Theorem idlval 33812

Description: The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

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

Assertion

Ref	Expression
idlval	|- ( R e. RingOps -> ( Idl ` R ) = { i e. ~P X | ( Z e. i /\ A. x e. i ( A. y e. i ( x G y ) e. i /\ A. z e. X ( ( z H x ) e. i /\ ( x H z ) e. i ) ) ) } )

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

Allowed substitution hints:   G( x, y, z)   H( x, y, z)   X( x, y)   Z( x, y, z)

Proof of Theorem idlval

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . 7 |- ( r = R -> ( 1st ` r ) = ( 1st ` R ) )
2		idlval.1	. . . . . . 7 |- G = ( 1st ` R )
3	1, 2	syl6eqr 2674	. . . . . 6 |- ( r = R -> ( 1st ` r ) = G )
4	3	rneqd 5353	. . . . 5 |- ( r = R -> ran ( 1st ` r ) = ran G )
5		idlval.3	. . . . 5 |- X = ran G
6	4, 5	syl6eqr 2674	. . . 4 |- ( r = R -> ran ( 1st ` r ) = X )
7	6	pweqd 4163	. . 3 |- ( r = R -> ~P ran ( 1st ` r ) = ~P X )
8	3	fveq2d 6195	. . . . . 6 |- ( r = R -> (GId ` ( 1st ` r ) ) = (GId ` G ) )
9		idlval.4	. . . . . 6 |- Z = (GId ` G )
10	8, 9	syl6eqr 2674	. . . . 5 |- ( r = R -> (GId ` ( 1st ` r ) ) = Z )
11	10	eleq1d 2686	. . . 4 |- ( r = R -> ( (GId ` ( 1st ` r ) ) e. i <-> Z e. i ) )
12	3	oveqd 6667	. . . . . . . 8 |- ( r = R -> ( x ( 1st ` r ) y ) = ( x G y ) )
13	12	eleq1d 2686	. . . . . . 7 |- ( r = R -> ( ( x ( 1st ` r ) y ) e. i <-> ( x G y ) e. i ) )
14	13	ralbidv 2986	. . . . . 6 |- ( r = R -> ( A. y e. i ( x ( 1st ` r ) y ) e. i <-> A. y e. i ( x G y ) e. i ) )
15		fveq2 6191	. . . . . . . . . . 11 |- ( r = R -> ( 2nd ` r ) = ( 2nd ` R ) )
16		idlval.2	. . . . . . . . . . 11 |- H = ( 2nd ` R )
17	15, 16	syl6eqr 2674	. . . . . . . . . 10 |- ( r = R -> ( 2nd ` r ) = H )
18	17	oveqd 6667	. . . . . . . . 9 |- ( r = R -> ( z ( 2nd ` r ) x ) = ( z H x ) )
19	18	eleq1d 2686	. . . . . . . 8 |- ( r = R -> ( ( z ( 2nd ` r ) x ) e. i <-> ( z H x ) e. i ) )
20	17	oveqd 6667	. . . . . . . . 9 |- ( r = R -> ( x ( 2nd ` r ) z ) = ( x H z ) )
21	20	eleq1d 2686	. . . . . . . 8 |- ( r = R -> ( ( x ( 2nd ` r ) z ) e. i <-> ( x H z ) e. i ) )
22	19, 21	anbi12d 747	. . . . . . 7 |- ( r = R -> ( ( ( z ( 2nd ` r ) x ) e. i /\ ( x ( 2nd ` r ) z ) e. i ) <-> ( ( z H x ) e. i /\ ( x H z ) e. i ) ) )
23	6, 22	raleqbidv 3152	. . . . . 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 ( ( z H x ) e. i /\ ( x H z ) e. i ) ) )
24	14, 23	anbi12d 747	. . . . 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 ( x G y ) e. i /\ A. z e. X ( ( z H x ) e. i /\ ( x H z ) e. i ) ) ) )
25	24	ralbidv 2986	. . . 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 ( x G y ) e. i /\ A. z e. X ( ( z H x ) e. i /\ ( x H z ) e. i ) ) ) )
26	11, 25	anbi12d 747	. . 3 |- ( r = R -> ( ( (GId ` ( 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 ( x G y ) e. i /\ A. z e. X ( ( z H x ) e. i /\ ( x H z ) e. i ) ) ) ) )
27	7, 26	rabeqbidv 3195	. 2 |- ( r = R -> { i e. ~P ran ( 1st ` r ) | ( (GId ` ( 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. ~P X | ( Z e. i /\ A. x e. i ( A. y e. i ( x G y ) e. i /\ A. z e. X ( ( z H x ) e. i /\ ( x H z ) e. i ) ) ) } )
28		df-idl 33809	. 2 |- Idl = ( r e. RingOps |-> { i e. ~P ran ( 1st ` r ) | ( (GId ` ( 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 ) ) ) } )
29		fvex 6201	. . . . . . 7 |- ( 1st ` R ) e. _V
30	2, 29	eqeltri 2697	. . . . . 6 |- G e. _V
31	30	rnex 7100	. . . . 5 |- ran G e. _V
32	5, 31	eqeltri 2697	. . . 4 |- X e. _V
33	32	pwex 4848	. . 3 |- ~P X e. _V
34	33	rabex 4813	. 2 |- { i e. ~P X | ( Z e. i /\ A. x e. i ( A. y e. i ( x G y ) e. i /\ A. z e. X ( ( z H x ) e. i /\ ( x H z ) e. i ) ) ) } e. _V
35	27, 28, 34	fvmpt 6282	1 |- ( R e. RingOps -> ( Idl ` R ) = { i e. ~P X | ( Z e. i /\ A. x e. i ( A. y e. i ( x G y ) e. i /\ A. z e. X ( ( z H x ) e. i /\ ( x H z ) e. i ) ) ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   ~Pcpw 4158   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167  GIdcgi 27344   RingOpscrngo 33693   Idlcidl 33806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-idl 33809

This theorem is referenced by:  isidl  33813