Step |
Hyp |
Ref |
Expression |
1 |
|
mxidlprm.1 |
|
2 |
|
crngring |
|
3 |
2
|
adantr |
Could not format ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> R e. Ring ) : No typesetting found for |- ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> R e. Ring ) with typecode |- |
4 |
|
eqid |
|
5 |
4
|
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 |- |
6 |
2 5
|
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 |- |
7 |
4
|
mxidlnr |
Could not format ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> M =/= ( Base ` R ) ) : No typesetting found for |- ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) -> M =/= ( Base ` R ) ) with typecode |- |
8 |
2 7
|
sylan |
Could not format ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> M =/= ( Base ` R ) ) : No typesetting found for |- ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> M =/= ( Base ` R ) ) with typecode |- |
9 |
|
simpllr |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( 1r ` R ) = ( u ( +g ` R ) k ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( 1r ` R ) = ( u ( +g ` R ) k ) ) with typecode |- |
10 |
|
simpr |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> k = ( a ( .r ` R ) x ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> k = ( a ( .r ` R ) x ) ) with typecode |- |
11 |
10
|
oveq2d |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( u ( +g ` R ) k ) = ( u ( +g ` R ) ( a ( .r ` R ) x ) ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( u ( +g ` R ) k ) = ( u ( +g ` R ) ( a ( .r ` R ) x ) ) ) with typecode |- |
12 |
9 11
|
eqtrd |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( 1r ` R ) = ( u ( +g ` R ) ( a ( .r ` R ) x ) ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( 1r ` R ) = ( u ( +g ` R ) ( a ( .r ` R ) x ) ) ) with typecode |- |
13 |
12
|
oveq1d |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( ( 1r ` R ) ( .r ` R ) y ) = ( ( u ( +g ` R ) ( a ( .r ` R ) x ) ) ( .r ` R ) y ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( ( 1r ` R ) ( .r ` R ) y ) = ( ( u ( +g ` R ) ( a ( .r ` R ) x ) ) ( .r ` R ) y ) ) with typecode |- |
14 |
3
|
ad4antr |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> R e. Ring ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> R e. Ring ) with typecode |- |
15 |
14
|
ad5antr |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> R e. Ring ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> R e. Ring ) with typecode |- |
16 |
|
simp-8r |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> y e. ( Base ` R ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> y e. ( Base ` R ) ) with typecode |- |
17 |
|
eqid |
|
18 |
|
eqid |
|
19 |
4 17 18
|
ringlidm |
|
20 |
15 16 19
|
syl2anc |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( ( 1r ` R ) ( .r ` R ) y ) = y ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( ( 1r ` R ) ( .r ` R ) y ) = y ) with typecode |- |
21 |
|
eqid |
|
22 |
4 21
|
lidlss |
|
23 |
6 22
|
syl |
Could not format ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> M C_ ( Base ` R ) ) : No typesetting found for |- ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> M C_ ( Base ` R ) ) with typecode |- |
24 |
23
|
ad4antr |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> M C_ ( Base ` R ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> M C_ ( Base ` R ) ) with typecode |- |
25 |
24
|
ad5antr |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> M C_ ( Base ` R ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> M C_ ( Base ` R ) ) with typecode |- |
26 |
|
simp-5r |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> u e. M ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> u e. M ) with typecode |- |
27 |
25 26
|
sseldd |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> u e. ( Base ` R ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> u e. ( Base ` R ) ) with typecode |- |
28 |
|
simplr |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> a e. ( Base ` R ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> a e. ( Base ` R ) ) with typecode |- |
29 |
|
simp-4r |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> x e. ( Base ` R ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> x e. ( Base ` R ) ) with typecode |- |
30 |
29
|
