Metamath Proof Explorer


Theorem zarclssn

Description: The closed points of Zariski topology are the maximal ideals. (Contributed by Thierry Arnoux, 16-Jun-2024)

Ref Expression
Hypotheses zarclsx.1 No typesetting found for |- V = ( i e. ( LIdeal ` R ) |-> { j e. ( PrmIdeal ` R ) | i C_ j } ) with typecode |-
zarclssn.1 B = LIdeal R
Assertion zarclssn Could not format assertion : No typesetting found for |- ( ( R e. CRing /\ M e. B ) -> ( { M } = ( V ` M ) <-> M e. ( MaxIdeal ` R ) ) ) with typecode |-

Proof

Step Hyp Ref Expression
1 zarclsx.1 Could not format V = ( i e. ( LIdeal ` R ) |-> { j e. ( PrmIdeal ` R ) | i C_ j } ) : No typesetting found for |- V = ( i e. ( LIdeal ` R ) |-> { j e. ( PrmIdeal ` R ) | i C_ j } ) with typecode |-
2 zarclssn.1 B = LIdeal R
3 crngring R CRing R Ring
4 3 ad2antrr R CRing M B M = V M R Ring
5 simplr R CRing M B M = V M M B
6 5 2 eleqtrdi R CRing M B M = V M M LIdeal R
7 simpr R CRing M B M = V M M = V M
8 5 snn0d R CRing M B M = V M M
9 7 8 eqnetrrd R CRing M B M = V M V M
10 simpll R CRing M B M = V M R CRing
11 eqid Base R = Base R
12 1 11 zarcls1 R CRing M LIdeal R V M = M = Base R
13 12 necon3bid R CRing M LIdeal R V M M Base R
14 10 6 13 syl2anc R CRing M B M = V M V M M Base R
15 9 14 mpbid R CRing M B M = V M M Base R
16 simpr Could not format ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> j C_ m ) : No typesetting found for |- ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> j C_ m ) with typecode |-
17 10 ad5antr Could not format ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> R e. CRing ) : No typesetting found for |- ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> R e. CRing ) with typecode |-
18 simplr Could not format ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> m e. ( MaxIdeal ` R ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> m e. ( MaxIdeal ` R ) ) with typecode |-
19 eqid LSSum mulGrp R = LSSum mulGrp R
20 19 mxidlprm Could not format ( ( R e. CRing /\ m e. ( MaxIdeal ` R ) ) -> m e. ( PrmIdeal ` R ) ) : No typesetting found for |- ( ( R e. CRing /\ m e. ( MaxIdeal ` R ) ) -> m e. ( PrmIdeal ` R ) ) with typecode |-
21 17 18 20 syl2anc Could not format ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> m e. ( PrmIdeal ` R ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> m e. ( PrmIdeal ` R ) ) with typecode |-
22 simp-4r Could not format ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> M C_ j ) : No typesetting found for |- ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> M C_ j ) with typecode |-
23 22 16 sstrd Could not format ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> M C_ m ) : No typesetting found for |- ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> M C_ m ) with typecode |-
24 1 a1i Could not format ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) -> V = ( i e. ( LIdeal ` R ) |-> { j e. ( PrmIdeal ` R ) | i C_ j } ) ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) -> V = ( i e. ( LIdeal ` R ) |-> { j e. ( PrmIdeal ` R ) | i C_ j } ) ) with typecode |-
25 sseq1 i = M i j M j
26 25 rabbidv Could not format ( i = M -> { j e. ( PrmIdeal ` R ) | i C_ j } = { j e. ( PrmIdeal ` R ) | M C_ j } ) : No typesetting found for |- ( i = M -> { j e. ( PrmIdeal ` R ) | i C_ j } = { j e. ( PrmIdeal ` R ) | M C_ j } ) with typecode |-
