Theorem 0idl 33824

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 2622	. . . 4 |- ran G = ran G
3		0idl.2	. . . 4 |- Z = (GId ` G )
4	1, 2, 3	rngo0cl 33718	. . 3 |- ( R e. RingOps -> Z e. ran G )
5	4	snssd 4340	. 2 |- ( R e. RingOps -> { Z } C_ ran G )
6		fvex 6201	. . . . 5 |- (GId ` G ) e. _V
7	3, 6	eqeltri 2697	. . . 4 |- Z e. _V
8	7	snid 4208	. . 3 |- Z e. { Z }
9	8	a1i 11	. 2 |- ( R e. RingOps -> Z e. { Z } )
10		velsn 4193	. . . 4 |- ( x e. { Z } <-> x = Z )
11		velsn 4193	. . . . . . . 8 |- ( y e. { Z } <-> y = Z )
12	1, 2, 3	rngo0rid 33719	. . . . . . . . . . 11 |- ( ( R e. RingOps /\ Z e. ran G ) -> ( Z G Z ) = Z )
13	4, 12	mpdan 702	. . . . . . . . . 10 |- ( R e. RingOps -> ( Z G Z ) = Z )
14		ovex 6678	. . . . . . . . . . 11 |- ( Z G Z ) e. _V
15	14	elsn 4192	. . . . . . . . . 10 |- ( ( Z G Z ) e. { Z } <-> ( Z G Z ) = Z )
16	13, 15	sylibr 224	. . . . . . . . 9 |- ( R e. RingOps -> ( Z G Z ) e. { Z } )
17		oveq2 6658	. . . . . . . . . 10 |- ( y = Z -> ( Z G y ) = ( Z G Z ) )
18	17	eleq1d 2686	. . . . . . . . 9 |- ( y = Z -> ( ( Z G y ) e. { Z } <-> ( Z G Z ) e. { Z } ) )
19	16, 18	syl5ibrcom 237	. . . . . . . 8 |- ( R e. RingOps -> ( y = Z -> ( Z G y ) e. { Z } ) )
20	11, 19	syl5bi 232	. . . . . . 7 |- ( R e. RingOps -> ( y e. { Z } -> ( Z G y ) e. { Z } ) )
21	20	ralrimiv 2965	. . . . . 6 |- ( R e. RingOps -> A. y e. { Z } ( Z G y ) e. { Z } )
22		eqid 2622	. . . . . . . . . 10 |- ( 2nd ` R ) = ( 2nd ` R )
23	3, 2, 1, 22	rngorz 33722	. . . . . . . . 9 |- ( ( R e. RingOps /\ z e. ran G ) -> ( z ( 2nd ` R ) Z ) = Z )
24		ovex 6678	. . . . . . . . . 10 |- ( z ( 2nd ` R ) Z ) e. _V
25	24	elsn 4192	. . . . . . . . 9 |- ( ( z ( 2nd ` R ) Z ) e. { Z } <-> ( z ( 2nd ` R ) Z ) = Z )
26	23, 25	sylibr 224	. . . . . . . 8 |- ( ( R e. RingOps /\ z e. ran G ) -> ( z ( 2nd ` R ) Z ) e. { Z } )
27	3, 2, 1, 22	rngolz 33721	. . . . . . . . 9 |- ( ( R e. RingOps /\ z e. ran G ) -> ( Z ( 2nd ` R ) z ) = Z )
28		ovex 6678	. . . . . . . . . 10 |- ( Z ( 2nd ` R ) z ) e. _V
29	28	elsn 4192	. . . . . . . . 9 |- ( ( Z ( 2nd ` R ) z ) e. { Z } <-> ( Z ( 2nd ` R ) z ) = Z )
30	27, 29	sylibr 224	. . . . . . . 8 |- ( ( R e. RingOps /\ z e. ran G ) -> ( Z ( 2nd ` R ) z ) e. { Z } )
31	26, 30	jca 554	. . . . . . 7 |- ( ( R e. RingOps /\ z e. ran G ) -> ( ( z ( 2nd ` R ) Z ) e. { Z } /\ ( Z ( 2nd ` R ) z ) e. { Z } ) )
32	31	ralrimiva 2966	. . . . . 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 554	. . . . 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 6657	. . . . . . . 8 |- ( x = Z -> ( x G y ) = ( Z G y ) )
35	34	eleq1d 2686	. . . . . . 7 |- ( x = Z -> ( ( x G y ) e. { Z } <-> ( Z G y ) e. { Z } ) )
36	35	ralbidv 2986	. . . . . 6 |- ( x = Z -> ( A. y e. { Z } ( x G y ) e. { Z } <-> A. y e. { Z } ( Z G y ) e. { Z } ) )
37		oveq2 6658	. . . . . . . . 9 |- ( x = Z -> ( z ( 2nd ` R ) x ) = ( z ( 2nd ` R ) Z ) )
38	37	eleq1d 2686	. . . . . . . 8 |- ( x = Z -> ( ( z ( 2nd ` R ) x ) e. { Z } <-> ( z ( 2nd ` R ) Z ) e. { Z } ) )
39		oveq1 6657	. . . . . . . . 9 |- ( x = Z -> ( x ( 2nd ` R ) z ) = ( Z ( 2nd ` R ) z ) )
40	39	eleq1d 2686	. . . . . . . 8 |- ( x = Z -> ( ( x ( 2nd ` R ) z ) e. { Z } <-> ( Z ( 2nd ` R ) z ) e. { Z } ) )
41	38, 40	anbi12d 747	. . . . . . 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 2986	. . . . . 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 747	. . . . 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 237	. . . 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 232	. . 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 2965	. 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 33813	. 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 1245	1 |- ( R e. RingOps -> { Z } e. ( Idl ` R ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   {csn 4177   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-rep 4771  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-rngo 33694  df-idl 33809

This theorem is referenced by:  0rngo  33826  divrngidl  33827  smprngopr  33851  isdmn3  33873