Theorem 0idl 26649

Description: The set containing only 0 is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
0idl.1	|- G = ( 1st ` R )
0idl.2	|- Z = (GId ` G )

Assertion

Ref	Expression
0idl	|- ( R e. RingOps -> { Z } e. ( Idl ` R ) )

Proof of Theorem 0idl

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0idl.1	. . . 4 |- G = ( 1st ` R )
2		eqid 2438	. . . 4 |- ran G = ran G
3		0idl.2	. . . 4 |- Z = (GId ` G )
4	1, 2, 3	rngo0cl 21991	. . 3 |- ( R e. RingOps -> Z e. ran G )
5	4	snssd 3945	. 2 |- ( R e. RingOps -> { Z } C_ ran G )
6		fvex 5745	. . . . 5 |- (GId ` G ) e. _V
7	3, 6	eqeltri 2508	. . . 4 |- Z e. _V
8	7	snid 3843	. . 3 |- Z e. { Z }
9	8	a1i 11	. 2 |- ( R e. RingOps -> Z e. { Z } )
10		elsn 3831	. . . 4 |- ( x e. { Z } <-> x = Z )
11		elsn 3831	. . . . . . . 8 |- ( y e. { Z } <-> y = Z )
12	1, 2, 3	rngo0rid 21992	. . . . . . . . . . 11 |- ( ( R e. RingOps /\ Z e. ran G ) -> ( Z G Z ) = Z )
13	4, 12	mpdan 651	. . . . . . . . . 10 |- ( R e. RingOps -> ( Z G Z ) = Z )
14		ovex 6109	. . . . . . . . . . 11 |- ( Z G Z ) e. _V
15	14	elsnc 3839	. . . . . . . . . 10 |- ( ( Z G Z ) e. { Z } <-> ( Z G Z ) = Z )
16	13, 15	sylibr 205	. . . . . . . . 9 |- ( R e. RingOps -> ( Z G Z ) e. { Z } )
17		oveq2 6092	. . . . . . . . . 10 |- ( y = Z -> ( Z G y ) = ( Z G Z ) )
18	17	eleq1d 2504	. . . . . . . . 9 |- ( y = Z -> ( ( Z G y ) e. { Z } <-> ( Z G Z ) e. { Z } ) )
19	16, 18	syl5ibrcom 215	. . . . . . . 8 |- ( R e. RingOps -> ( y = Z -> ( Z G y ) e. { Z } ) )
20	11, 19	syl5bi 210	. . . . . . 7 |- ( R e. RingOps -> ( y e. { Z } -> ( Z G y ) e. { Z } ) )
21	20	ralrimiv 2790	. . . . . 6 |- ( R e. RingOps -> A. y e. { Z } ( Z G y ) e. { Z } )
22		eqid 2438	. . . . . . . . . 10 |- ( 2nd ` R ) = ( 2nd ` R )
23	3, 2, 1, 22	rngorz 21995	. . . . . . . . 9 |- ( ( R e. RingOps /\ z e. ran G ) -> ( z ( 2nd ` R ) Z ) = Z )
24		ovex 6109	. . . . . . . . . 10 |- ( z ( 2nd ` R ) Z ) e. _V
25	24	elsnc 3839	. . . . . . . . 9 |- ( ( z ( 2nd ` R ) Z ) e. { Z } <-> ( z ( 2nd ` R ) Z ) = Z )
26	23, 25	sylibr 205	. . . . . . . 8 |- ( ( R e. RingOps /\ z e. ran G ) -> ( z ( 2nd ` R ) Z ) e. { Z } )
27	3, 2, 1, 22	rngolz 21994	. . . . . . . . 9 |- ( ( R e. RingOps /\ z e. ran G ) -> ( Z ( 2nd ` R ) z ) = Z )
28		ovex 6109	. . . . . . . . . 10 |- ( Z ( 2nd ` R ) z ) e. _V
29	28	elsnc 3839	. . . . . . . . 9 |- ( ( Z ( 2nd ` R ) z ) e. { Z } <-> ( Z ( 2nd ` R ) z ) = Z )
30	27, 29	sylibr 205	. . . . . . . 8 |- ( ( R e. RingOps /\ z e. ran G ) -> ( Z ( 2nd ` R ) z ) e. { Z } )
31	26, 30	jca 520	. . . . . . 7 |- ( ( R e. RingOps /\ z e. ran G ) -> ( ( z ( 2nd ` R ) Z ) e. { Z } /\ ( Z ( 2nd ` R ) z ) e. { Z } ) )
32	31	ralrimiva 2791	. . . . . 6 |- ( R e. RingOps -> A. z e. ran G ( ( z ( 2nd ` R ) Z ) e. { Z } /\ ( Z ( 2nd ` R ) z ) e. { Z } ) )
33	21, 32	jca 520	. . . . 5 |- ( R e. RingOps -> ( A. y e. { Z } ( Z G y ) e. { Z } /\ A. z e. ran G ( ( z ( 2nd ` R ) Z ) e. { Z } /\ ( Z ( 2nd ` R ) z ) e. { Z } ) ) )