27 26 adantl Could not format ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ i = M ) -> { j e. ( PrmIdeal ` R ) | i C_ j } = { j e. ( PrmIdeal ` R ) | M C_ j } ) : No typesetting found for |- ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ i = M ) -> { j e. ( PrmIdeal ` R ) | i C_ j } = { j e. ( PrmIdeal ` R ) | M C_ j } ) with typecode |-
28 fvex Could not format ( PrmIdeal ` R ) e. _V : No typesetting found for |- ( PrmIdeal ` R ) e. _V with typecode |-
29 28 rabex Could not format { j e. ( PrmIdeal ` R ) | M C_ j } e. _V : No typesetting found for |- { j e. ( PrmIdeal ` R ) | M C_ j } e. _V with typecode |-
30 29 a1i Could not format ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) -> { j e. ( PrmIdeal ` R ) | M C_ j } e. _V ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) -> { j e. ( PrmIdeal ` R ) | M C_ j } e. _V ) with typecode |-
31 24 27 6 30 fvmptd Could not format ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) -> ( V ` M ) = { j e. ( PrmIdeal ` R ) | M C_ j } ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) -> ( V ` M ) = { j e. ( PrmIdeal ` R ) | M C_ j } ) with typecode |-
32 7 31 eqtr2d Could not format ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) -> { j e. ( PrmIdeal ` R ) | M C_ j } = { M } ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) -> { j e. ( PrmIdeal ` R ) | M C_ j } = { M } ) with typecode |-
33 rabeqsn Could not format ( { j e. ( PrmIdeal ` R ) | M C_ j } = { M } <-> A. j ( ( j e. ( PrmIdeal ` R ) /\ M C_ j ) <-> j = M ) ) : No typesetting found for |- ( { j e. ( PrmIdeal ` R ) | M C_ j } = { M } <-> A. j ( ( j e. ( PrmIdeal ` R ) /\ M C_ j ) <-> j = M ) ) with typecode |-
34 32 33 sylib Could not format ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) -> A. j ( ( j e. ( PrmIdeal ` R ) /\ M C_ j ) <-> j = M ) ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) -> A. j ( ( j e. ( PrmIdeal ` R ) /\ M C_ j ) <-> j = M ) ) with typecode |-
35 34 ad5antr Could not format ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> A. j ( ( j e. ( PrmIdeal ` R ) /\ M C_ j ) <-> j = M ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> A. j ( ( j e. ( PrmIdeal ` R ) /\ M C_ j ) <-> j = M ) ) with typecode |-
36 vex m V
37 eleq1w Could not format ( j = m -> ( j e. ( PrmIdeal ` R ) <-> m e. ( PrmIdeal ` R ) ) ) : No typesetting found for |- ( j = m -> ( j e. ( PrmIdeal ` R ) <-> m e. ( PrmIdeal ` R ) ) ) with typecode |-
38 sseq2 j = m M j M m
39 37 38 anbi12d Could not format ( j = m -> ( ( j e. ( PrmIdeal ` R ) /\ M C_ j ) <-> ( m e. ( PrmIdeal ` R ) /\ M C_ m ) ) ) : No typesetting found for |- ( j = m -> ( ( j e. ( PrmIdeal ` R ) /\ M C_ j ) <-> ( m e. ( PrmIdeal ` R ) /\ M C_ m ) ) ) with typecode |-
40 eqeq1 j = m j = M m = M
41 39 40 bibi12d Could not format ( j = m -> ( ( ( j e. ( PrmIdeal ` R ) /\ M C_ j ) <-> j = M ) <-> ( ( m e. ( PrmIdeal ` R ) /\ M C_ m ) <-> m = M ) ) ) : No typesetting found for |- ( j = m -> ( ( ( j e. ( PrmIdeal ` R ) /\ M C_ j ) <-> j = M ) <-> ( ( m e. ( PrmIdeal ` R ) /\ M C_ m ) <-> m = M ) ) ) with typecode |-
42 36 41 spcv Could not format ( A. j ( ( j e. ( PrmIdeal ` R ) /\ M C_ j ) <-> j = M ) -> ( ( m e. ( PrmIdeal ` R ) /\ M C_ m ) <-> m = M ) ) : No typesetting found for |- ( A. j ( ( j e. ( PrmIdeal ` R ) /\ M C_ j ) <-> j = M ) -> ( ( m e. ( PrmIdeal ` R ) /\ M C_ m ) <-> m = M ) ) with typecode |-
43 35 42 syl Could not format ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> ( ( m e. ( PrmIdeal ` R ) /\ M C_ m ) <-> m = M ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> ( ( m e. ( PrmIdeal ` R ) /\ M C_ m ) <-> m = M ) ) with typecode |-
