Step |
Hyp |
Ref |
Expression |
1 |
|
ssmxidl.1 |
|- B = ( Base ` R ) |
2 |
|
neeq1 |
|- ( p = I -> ( p =/= B <-> I =/= B ) ) |
3 |
|
sseq2 |
|- ( p = I -> ( I C_ p <-> I C_ I ) ) |
4 |
2 3
|
anbi12d |
|- ( p = I -> ( ( p =/= B /\ I C_ p ) <-> ( I =/= B /\ I C_ I ) ) ) |
5 |
|
simp2 |
|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) -> I e. ( LIdeal ` R ) ) |
6 |
|
simp3 |
|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) -> I =/= B ) |
7 |
|
ssidd |
|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) -> I C_ I ) |
8 |
6 7
|
jca |
|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) -> ( I =/= B /\ I C_ I ) ) |
9 |
4 5 8
|
elrabd |
|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) -> I e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } ) |
10 |
9
|
ne0d |
|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) -> { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } =/= (/) ) |
11 |
|
eqid |
|- { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } = { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } |
12 |
|
simpl1 |
|- ( ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) /\ ( z C_ { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } /\ z =/= (/) /\ [C.] Or z ) ) -> R e. Ring ) |
13 |
|
simpl2 |
|- ( ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) /\ ( z C_ { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } /\ z =/= (/) /\ [C.] Or z ) ) -> I e. ( LIdeal ` R ) ) |
14 |
|
simpl3 |
|- ( ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) /\ ( z C_ { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } /\ z =/= (/) /\ [C.] Or z ) ) -> I =/= B ) |
15 |
|
simpr1 |
|- ( ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) /\ ( z C_ { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } /\ z =/= (/) /\ [C.] Or z ) ) -> z C_ { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } ) |
16 |
|
simpr2 |
|- ( ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) /\ ( z C_ { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } /\ z =/= (/) /\ [C.] Or z ) ) -> z =/= (/) ) |
17 |
|
simpr3 |
|- ( ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) /\ ( z C_ { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } /\ z =/= (/) /\ [C.] Or z ) ) -> [C.] Or z ) |
18 |
1 11 12 13 14 15 16 17
|
ssmxidllem |
|- ( ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) /\ ( z C_ { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } /\ z =/= (/) /\ [C.] Or z ) ) -> U. z e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } ) |
19 |
18
|
ex |
|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) -> ( ( z C_ { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } /\ z =/= (/) /\ [C.] Or z ) -> U. z e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } ) ) |
20 |
19
|
alrimiv |
|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) -> A. z ( ( z C_ { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } /\ z =/= (/) /\ [C.] Or z ) -> U. z e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } ) ) |
21 |
|
fvex |
|- ( LIdeal ` R ) e. _V |
22 |
21
|
rabex |
|- { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } e. _V |
23 |
22
|
zornn0 |
|- ( ( { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } =/= (/) /\ A. z ( ( z C_ { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } /\ z =/= (/) /\ [C.] Or z ) -> U. z e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } ) ) -> 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 ) |
24 |
10 20 23
|
syl2anc |
|- ( ( 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 ) |
25 |
|
neeq1 |
|- ( p = m -> ( p =/= B <-> m =/= B ) ) |
26 |
|
sseq2 |
|- ( p = m -> ( I C_ p <-> I C_ m ) ) |
27 |
25 26
|
anbi12d |
|- ( p = m -> ( ( p =/= B /\ I C_ p ) <-> ( m =/= B /\ I C_ m ) ) ) |
28 |
27
|
elrab |
|- ( m e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } <-> ( m e. ( LIdeal ` R ) /\ ( m =/= B /\ I C_ m ) ) ) |
29 |
28
|
anbi2i |
|- ( ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) /\ m e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } ) <-> ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) /\ ( m e. ( LIdeal ` R ) /\ ( m =/= B /\ I C_ m ) ) ) ) |
30 |
|
simpll1 |
|- ( ( ( ( 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 ) -> R e. Ring ) |
31 |
|
simplrl |
|- ( ( ( ( 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. ( LIdeal ` R ) ) |
32 |
|
simplr |
|- ( ( ( ( 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. ( LIdeal ` R ) /\ ( m =/= B /\ I C_ m ) ) ) |
