Theorem idlval 16161

Description: The class of ideals of a ring.

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
idlval	|- (R e. Ring -> (Idl` R) = {i e. ~PX | (Z e. i /\ A.x e. i (A.y e. i (xGy) e. i /\ A.z e. X ((zHx) e. i /\ (xHz) e. i)))})

Distinct variable groups:   x,R,y,z,i   z,X,i   i,Z   i,G   i,H

Proof of Theorem idlval

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . . . . 7 |- (r = R -> (1st` r) = (1st` R))
2		idlval.1	. . . . . . 7 |- G = (1st` R)
3	1, 2	syl6eqr 1946	. . . . . 6 |- (r = R -> (1st` r) = G)
4	3	rneqd 4188	. . . . 5 |- (r = R -> ran (1st` r) = ran G)
5		idlval.3	. . . . 5 |- X = ran G
6	4, 5	syl6eqr 1946	. . . 4 |- (r = R -> ran (1st` r) = X)
7		pweq 3036	. . . 4 |- (ran (1st` r) = X -> ~Pran (1st` r) = ~PX)
8	6, 7	syl 12	. . 3 |- (r = R -> ~Pran (1st` r) = ~PX)
9	3	fveq2d 4685	. . . . . 6 |- (r = R -> (Id` (1st` r)) = (Id` G))
10		idlval.4	. . . . . 6 |- Z = (Id` G)
11	9, 10	syl6eqr 1946	. . . . 5 |- (r = R -> (Id` (1st` r)) = Z)
12	11	eleq1d 1963	. . . 4 |- (r = R -> ((Id` (1st` r)) e. i <-> Z e. i))
13	3	opreqd 4899	. . . . . . . 8 |- (r = R -> (x(1st` r)y) = (xGy))
14	13	eleq1d 1963	. . . . . . 7 |- (r = R -> ((x(1st` r)y) e. i <-> (xGy) e. i))
15	14	ralbidv 2123	. . . . . 6 |- (r = R -> (A.y e. i (x(1st` r)y) e. i <-> A.y e. i (xGy) e. i))
16		fveq2 4681	. . . . . . . . . . 11 |- (r = R -> (2nd` r) = (2nd` R))
17		idlval.2	. . . . . . . . . . 11 |- H = (2nd` R)
18	16, 17	syl6eqr 1946	. . . . . . . . . 10 |- (r = R -> (2nd` r) = H)
19	18	opreqd 4899	. . . . . . . . 9 |- (r = R -> (z(2nd` r)x) = (zHx))
20	19	eleq1d 1963	. . . . . . . 8 |- (r = R -> ((z(2nd` r)x) e. i <-> (zHx) e. i))
21	18	opreqd 4899	. . . . . . . . 9 |- (r = R -> (x(2nd` r)z) = (xHz))
22	21	eleq1d 1963	. . . . . . . 8 |- (r = R -> ((x(2nd` r)z) e. i <-> (xHz) e. i))
23	20, 22	anbi12d 690	. . . . . . 7 |- (r = R -> (((z(2nd` r)x) e. i /\ (x(2nd` r)z) e. i) <-> ((zHx) e. i /\ (xHz) e. i)))
24	6, 23	raleqbidv 2274	. . . . . 6 |- (r = R -> (A.z e. ran (1st` r)((z(2nd` r)x) e. i /\ (x(2nd` r)z) e. i) <-> A.z e. X ((zHx) e. i /\ (xHz) e. i)))
25	15, 24	anbi12d 690	. . . . 5 |- (r = R -> ((A.y e. i (x(1st` r)y) e. i /\ A.z e. ran (1st` r)((z(2nd` r)x) e. i /\ (x(2nd` r)z) e. i)) <-> (A.y e. i (xGy) e. i /\ A.z e. X ((zHx) e. i /\ (xHz) e. i))))
26	25	ralbidv 2123	. . . 4 |- (r = R -> (A.x e. i (A.y e. i (x(1st` r)y) e. i /\ A.z e. ran (1st` r)((z(2nd` r)x) e. i /\ (x(2nd` r)z) e. i)) <-> A.x e. i (A.y e. i (xGy) e. i /\ A.z e. X ((zHx) e. i /\ (xHz) e. i))))
27	12, 26	anbi12d 690	. . 3 |- (r = R -> (((Id` (1st` r)) e. i /\ A.x e. i (A.y e. i (x(1st` r)y) e. i /\ A.z e. ran (1st` r)((z(2nd` r)x) e. i /\ (x(2nd` r)z) e. i))) <-> (Z e. i /\ A.x e. i (A.y e. i (xGy) e. i /\ A.z e. X ((zHx) e. i /\ (xHz) e. i)))))
28	8, 27	rabeqbidv 2290	. 2 |- (r = R -> {i e. ~Pran (1st` r) | ((Id` (1st` r)) e. i /\ A.x e. i (A.y e. i (x(1st` r)y) e. i /\ A.z e. ran (1st` r)((z(2nd` r)x) e. i /\ (x(2nd` r)z) e. i)))} = {i e. ~PX | (Z e. i /\ A.x e. i (A.y e. i (xGy) e. i /\ A.z e. X ((zHx) e. i /\ (xHz) e. i)))})
29		df-idl 16158	. 2 |- Idl = {<.r, s>. | (r e. Ring /\ s = {i e. ~Pran (1st` r) | ((Id` (1st` r)) e. i /\ A.x e. i (A.y e. i (x(1st` r)y) e. i /\ A.z e. ran (1st` r)((z(2nd` r)x) e. i /\ (x(2nd` r)z) e. i)))})}
30		fvex 4689	. . . . . . 7 |- (1st` R) e. _V
31	2, 30	eqeltri 1967	. . . . . 6 |- G e. _V
32	31	rnex 4209	. . . . 5 |- ran G e. _V
33	5, 32	eqeltri 1967	. . . 4 |- X e. _V
34	33	pwex 3487	. . 3 |- ~PX e. _V
35	34	rabex 3461	. 2 |- {i e. ~PX | (Z e. i /\ A.x e. i (A.y e. i (xGy) e. i /\ A.z e. X ((zHx) e. i /\ (xHz) e. i)))} e. _V
36	28, 29, 35	fvopab4 4743	1 |- (R e. Ring -> (Idl` R) = {i e. ~PX | (Z e. i /\ A.x e. i (A.y e. i (xGy) e. i /\ A.z e. X ((zHx) e. i /\ (xHz) e. i)))})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108  _Vcvv 2292  ~Pcpw 3032  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Idcgi 9312  Ringcring 9463  Idlcidl 16155

This theorem is referenced by:  isidl 16162

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-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-idl 16158

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