. X -> E.x e. X ((yHx) = y /\ (xHy) = y)))
18	17	com12 11	. . 3 |- (y e. X -> (R e. Ring -> E.x e. X ((yHx) = y /\ (xHy) = y)))
19	8, 18	vtoclga 1855	. 2 |- (A e. X -> (R e. Ring -> E.x e. X ((AHx) = A /\ (xHA) = A)))
20	19	impcom 351	1 |- ((R e. Ring /\ A e. X) -> E.x e. X ((AHx) = A /\ (xHA) = A))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  E.wrex 1649   X. cxp 3174  ran crn 3177  -->wf 3184  ` cfv 3188  (class class class)co 3969  1stc1st 4083  2ndc2nd 4084  Abelcabl 8095  Ringcring 8135

This theorem is referenced by:  ring2 8145

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-1st 4085  df-2nd 4086  df-ring 8136

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