44 21 23 43 mpbi2and Could not format ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> m = M ) : No typesetting found for |- ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> m = M ) with typecode |-
45 16 44 sseqtrd Could not format ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> j C_ M ) : No typesetting found for |- ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> j C_ M ) with typecode |-
46 45 22 eqssd Could not format ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> j = M ) : No typesetting found for |- ( ( ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) /\ m e. ( MaxIdeal ` R ) ) /\ j C_ m ) -> j = M ) with typecode |-
47 3 ad5antr R CRing M B M = V M j LIdeal R M j ¬ j = Base R R Ring
48 simpllr R CRing M B M = V M j LIdeal R M j ¬ j = Base R j LIdeal R
49 simpr R CRing M B M = V M j LIdeal R M j ¬ j = Base R ¬ j = Base R
50 49 neqned R CRing M B M = V M j LIdeal R M j ¬ j = Base R j Base R
51 11 ssmxidl Could not format ( ( R e. Ring /\ j e. ( LIdeal ` R ) /\ j =/= ( Base ` R ) ) -> E. m e. ( MaxIdeal ` R ) j C_ m ) : No typesetting found for |- ( ( R e. Ring /\ j e. ( LIdeal ` R ) /\ j =/= ( Base ` R ) ) -> E. m e. ( MaxIdeal ` R ) j C_ m ) with typecode |-
52 47 48 50 51 syl3anc Could not format ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) -> E. m e. ( MaxIdeal ` R ) j C_ m ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) /\ j e. ( LIdeal ` R ) ) /\ M C_ j ) /\ -. j = ( Base ` R ) ) -> E. m e. ( MaxIdeal ` R ) j C_ m ) with typecode |-
53 46 52 r19.29a R CRing M B M = V M j LIdeal R M j ¬ j = Base R j = M
54 53 ex R CRing M B M = V M j LIdeal R M j ¬ j = Base R j = M
55 54 orrd R CRing M B M = V M j LIdeal R M j j = Base R j = M
56 55 orcomd R CRing M B M = V M j LIdeal R M j j = M j = Base R
57 56 ex R CRing M B M = V M j LIdeal R M j j = M j = Base R
58 57 ralrimiva R CRing M B M = V M j LIdeal R M j j = M j = Base R
59 6 15 58 3jca R CRing M B M = V M M LIdeal R M Base R j LIdeal R M j j = M j = Base R
60 11 ismxidl Could not format ( R e. Ring -> ( M e. ( MaxIdeal ` R ) <-> ( M e. ( LIdeal ` R ) /\ M =/= ( Base ` R ) /\ A. j e. ( LIdeal ` R ) ( M C_ j -> ( j = M \/ j = ( Base ` R ) ) ) ) ) ) : No typesetting found for |- ( R e. Ring -> ( M e. ( MaxIdeal ` R ) <-> ( M e. ( LIdeal ` R ) /\ M =/= ( Base ` R ) /\ A. j e. ( LIdeal ` R ) ( M C_ j -> ( j = M \/ j = ( Base ` R ) ) ) ) ) ) with typecode |-
61 60 biimpar Could not format ( ( R e. Ring /\ ( M e. ( LIdeal ` R ) /\ M =/= ( Base ` R ) /\ A. j e. ( LIdeal ` R ) ( M C_ j -> ( j = M \/ j = ( Base ` R ) ) ) ) ) -> M e. ( MaxIdeal ` R ) ) : No typesetting found for |- ( ( R e. Ring /\ ( M e. ( LIdeal ` R ) /\ M =/= ( Base ` R ) /\ A. j e. ( LIdeal ` R ) ( M C_ j -> ( j = M \/ j = ( Base ` R ) ) ) ) ) -> M e. ( MaxIdeal ` R ) ) with typecode |-
62 4 59 61 syl2anc Could not format ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) -> M e. ( MaxIdeal ` R ) ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. B ) /\ { M } = ( V ` M ) ) -> M e. ( MaxIdeal ` R ) ) with typecode |-
63 1 a1i Could not format ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> V = ( i e. ( LIdeal ` R ) |-> { j e. ( PrmIdeal ` R ) | i C_ j } ) ) : No typesetting found for |- ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> V = ( i e. ( LIdeal ` R ) |-> { j e. ( PrmIdeal ` R ) | i C_ j } ) ) with typecode |-