ad5antr |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> x e. ( Base ` R ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> x e. ( Base ` R ) ) with typecode |- |
31 |
4 17
|
ringcl |
|
32 |
15 28 30 31
|
syl3anc |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( a ( .r ` R ) x ) e. ( Base ` R ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( a ( .r ` R ) x ) e. ( Base ` R ) ) with typecode |- |
33 |
|
eqid |
|
34 |
4 33 17
|
ringdir |
|
35 |
15 27 32 16 34
|
syl13anc |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( ( u ( +g ` R ) ( a ( .r ` R ) x ) ) ( .r ` R ) y ) = ( ( u ( .r ` R ) y ) ( +g ` R ) ( ( a ( .r ` R ) x ) ( .r ` R ) y ) ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( ( u ( +g ` R ) ( a ( .r ` R ) x ) ) ( .r ` R ) y ) = ( ( u ( .r ` R ) y ) ( +g ` R ) ( ( a ( .r ` R ) x ) ( .r ` R ) y ) ) ) with typecode |- |
36 |
13 20 35
|
3eqtr3d |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> y = ( ( u ( .r ` R ) y ) ( +g ` R ) ( ( a ( .r ` R ) x ) ( .r ` R ) y ) ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> y = ( ( u ( .r ` R ) y ) ( +g ` R ) ( ( a ( .r ` R ) x ) ( .r ` R ) y ) ) ) with typecode |- |
37 |
|
simp-5r |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> M e. ( MaxIdeal ` R ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> M e. ( MaxIdeal ` R ) ) with typecode |- |
38 |
14 37 5
|
syl2anc |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> M e. ( LIdeal ` R ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> M e. ( LIdeal ` R ) ) with typecode |- |
39 |
38
|
ad5antr |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> M e. ( LIdeal ` R ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> M e. ( LIdeal ` R ) ) with typecode |- |
40 |
|
simp-10l |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> R e. CRing ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> R e. CRing ) with typecode |- |
41 |
4 17
|
crngcom |
|
42 |
40 16 27 41
|
syl3anc |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( y ( .r ` R ) u ) = ( u ( .r ` R ) y ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( y ( .r ` R ) u ) = ( u ( .r ` R ) y ) ) with typecode |- |
43 |
21 4 17
|
lidlmcl |
|
44 |
15 39 16 26 43
|
syl22anc |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( y ( .r ` R ) u ) e. M ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( y ( .r ` R ) u ) e. M ) with typecode |- |
45 |
42 44
|
eqeltrrd |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( u ( .r ` R ) y ) e. M ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( u ( .r ` R ) y ) e. M ) with typecode |- |
46 |
4 17
|
ringass |
|
47 |
15 28 30 16 46
|
syl13anc |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( ( a ( .r ` R ) x ) ( .r ` R ) y ) = ( a ( .r ` R ) ( x ( .r ` R ) y ) ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( ( a ( .r ` R ) x ) ( .r ` R ) y ) = ( a ( .r ` R ) ( x ( .r ` R ) y ) ) ) with typecode |- |
48 |
|
simp-7r |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( x ( .r ` R ) y ) e. M ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( x ( .r ` R ) y ) e. M ) with typecode |- |
49 |
21 4 17
|
lidlmcl |
|
50 |
15 39 28 48 49
|
syl22anc |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( a ( .r ` R ) ( x ( .r ` R ) y ) ) e. M ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( a ( .r ` R ) ( x ( .r ` R ) y ) ) e. M ) with typecode |- |
51 |
47 50
|
eqeltrd |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( ( a ( .r ` R ) x ) ( .r ` R ) y ) e. M ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( ( a ( .r ` R ) x ) ( .r ` R ) y ) e. M ) with typecode |- |
52 |
21 33
|
lidlacl |
|
53 |
15 39 45 51 52
|
syl22anc |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( ( u ( .r ` R ) y ) ( +g ` R ) ( ( a ( .r ` R ) x ) ( .r ` R ) y ) ) e. M ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> ( ( u ( .r ` R ) y ) ( +g ` R ) ( ( a ( .r ` R ) x ) ( .r ` R ) y ) ) e. M ) with typecode |- |
54 |
36 53
|
eqeltrd |
Could not format ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> y e. M ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) /\ a e. ( Base ` R ) ) /\ k = ( a ( .r ` R ) x ) ) -> y e. M ) with typecode |- |
55 |
|
simplr |
Could not format ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) -> k e. ( ( Base ` R ) .X. { x } ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) -> k e. ( ( Base ` R ) .X. { x } ) ) with typecode |- |