34		oveq1 6091	. . . . . . . 8 |- ( x = Z -> ( x G y ) = ( Z G y ) )
35	34	eleq1d 2504	. . . . . . 7 |- ( x = Z -> ( ( x G y ) e. { Z } <-> ( Z G y ) e. { Z } ) )
36	35	ralbidv 2727	. . . . . 6 |- ( x = Z -> ( A. y e. { Z } ( x G y ) e. { Z } <-> A. y e. { Z } ( Z G y ) e. { Z } ) )
37		oveq2 6092	. . . . . . . . 9 |- ( x = Z -> ( z ( 2nd ` R ) x ) = ( z ( 2nd ` R ) Z ) )
38	37	eleq1d 2504	. . . . . . . 8 |- ( x = Z -> ( ( z ( 2nd ` R ) x ) e. { Z } <-> ( z ( 2nd ` R ) Z ) e. { Z } ) )
39		oveq1 6091	. . . . . . . . 9 |- ( x = Z -> ( x ( 2nd ` R ) z ) = ( Z ( 2nd ` R ) z ) )
40	39	eleq1d 2504	. . . . . . . 8 |- ( x = Z -> ( ( x ( 2nd ` R ) z ) e. { Z } <-> ( Z ( 2nd ` R ) z ) e. { Z } ) )
41	38, 40	anbi12d 693	. . . . . . 7 |- ( x = Z -> ( ( ( z ( 2nd ` R ) x ) e. { Z } /\ ( x ( 2nd ` R ) z ) e. { Z } ) <-> ( ( z ( 2nd ` R ) Z ) e. { Z } /\ ( Z ( 2nd ` R ) z ) e. { Z } ) ) )
42	41	ralbidv 2727	. . . . . 6 |- ( x = Z -> ( A. z e. ran G ( ( z ( 2nd ` R ) x ) e. { Z } /\ ( x ( 2nd ` R ) z ) e. { Z } ) <-> A. z e. ran G ( ( z ( 2nd ` R ) Z ) e. { Z } /\ ( Z ( 2nd ` R ) z ) e. { Z } ) ) )
43	36, 42	anbi12d 693	. . . . 5 |- ( x = Z -> ( ( A. y e. { Z } ( x G y ) e. { Z } /\ A. z e. ran G ( ( z ( 2nd ` R ) x ) e. { Z } /\ ( x ( 2nd ` R ) z ) e. { Z } ) ) <-> ( A. y e. { Z } ( Z G y ) e. { Z } /\ A. z e. ran G ( ( z ( 2nd ` R ) Z ) e. { Z } /\ ( Z ( 2nd ` R ) z ) e. { Z } ) ) ) )
44	33, 43	syl5ibrcom 215	. . . 4 |- ( R e. RingOps -> ( x = Z -> ( A. y e. { Z } ( x G y ) e. { Z } /\ A. z e. ran G ( ( z ( 2nd ` R ) x ) e. { Z } /\ ( x ( 2nd ` R ) z ) e. { Z } ) ) ) )
45	10, 44	syl5bi 210	. . 3 |- ( R e. RingOps -> ( x e. { Z } -> ( A. y e. { Z } ( x G y ) e. { Z } /\ A. z e. ran G ( ( z ( 2nd ` R ) x ) e. { Z } /\ ( x ( 2nd ` R ) z ) e. { Z } ) ) ) )
46	45	ralrimiv 2790	. 2 |- ( R e. RingOps -> A. x e. { Z } ( A. y e. { Z } ( x G y ) e. { Z } /\ A. z e. ran G ( ( z ( 2nd ` R ) x ) e. { Z } /\ ( x ( 2nd ` R ) z ) e. { Z } ) ) )
47	1, 22, 2, 3	isidl 26638	. 2 |- ( R e. RingOps -> ( { Z } e. ( Idl ` R ) <-> ( { Z } C_ ran G /\ Z e. { Z } /\ A. x e. { Z } ( A. y e. { Z } ( x G y ) e. { Z } /\ A. z e. ran G ( ( z ( 2nd ` R ) x ) e. { Z } /\ ( x ( 2nd ` R ) z ) e. { Z } ) ) ) ) )
48	5, 9, 46, 47	mpbir3and 1138	1 |- ( R e. RingOps -> { Z } e. ( Idl ` R ) )

Syntax hints:   -> wi 4   /\ wa 360   = wceq 1653   e. wcel 1726   A.wral 2707   _Vcvv 2958   C_ wss 3322   {csn 3816   ran crn 4882   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351  GIdcgi 21780   RingOpscrngo 21968   Idlcidl 26631

This theorem is referenced by:  0rngo  26651  divrngidl  26652  smprngopr  26676  isdmn3  26698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-1st 6352  df-2nd 6353  df-riota 6552  df-grpo 21784  df-gid 21785  df-ginv 21786  df-ablo 21875  df-rngo 21969  df-idl 26634