Theorem 0idl 26529

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 2408	. . . 4 |- ran G = ran G
3		0idl.2	. . . 4 |- Z = (GId ` G )
4	1, 2, 3	rngo0cl 21943	. . 3 |- ( R e. RingOps -> Z e. ran G )
5	4	snssd 3907	. 2 |- ( R e. RingOps -> { Z } C_ ran G )
6		fvex 5705	. . . . 5 |- (GId ` G ) e. _V
7	3, 6	eqeltri 2478	. . . 4 |- Z e. _V
8	7	snid 3805	. . 3 |- Z e. { Z }
9	8	a1i 11	. 2 |- ( R e. RingOps -> Z e. { Z } )
10		elsn 3793	. . . 4 |- ( x e. { Z } <-> x = Z )
11		elsn 3793	. . . . . . . 8 |- ( y e. { Z } <-> y = Z )
12	1, 2, 3	rngo0rid 21944	. . . . . . . . . . 11 |- ( ( R e. RingOps /\ Z e. ran G ) -> ( Z G Z ) = Z )
13	4, 12	mpdan 650	. . . . . . . . . 10 |- ( R e. RingOps -> ( Z G Z ) = Z )
14		ovex 6069	. . . . . . . . . . 11 |- ( Z G Z ) e. _V
15	14	elsnc 3801	. . . . . . . . . 10 |- ( ( Z G Z ) e. { Z } <-> ( Z G Z ) = Z )
16	13, 15	sylibr 204	. . . . . . . . 9 |- ( R e. RingOps -> ( Z G Z ) e. { Z } )
17		oveq2 6052	. . . . . . . . . 10 |- ( y = Z -> ( Z G y ) = ( Z G Z ) )
18	17	eleq1d 2474	. . . . . . . . 9 |- ( y = Z -> ( ( Z G y ) e. { Z } <-> ( Z G Z ) e. { Z } ) )
19	16, 18	syl5ibrcom 214	. . . . . . . 8 |- ( R e. RingOps -> ( y = Z -> ( Z G y ) e. { Z } ) )
20	11, 19	syl5bi 209	. . . . . . 7 |- ( R e. RingOps -> ( y e. { Z } -> ( Z G y ) e. { Z } ) )
21	20	ralrimiv 2752	. . . . . 6 |- ( R e. RingOps -> A. y e. { Z } ( Z G y ) e. { Z } )
22		eqid 2408	. . . . . . . . . 10 |- ( 2nd ` R ) = ( 2nd ` R )
23	3, 2, 1, 22	rngorz 21947	. . . . . . . . 9 |- ( ( R e. RingOps /\ z e. ran G ) -> ( z ( 2nd ` R ) Z ) = Z )
24		ovex 6069	. . . . . . . . . 10 |- ( z ( 2nd ` R ) Z ) e. _V
25	24	elsnc 3801	. . . . . . . . 9 |- ( ( z ( 2nd ` R ) Z ) e. { Z } <-> ( z ( 2nd ` R ) Z ) = Z )
26	23, 25	sylibr 204	. . . . . . . 8 |- ( ( R e. RingOps /\ z e. ran G ) -> ( z ( 2nd ` R ) Z ) e. { Z } )
27	3, 2, 1, 22	rngolz 21946	. . . . . . . . 9 |- ( ( R e. RingOps /\ z e. ran G ) -> ( Z ( 2nd ` R ) z ) = Z )
28		ovex 6069	. . . . . . . . . 10 |- ( Z ( 2nd ` R ) z ) e. _V
29	28	elsnc 3801	. . . . . . . . 9 |- ( ( Z ( 2nd ` R ) z ) e. { Z } <-> ( Z ( 2nd ` R ) z ) = Z )
30	27, 29	sylibr 204	. . . . . . . 8 |- ( ( R e. RingOps /\ z e. ran G ) -> ( Z ( 2nd ` R ) z ) e. { Z } )
31	26, 30	jca 519	. . . . . . 7 |- ( ( R e. RingOps /\ z e. ran G ) -> ( ( z ( 2nd ` R ) Z ) e. { Z } /\ ( Z ( 2nd ` R ) z ) e. { Z } ) )
32	31	ralrimiva 2753	. . . . . 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 519	. . . . 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 6051	. . . . . . . 8 |- ( x = Z -> ( x G y ) = ( Z G y ) )
35	34	eleq1d 2474	. . . . . . 7 |- ( x = Z -> ( ( x G y ) e. { Z } <-> ( Z G y ) e. { Z } ) )
36	35	ralbidv 2690	. . . . . 6 |- ( x = Z -> ( A. y e. { Z } ( x G y ) e. { Z } <-> A. y e. { Z } ( Z G y ) e. { Z } ) )
37		oveq2 6052	. . . . . . . . 9 |- ( x = Z -> ( z ( 2nd ` R ) x ) = ( z ( 2nd ` R ) Z ) )
38	37	eleq1d 2474	. . . . . . . 8 |- ( x = Z -> ( ( z ( 2nd ` R ) x ) e. { Z } <-> ( z ( 2nd ` R ) Z ) e. { Z } ) )
39		oveq1 6051	. . . . . . . . 9 |- ( x = Z -> ( x ( 2nd ` R ) z ) = ( Z ( 2nd ` R ) z ) )
40	39	eleq1d 2474	. . . . . . . 8 |- ( x = Z -> ( ( x ( 2nd ` R ) z ) e. { Z } <-> ( Z ( 2nd ` R ) z ) e. { Z } ) )
41	38, 40	anbi12d 692	. . . . . . 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 2690	. . . . . 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 692	. . . . 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 214	. . . 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 209	. . 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 2752	. 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 26518	. 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 1137	1 |- ( R e. RingOps -> { Z } e. ( Idl ` R ) )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1649   e. wcel 1721   A.wral 2670   _Vcvv 2920   C_ wss 3284   {csn 3778   ran crn 4842   ` cfv 5417  (class class class)co 6044   1stc1st 6310   2ndc2nd 6311  GIdcgi 21732   RingOpscrngo 21920   Idlcidl 26511

This theorem is referenced by:  0rngo  26531  divrngidl  26532  smprngopr  26556  isdmn3  26578

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-1st 6312  df-2nd 6313  df-riota 6512  df-grpo 21736  df-gid 21737  df-ginv 21738  df-ablo 21827  df-rngo 21921  df-idl 26514