Theorem isidlc 16163

Description: The predicate "is an ideal of the commutative ring R."

Hypotheses

Ref	Expression
idlval.1	|- G = (1st` R)
idlval.2	|- H = (2nd` R)
idlval.3	|- X = ran G
idlval.4	|- Z = (Id` G)

Assertion

Ref	Expression
isidlc	|- (R e. CRing -> (I e. (Idl` R) <-> (I C_ X /\ Z e. I /\ A.x e. I (A.y e. I (xGy) e. I /\ A.z e. X (zHx) e. I))))

Distinct variable groups:   x,R,y,z   z,X   x,I,y,z   x,X

Proof of Theorem isidlc

Step	Hyp	Ref	Expression
1		crngrng 16148	. . 3 |- (R e. CRing -> R e. Ring)
2		idlval.1	. . . 4 |- G = (1st` R)
3		idlval.2	. . . 4 |- H = (2nd` R)
4		idlval.3	. . . 4 |- X = ran G
5		idlval.4	. . . 4 |- Z = (Id` G)
6	2, 3, 4, 5	isidl 16162	. . 3 |- (R e. Ring -> (I e. (Idl` R) <-> (I C_ X /\ Z e. I /\ A.x e. I (A.y e. I (xGy) e. I /\ A.z e. X ((zHx) e. I /\ (xHz) e. I)))))
7	1, 6	syl 12	. 2 |- (R e. CRing -> (I e. (Idl` R) <-> (I C_ X /\ Z e. I /\ A.x e. I (A.y e. I (xGy) e. I /\ A.z e. X ((zHx) e. I /\ (xHz) e. I)))))
8	2, 3, 4	crngcom 16149	. . . . . . . . . . . . . . 15 |- ((R e. CRing /\ x e. X /\ z e. X) -> (xHz) = (zHx))
9	8	eleq1d 1963	. . . . . . . . . . . . . 14 |- ((R e. CRing /\ x e. X /\ z e. X) -> ((xHz) e. I <-> (zHx) e. I))
10	9	biimprd 171	. . . . . . . . . . . . 13 |- ((R e. CRing /\ x e. X /\ z e. X) -> ((zHx) e. I -> (xHz) e. I))
11	10	3expa 1067	. . . . . . . . . . . 12 |- (((R e. CRing /\ x e. X) /\ z e. X) -> ((zHx) e. I -> (xHz) e. I))
12		pm4.71 697	. . . . . . . . . . . 12 |- (((zHx) e. I -> (xHz) e. I) <-> ((zHx) e. I <-> ((zHx) e. I /\ (xHz) e. I)))
13	11, 12	sylib 215	. . . . . . . . . . 11 |- (((R e. CRing /\ x e. X) /\ z e. X) -> ((zHx) e. I <-> ((zHx) e. I /\ (xHz) e. I)))
14	13	bicomd 580	. . . . . . . . . 10 |- (((R e. CRing /\ x e. X) /\ z e. X) -> (((zHx) e. I /\ (xHz) e. I) <-> (zHx) e. I))
15	14	ralbidva 2119	. . . . . . . . 9 |- ((R e. CRing /\ x e. X) -> (A.z e. X ((zHx) e. I /\ (xHz) e. I) <-> A.z e. X (zHx) e. I))
16	15	anbi2d 678	. . . . . . . 8 |- ((R e. CRing /\ x e. X) -> ((A.y e. I (xGy) e. I /\ A.z e. X ((zHx) e. I /\ (xHz) e. I)) <-> (A.y e. I (xGy) e. I /\ A.z e. X (zHx) e. I)))
17		ssel2 2616	. . . . . . . 8 |- ((I C_ X /\ x e. I) -> x e. X)
18	16, 17	sylan2 500	. . . . . . 7 |- ((R e. CRing /\ (I C_ X /\ x e. I)) -> ((A.y e. I (xGy) e. I /\ A.z e. X ((zHx) e. I /\ (xHz) e. I)) <-> (A.y e. I (xGy) e. I /\ A.z e. X (zHx) e. I)))
19	18	anassrs 489	. . . . . 6 |- (((R e. CRing /\ I C_ X) /\ x e. I) -> ((A.y e. I (xGy) e. I /\ A.z e. X ((zHx) e. I /\ (xHz) e. I)) <-> (A.y e. I (xGy) e. I /\ A.z e. X (zHx) e. I)))
20	19	ralbidva 2119	. . . . 5 |- ((R e. CRing /\ I C_ X) -> (A.x e. I (A.y e. I (xGy) e. I /\ A.z e. X ((zHx) e. I /\ (xHz) e. I)) <-> A.x e. I (A.y e. I (xGy) e. I /\ A.z e. X (zHx) e. I)))
21	20	adantrr 431	. . . 4 |- ((R e. CRing /\ (I C_ X /\ Z e. I)) -> (A.x e. I (A.y e. I (xGy) e. I /\ A.z e. X ((zHx) e. I /\ (xHz) e. I)) <-> A.x e. I (A.y e. I (xGy) e. I /\ A.z e. X (zHx) e. I)))
22	21	pm5.32da 711	. . 3 |- (R e. CRing -> (((I C_ X /\ Z e. I) /\ A.x e. I (A.y e. I (xGy) e. I /\ A.z e. X ((zHx) e. I /\ (xHz) e. I))) <-> ((I C_ X /\ Z e. I) /\ A.x e. I (A.y e. I (xGy) e. I /\ A.z e. X (zHx) e. I))))
23		df-3an 860	. . 3 |- ((I C_ X /\ Z e. I /\ A.x e. I (A.y e. I (xGy) e. I /\ A.z e. X ((zHx) e. I /\ (xHz) e. I))) <-> ((I C_ X /\ Z e. I) /\ A.x e. I (A.y e. I (xGy) e. I /\ A.z e. X ((zHx) e. I /\ (xHz) e. I))))
24		df-3an 860	. . 3 |- ((I C_ X /\ Z e. I /\ A.x e. I (A.y e. I (xGy) e. I /\ A.z e. X (zHx) e. I)) <-> ((I C_ X /\ Z e. I) /\ A.x e. I (A.y e. I (xGy) e. I /\ A.z e. X (zHx) e. I)))
25	22, 23, 24	3bitr4g 614	. 2 |- (R e. CRing -> ((I C_ X /\ Z e. I /\ A.x e. I (A.y e. I (xGy) e. I /\ A.z e. X ((zHx) e. I /\ (xHz) e. I))) <-> (I C_ X /\ Z e. I /\ A.x e. I (A.y e. I (xGy) e. I /\ A.z e. X (zHx) e. I))))
26	7, 25	bitrd 587	1 |- (R e. CRing -> (I e. (Idl` R) <-> (I C_ X /\ Z e. I /\ A.x e. I (A.y e. I (xGy) e. I /\ A.z e. X (zHx) e. I))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Idcgi 9312  Ringcring 9463  CRingccring 16143  Idlcidl 16155

This theorem is referenced by:  prnc 16215

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-rab 2112  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-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-ring 9464  df-com2 10395  df-cring 16144  df-idl 16158

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