Metamath Proof Explorer


Theorem ssmxidl

Description: Let R be a ring, and let I be a proper ideal of R . Then there is a maximal ideal of R containing I . (Contributed by Thierry Arnoux, 10-Apr-2024)

Ref Expression
Hypothesis ssmxidl.1 B = Base R
Assertion ssmxidl Could not format assertion : No typesetting found for |- ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) -> E. m e. ( MaxIdeal ` R ) I C_ m ) with typecode |-

Proof

Step Hyp Ref Expression
1 ssmxidl.1 B = Base R
2 neeq1 p = I p B I B
3 sseq2 p = I I p I I
4 2 3 anbi12d p = I p B I p I B I I
5 simp2 R Ring I LIdeal R I B I LIdeal R
6 simp3 R Ring I LIdeal R I B I B
7 ssidd R Ring I LIdeal R I B I I
8 6 7 jca R Ring I LIdeal R I B I B I I
9 4 5 8 elrabd R Ring I LIdeal R I B I p LIdeal R | p B I p
10 9 ne0d R Ring I LIdeal R I B p LIdeal R | p B I p
11 eqid p LIdeal R | p B I p = p LIdeal R | p B I p
12 simpl1 R Ring I LIdeal R I B z p LIdeal R | p B I p z [⊂] Or z R Ring
13 simpl2 R Ring I LIdeal R I B z p LIdeal R | p B I p z [⊂] Or z I LIdeal R
14 simpl3 R Ring I LIdeal R I B z p LIdeal R | p B I p z [⊂] Or z I B
15 simpr1 R Ring I LIdeal R I B z p LIdeal R | p B I p z [⊂] Or z z p LIdeal R | p B I p
16 simpr2 R Ring I LIdeal R I B z p LIdeal R | p B I p z [⊂] Or z z
17 simpr3 R Ring I LIdeal R I B z p LIdeal R | p B I p z [⊂] Or z [⊂] Or z
18 1 11 12 13 14 15 16 17 ssmxidllem R Ring I LIdeal R I B z p LIdeal R | p B I p z [⊂] Or z z p LIdeal R | p B I p
19 18 ex R Ring I LIdeal R I B z p LIdeal R | p B I p z [⊂] Or z z p LIdeal R | p B I p
20 19 alrimiv R Ring I LIdeal R I B z z p LIdeal R | p B I p z [⊂] Or z z p LIdeal R | p B I p
21 fvex LIdeal R V
22 21 rabex p LIdeal R | p B I p V
23 22 zornn0 p LIdeal R | p B I p z z p LIdeal R | p B I p z [⊂] Or z z p LIdeal R | p B I p m p LIdeal R | p B I p j p LIdeal R | p B I p ¬ m j
24 10 20 23 syl2anc R Ring I LIdeal R I B m p LIdeal R | p B I p j p LIdeal R | p B I p ¬ m j
25 neeq1 p = m p B m B
26 sseq2 p = m I p I m
27 25 26 anbi12d p = m p B I p m B I m
28 27 elrab m p LIdeal R | p B I p m LIdeal R m B I m
29 28 anbi2i R Ring I LIdeal R I B m p LIdeal R | p B I p R Ring I LIdeal R I B m LIdeal R m B I m
30 simpll1 R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j R Ring
31 simplrl R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j m LIdeal R
32 simplr R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j m LIdeal R m B I m
33 32 simprld R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j m B
34 psseq2 j = k m j m k
35 34 notbid j = k ¬ m j ¬ m k
36 simp-4r R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j k LIdeal R m k ¬ k = B j p LIdeal R | p B I p ¬ m j
37 neeq1 p = k p B k B
38 sseq2 p = k I p I k
39 37 38 anbi12d p = k p B I p k B I k
40 simpllr R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j k LIdeal R m k ¬ k = B k LIdeal R
41 simpr R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j k LIdeal R m k ¬ k = B ¬ k = B
42 41 neqned R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j k LIdeal R m k ¬ k = B k B
43 simp-5r R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j k LIdeal R m k ¬ k = B m LIdeal R m B I m
44 43 simprrd R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j k LIdeal R m k ¬ k = B I m
45 simplr R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j k LIdeal R m k ¬ k = B m k
46 44 45 sstrd R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j k LIdeal R m k ¬ k = B I k
47 42 46 jca R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j k LIdeal R m k ¬ k = B k B I k
48 39 40 47 elrabd R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j k LIdeal R m k ¬ k = B k p LIdeal R | p B I p
49 35 36 48 rspcdva R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j k LIdeal R m k ¬ k = B ¬ m k
50 npss ¬ m k m k m = k
51 50 biimpi ¬ m k m k m = k
52 49 45 51 sylc R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j k LIdeal R m k ¬ k = B m = k
53 52 equcomd R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j k LIdeal R m k ¬ k = B k = m
54 53 ex R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j k LIdeal R m k ¬ k = B k = m