33 |
32
|
simprld |
|- ( ( ( ( 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 =/= B ) |
34 |
|
psseq2 |
|- ( j = k -> ( m C. j <-> m C. k ) ) |
35 |
34
|
notbid |
|- ( j = k -> ( -. m C. j <-> -. m C. k ) ) |
36 |
|
simp-4r |
|- ( ( ( ( ( ( ( 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 ) /\ k e. ( LIdeal ` R ) ) /\ m C_ k ) /\ -. k = B ) -> A. j e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } -. m C. j ) |
37 |
|
neeq1 |
|- ( p = k -> ( p =/= B <-> k =/= B ) ) |
38 |
|
sseq2 |
|- ( p = k -> ( I C_ p <-> I C_ k ) ) |
39 |
37 38
|
anbi12d |
|- ( p = k -> ( ( p =/= B /\ I C_ p ) <-> ( k =/= B /\ I C_ k ) ) ) |
40 |
|
simpllr |
|- ( ( ( ( ( ( ( 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 ) /\ k e. ( LIdeal ` R ) ) /\ m C_ k ) /\ -. k = B ) -> k e. ( LIdeal ` R ) ) |
41 |
|
simpr |
|- ( ( ( ( ( ( ( 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 ) /\ k e. ( LIdeal ` R ) ) /\ m C_ k ) /\ -. k = B ) -> -. k = B ) |
42 |
41
|
neqned |
|- ( ( ( ( ( ( ( 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 ) /\ k e. ( LIdeal ` R ) ) /\ m C_ k ) /\ -. k = B ) -> k =/= B ) |
43 |
|
simp-5r |
|- ( ( ( ( ( ( ( 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 ) /\ k e. ( LIdeal ` R ) ) /\ m C_ k ) /\ -. k = B ) -> ( m e. ( LIdeal ` R ) /\ ( m =/= B /\ I C_ m ) ) ) |
44 |
43
|
simprrd |
|- ( ( ( ( ( ( ( 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 ) /\ k e. ( LIdeal ` R ) ) /\ m C_ k ) /\ -. k = B ) -> I C_ m ) |
45 |
|
simplr |
|- ( ( ( ( ( ( ( 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 ) /\ k e. ( LIdeal ` R ) ) /\ m C_ k ) /\ -. k = B ) -> m C_ k ) |
46 |
44 45
|
sstrd |
|- ( ( ( ( ( ( ( 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 ) /\ k e. ( LIdeal ` R ) ) /\ m C_ k ) /\ -. k = B ) -> I C_ k ) |
47 |
42 46
|
jca |
|- ( ( ( ( ( ( ( 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 ) /\ k e. ( LIdeal ` R ) ) /\ m C_ k ) /\ -. k = B ) -> ( k =/= B /\ I C_ k ) ) |
48 |
39 40 47
|
elrabd |
|- ( ( ( ( ( ( ( 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 ) /\ k e. ( LIdeal ` R ) ) /\ m C_ k ) /\ -. k = B ) -> k e. { p e. ( LIdeal ` R ) | ( p =/= B /\ I C_ p ) } ) |
49 |
35 36 48
|
rspcdva |
|- ( ( ( ( ( ( ( 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 ) /\ k e. ( LIdeal ` R ) ) /\ m C_ k ) /\ -. k = B ) -> -. m C. k ) |
50 |
|
npss |
|- ( -. m C. k <-> ( m C_ k -> m = k ) ) |
51 |
50
|
biimpi |
|- ( -. m C. k -> ( m C_ k -> m = k ) ) |
52 |
49 45 51
|
sylc |
|- ( ( ( ( ( ( ( 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 ) /\ k e. ( LIdeal ` R ) ) /\ m C_ k ) /\ -. k = B ) -> m = k ) |
53 |
52
|
equcomd |
|- ( ( ( ( ( ( ( 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 ) /\ k e. ( LIdeal ` R ) ) /\ m C_ k ) /\ -. k = B ) -> k = m ) |
54 |
53
|
ex |
|- ( ( ( ( ( ( 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 ) /\ k e. ( LIdeal ` R ) ) /\ m C_ k ) -> ( -. k = B -> k = m ) ) |
55 |
54
|
orrd |
|- ( ( ( ( ( ( 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 ) /\ k e. ( LIdeal ` R ) ) /\ m C_ k ) -> ( k = B \/ k = m ) ) |
56 |
55
|
orcomd |
|- ( ( ( ( ( ( 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 ) /\ k e. ( LIdeal ` R ) ) /\ m C_ k ) -> ( k = m \/ k = B ) ) |
57 |
56
|
ex |
|- ( ( ( ( ( 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 ) /\ k e. ( LIdeal ` R ) ) -> ( m C_ k -> ( k = m \/ k = B ) ) ) |
58 |
57
|
ralrimiva |
|- ( ( ( ( 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 ) -> A. k e. ( LIdeal ` R ) ( m C_ k -> ( k = m \/ k = B ) ) ) |
59 |
1
|
ismxidl |
|- ( 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 ) ) ) ) ) |
60 |
59
|
biimpar |
|- ( ( 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 ) ) |
61 |
30 31 33 58 60
|
syl13anc |
|- ( ( ( ( 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 ) ) |
62 |
32
|
simprrd |
|- ( ( ( ( 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 ) -> I C_ m ) |
63 |
61 62
|
jca |
|- ( ( ( ( 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 ) ) |
64 |
29 63
|
sylanb |
|- ( ( ( ( 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 ) ) |
65 |
64
|
expl |
|- ( ( 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 ) ) ) |
66 |
65
|
reximdv2 |
|- ( ( 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 ) ) |
67 |
24 66
|
mpd |
|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) /\ I =/= B ) -> E. m e. ( MaxIdeal ` R ) I C_ m ) |