64 26 adantl Could not format ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ i = M ) -> { j e. ( PrmIdeal ` R ) | i C_ j } = { j e. ( PrmIdeal ` R ) | M C_ j } ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ i = M ) -> { j e. ( PrmIdeal ` R ) | i C_ j } = { j e. ( PrmIdeal ` R ) | M C_ j } ) with typecode |-
65 11 mxidlidl Could not format ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> M e. ( LIdeal ` R ) ) : No typesetting found for |- ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> M e. ( LIdeal ` R ) ) with typecode |-
66 3 65 sylan Could not format ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> M e. ( LIdeal ` R ) ) : No typesetting found for |- ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> M e. ( LIdeal ` R ) ) with typecode |-
67 29 a1i Could not format ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> { j e. ( PrmIdeal ` R ) | M C_ j } e. _V ) : No typesetting found for |- ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> { j e. ( PrmIdeal ` R ) | M C_ j } e. _V ) with typecode |-
68 63 64 66 67 fvmptd Could not format ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> ( V ` M ) = { j e. ( PrmIdeal ` R ) | M C_ j } ) : No typesetting found for |- ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> ( V ` M ) = { j e. ( PrmIdeal ` R ) | M C_ j } ) with typecode |-
69 3 ad2antrr Could not format ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> R e. Ring ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> R e. Ring ) with typecode |-
70 simplr Could not format ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> M e. ( MaxIdeal ` R ) ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> M e. ( MaxIdeal ` R ) ) with typecode |-
71 simprl Could not format ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> j e. ( PrmIdeal ` R ) ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> j e. ( PrmIdeal ` R ) ) with typecode |-
72 prmidlidl Could not format ( ( R e. Ring /\ j e. ( PrmIdeal ` R ) ) -> j e. ( LIdeal ` R ) ) : No typesetting found for |- ( ( R e. Ring /\ j e. ( PrmIdeal ` R ) ) -> j e. ( LIdeal ` R ) ) with typecode |-
73 69 71 72 syl2anc Could not format ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> j e. ( LIdeal ` R ) ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> j e. ( LIdeal ` R ) ) with typecode |-
74 simprr Could not format ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> M C_ j ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> M C_ j ) with typecode |-
75 73 74 jca Could not format ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> ( j e. ( LIdeal ` R ) /\ M C_ j ) ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> ( j e. ( LIdeal ` R ) /\ M C_ j ) ) with typecode |-
76 11 mxidlmax Could not format ( ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( LIdeal ` R ) /\ M C_ j ) ) -> ( j = M \/ j = ( Base ` R ) ) ) : No typesetting found for |- ( ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( LIdeal ` R ) /\ M C_ j ) ) -> ( j = M \/ j = ( Base ` R ) ) ) with typecode |-
77 69 70 75 76 syl21anc Could not format ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> ( j = M \/ j = ( Base ` R ) ) ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> ( j = M \/ j = ( Base ` R ) ) ) with typecode |-
78 eqid R = R
79 11 78 prmidlnr Could not format ( ( R e. Ring /\ j e. ( PrmIdeal ` R ) ) -> j =/= ( Base ` R ) ) : No typesetting found for |- ( ( R e. Ring /\ j e. ( PrmIdeal ` R ) ) -> j =/= ( Base ` R ) ) with typecode |-
80 69 71 79 syl2anc Could not format ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> j =/= ( Base ` R ) ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> j =/= ( Base ` R ) ) with typecode |-