55 54 orrd R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j k LIdeal R m k k = B k = m
56 55 orcomd R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j k LIdeal R m k k = m k = B
57 56 ex R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j k LIdeal R m k k = m k = B
58 57 ralrimiva R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j k LIdeal R m k k = m k = B
59 1 ismxidl Could not format ( R e. Ring -> ( m e. ( MaxIdeal ` R ) <-> ( m e. ( LIdeal ` R ) /\ m =/= B /\ A. k e. ( LIdeal ` R ) ( m C_ k -> ( k = m \/ k = B ) ) ) ) ) : No typesetting found for |- ( R e. Ring -> ( m e. ( MaxIdeal ` R ) <-> ( m e. ( LIdeal ` R ) /\ m =/= B /\ A. k e. ( LIdeal ` R ) ( m C_ k -> ( k = m \/ k = B ) ) ) ) ) with typecode |-
60 59 biimpar Could not format ( ( R e. Ring /\ ( m e. ( LIdeal ` R ) /\ m =/= B /\ A. k e. ( LIdeal ` R ) ( m C_ k -> ( k = m \/ k = B ) ) ) ) -> m e. ( MaxIdeal ` R ) ) : No typesetting found for |- ( ( R e. Ring /\ ( m e. ( LIdeal ` R ) /\ m =/= B /\ A. k e. ( LIdeal ` R ) ( m C_ k -> ( k = m \/ k = B ) ) ) ) -> m e. ( MaxIdeal ` R ) ) with typecode |-
61 30 31 33 58 60 syl13anc Could not format ( ( ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) /\ ( m e. ( LIdeal ` R ) /\ ( m =/= B /\ I C_ m ) ) ) /\ A. j e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } -. m C. j ) -> m e. ( MaxIdeal ` R ) ) : No typesetting found for |- ( ( ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) /\ ( m e. ( LIdeal ` R ) /\ ( m =/= B /\ I C_ m ) ) ) /\ A. j e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } -. m C. j ) -> m e. ( MaxIdeal ` R ) ) with typecode |-
62 32 simprrd R Ring I LIdeal R I B m LIdeal R m B I m j p LIdeal R | p B I p ¬ m j I m
63 61 62 jca Could not format ( ( ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) /\ ( m e. ( LIdeal ` R ) /\ ( m =/= B /\ I C_ m ) ) ) /\ A. j e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } -. m C. j ) -> ( m e. ( MaxIdeal ` R ) /\ I C_ m ) ) : No typesetting found for |- ( ( ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) /\ ( m e. ( LIdeal ` R ) /\ ( m =/= B /\ I C_ m ) ) ) /\ A. j e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } -. m C. j ) -> ( m e. ( MaxIdeal ` R ) /\ I C_ m ) ) with typecode |-
64 29 63 sylanb Could not format ( ( ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) /\ m e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } ) /\ A. j e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } -. m C. j ) -> ( m e. ( MaxIdeal ` R ) /\ I C_ m ) ) : No typesetting found for |- ( ( ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) /\ m e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } ) /\ A. j e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } -. m C. j ) -> ( m e. ( MaxIdeal ` R ) /\ I C_ m ) ) with typecode |-
65 64 expl Could not format ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) -> ( ( m e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } /\ A. j e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } -. m C. j ) -> ( m e. ( MaxIdeal ` R ) /\ I C_ m ) ) ) : No typesetting found for |- ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) -> ( ( m e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } /\ A. j e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } -. m C. j ) -> ( m e. ( MaxIdeal ` R ) /\ I C_ m ) ) ) with typecode |-
66 65 reximdv2 Could not format ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) -> ( E. m e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } A. j e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } -. m C. j -> E. m e. ( MaxIdeal ` R ) I C_ m ) ) : No typesetting found for |- ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) -> ( E. m e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } A. j e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } -. m C. j -> E. m e. ( MaxIdeal ` R ) I C_ m ) ) with typecode |-
67 24 66 mpd Could not format ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) -> E. m e. ( MaxIdeal ` R ) I C_ m ) : No typesetting found for |- ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) -> E. m e. ( MaxIdeal ` R ) I C_ m ) with typecode |-