Theorem ismaxidl 33839

Description: The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.)

Hypotheses

Ref	Expression
ismaxidl.1	|- G = ( 1st ` R )
ismaxidl.2	|- X = ran G

Assertion

Ref	Expression
ismaxidl	|- ( R e. RingOps -> ( M e. ( MaxIdl ` R ) <-> ( M e. ( Idl ` R ) /\ M =/= X /\ A. j e. ( Idl ` R ) ( M C_ j -> ( j = M \/ j = X ) ) ) ) )

Distinct variable groups:   R, j   j, M

Allowed substitution hints:   G( j)   X( j)

Proof of Theorem ismaxidl

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismaxidl.1	. . . 4 |- G = ( 1st ` R )
2		ismaxidl.2	. . . 4 |- X = ran G
3	1, 2	maxidlval 33838	. . 3 |- ( R e. RingOps -> ( MaxIdl ` R ) = { i e. ( Idl ` R ) | ( i =/= X /\ A. j e. ( Idl ` R ) ( i C_ j -> ( j = i \/ j = X ) ) ) } )
4	3	eleq2d 2687	. 2 |- ( R e. RingOps -> ( M e. ( MaxIdl ` R ) <-> M e. { i e. ( Idl ` R ) | ( i =/= X /\ A. j e. ( Idl ` R ) ( i C_ j -> ( j = i \/ j = X ) ) ) } ) )
5		neeq1 2856	. . . . 5 |- ( i = M -> ( i =/= X <-> M =/= X ) )
6		sseq1 3626	. . . . . . 7 |- ( i = M -> ( i C_ j <-> M C_ j ) )
7		eqeq2 2633	. . . . . . . 8 |- ( i = M -> ( j = i <-> j = M ) )
8	7	orbi1d 739	. . . . . . 7 |- ( i = M -> ( ( j = i \/ j = X ) <-> ( j = M \/ j = X ) ) )
9	6, 8	imbi12d 334	. . . . . 6 |- ( i = M -> ( ( i C_ j -> ( j = i \/ j = X ) ) <-> ( M C_ j -> ( j = M \/ j = X ) ) ) )
10	9	ralbidv 2986	. . . . 5 |- ( i = M -> ( A. j e. ( Idl ` R ) ( i C_ j -> ( j = i \/ j = X ) ) <-> A. j e. ( Idl ` R ) ( M C_ j -> ( j = M \/ j = X ) ) ) )
11	5, 10	anbi12d 747	. . . 4 |- ( i = M -> ( ( i =/= X /\ A. j e. ( Idl ` R ) ( i C_ j -> ( j = i \/ j = X ) ) ) <-> ( M =/= X /\ A. j e. ( Idl ` R ) ( M C_ j -> ( j = M \/ j = X ) ) ) ) )
12	11	elrab 3363	. . 3 |- ( M e. { i e. ( Idl ` R ) | ( i =/= X /\ A. j e. ( Idl ` R ) ( i C_ j -> ( j = i \/ j = X ) ) ) } <-> ( M e. ( Idl ` R ) /\ ( M =/= X /\ A. j e. ( Idl ` R ) ( M C_ j -> ( j = M \/ j = X ) ) ) ) )
13		3anass 1042	. . 3 |- ( ( M e. ( Idl ` R ) /\ M =/= X /\ A. j e. ( Idl ` R ) ( M C_ j -> ( j = M \/ j = X ) ) ) <-> ( M e. ( Idl ` R ) /\ ( M =/= X /\ A. j e. ( Idl ` R ) ( M C_ j -> ( j = M \/ j = X ) ) ) ) )
14	12, 13	bitr4i 267	. 2 |- ( M e. { i e. ( Idl ` R ) | ( i =/= X /\ A. j e. ( Idl ` R ) ( i C_ j -> ( j = i \/ j = X ) ) ) } <-> ( M e. ( Idl ` R ) /\ M =/= X /\ A. j e. ( Idl ` R ) ( M C_ j -> ( j = M \/ j = X ) ) ) )
15	4, 14	syl6bb 276	1 |- ( R e. RingOps -> ( M e. ( MaxIdl ` R ) <-> ( M e. ( Idl ` R ) /\ M =/= X /\ A. j e. ( Idl ` R ) ( M C_ j -> ( j = M \/ j = X ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   C_ wss 3574   ran crn 5115   ` cfv 5888   1stc1st 7166   RingOpscrngo 33693   Idlcidl 33806   MaxIdlcmaxidl 33808

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-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-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-maxidl 33811

This theorem is referenced by:  maxidlidl  33840  maxidlnr  33841  maxidlmax  33842