81 80 neneqd Could not format ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> -. j = ( Base ` R ) ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> -. j = ( Base ` R ) ) with typecode |-
82 77 81 olcnd Could not format ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> j = M ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) -> j = M ) with typecode |-
83 simpr Could not format ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ j = M ) -> j = M ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ j = M ) -> j = M ) with typecode |-
84 19 mxidlprm Could not format ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> M e. ( PrmIdeal ` R ) ) : No typesetting found for |- ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> M e. ( PrmIdeal ` R ) ) with typecode |-
85 84 adantr Could not format ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ j = M ) -> M e. ( PrmIdeal ` R ) ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ j = M ) -> M e. ( PrmIdeal ` R ) ) with typecode |-
86 83 85 eqeltrd Could not format ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ j = M ) -> j e. ( PrmIdeal ` R ) ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ j = M ) -> j e. ( PrmIdeal ` R ) ) with typecode |-
87 ssidd Could not format ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ j = M ) -> j C_ j ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ j = M ) -> j C_ j ) with typecode |-
88 83 87 eqsstrrd Could not format ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ j = M ) -> M C_ j ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ j = M ) -> M C_ j ) with typecode |-
89 86 88 jca Could not format ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ j = M ) -> ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ j = M ) -> ( j e. ( PrmIdeal ` R ) /\ M C_ j ) ) with typecode |-
90 82 89 impbida Could not format ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> ( ( j e. ( PrmIdeal ` R ) /\ M C_ j ) <-> j = M ) ) : No typesetting found for |- ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> ( ( j e. ( PrmIdeal ` R ) /\ M C_ j ) <-> j = M ) ) with typecode |-
91 90 alrimiv Could not format ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> A. j ( ( j e. ( PrmIdeal ` R ) /\ M C_ j ) <-> j = M ) ) : No typesetting found for |- ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> A. j ( ( j e. ( PrmIdeal ` R ) /\ M C_ j ) <-> j = M ) ) with typecode |-
92 91 33 sylibr Could not format ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> { j e. ( PrmIdeal ` R ) | M C_ j } = { M } ) : No typesetting found for |- ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> { j e. ( PrmIdeal ` R ) | M C_ j } = { M } ) with typecode |-
93 68 92 eqtr2d Could not format ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> { M } = ( V ` M ) ) : No typesetting found for |- ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> { M } = ( V ` M ) ) with typecode |-
94 93 adantlr Could not format ( ( ( R e. CRing /\ M e. B ) /\ M e. ( MaxIdeal ` R ) ) -> { M } = ( V ` M ) ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. B ) /\ M e. ( MaxIdeal ` R ) ) -> { M } = ( V ` M ) ) with typecode |-
95 62 94 impbida Could not format ( ( R e. CRing /\ M e. B ) -> ( { M } = ( V ` M ) <-> M e. ( MaxIdeal ` R ) ) ) : No typesetting found for |- ( ( R e. CRing /\ M e. B ) -> ( { M } = ( V ` M ) <-> M e. ( MaxIdeal ` R ) ) ) with typecode |-