56 |
|
eqid |
|
57 |
56 4
|
mgpbas |
|
58 |
56 17
|
mgpplusg |
|
59 |
|
fvexd |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( mulGrp ` R ) e. _V ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( mulGrp ` R ) e. _V ) with typecode |- |
60 |
|
ssidd |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( Base ` R ) C_ ( Base ` R ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( Base ` R ) C_ ( Base ` R ) ) with typecode |- |
61 |
57 58 1 59 60 29
|
elgrplsmsn |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( k e. ( ( Base ` R ) .X. { x } ) <-> E. a e. ( Base ` R ) k = ( a ( .r ` R ) x ) ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( k e. ( ( Base ` R ) .X. { x } ) <-> E. a e. ( Base ` R ) k = ( a ( .r ` R ) x ) ) ) with typecode |- |
62 |
61
|
ad3antrrr |
Could not format ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) -> ( k e. ( ( Base ` R ) .X. { x } ) <-> E. a e. ( Base ` R ) k = ( a ( .r ` R ) x ) ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) -> ( k e. ( ( Base ` R ) .X. { x } ) <-> E. a e. ( Base ` R ) k = ( a ( .r ` R ) x ) ) ) with typecode |- |
63 |
55 62
|
mpbid |
Could not format ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) -> E. a e. ( Base ` R ) k = ( a ( .r ` R ) x ) ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) -> E. a e. ( Base ` R ) k = ( a ( .r ` R ) x ) ) with typecode |- |
64 |
54 63
|
r19.29a |
Could not format ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) -> y e. M ) : No typesetting found for |- ( ( ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ u e. M ) /\ k e. ( ( Base ` R ) .X. { x } ) ) /\ ( 1r ` R ) = ( u ( +g ` R ) k ) ) -> y e. M ) with typecode |- |
65 |
4 18
|
ringidcl |
|
66 |
14 65
|
syl |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( 1r ` R ) e. ( Base ` R ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( 1r ` R ) e. ( Base ` R ) ) with typecode |- |
67 |
|
eqid |
|
68 |
|
eqid |
|
69 |
|
eqid |
|
70 |
69 21
|
lpiss |
|
71 |
14 70
|
syl |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( LPIdeal ` R ) C_ ( LIdeal ` R ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( LPIdeal ` R ) C_ ( LIdeal ` R ) ) with typecode |- |
72 |
4 56 1 68 14 29
|
lsmsnidl |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( ( Base ` R ) .X. { x } ) e. ( LPIdeal ` R ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( ( Base ` R ) .X. { x } ) e. ( LPIdeal ` R ) ) with typecode |- |
73 |
71 72
|
sseldd |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( ( Base ` R ) .X. { x } ) e. ( LIdeal ` R ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( ( Base ` R ) .X. { x } ) e. ( LIdeal ` R ) ) with typecode |- |
74 |
4 67 68 14 38 73
|
lsmidl |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) e. ( LIdeal ` R ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) e. ( LIdeal ` R ) ) with typecode |- |
75 |
|
rlmlmod |
|
76 |
14 75
|
syl |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( ringLMod ` R ) e. LMod ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( ringLMod ` R ) e. LMod ) with typecode |- |
77 |
|
rlmbas |
|
78 |
|
rspval |
|
79 |
77 78
|
lspssid |
|
80 |
76 24 79
|
syl2anc |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> M C_ ( ( RSpan ` R ) ` M ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> M C_ ( ( RSpan ` R ) ` M ) ) with typecode |- |
81 |
29
|
snssd |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> { x } C_ ( Base ` R ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> { x } C_ ( Base ` R ) ) with typecode |- |
82 |
4 56 1 14 60 81
|
ringlsmss |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( ( Base ` R ) .X. { x } ) C_ ( Base ` R ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( ( Base ` R ) .X. { x } ) C_ ( Base ` R ) ) with typecode |- |
83 |
24 82
|
unssd |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( M u. ( ( Base ` R ) .X. { x } ) ) C_ ( Base ` R ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( M u. ( ( Base ` R ) .X. { x } ) ) C_ ( Base ` R ) ) with typecode |- |
84 |
|
ssun1 |
|
85 |
84
|
a1i |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> M C_ ( M u. ( ( Base ` R ) .X. { x } ) ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> M C_ ( M u. ( ( Base ` R ) .X. { x } ) ) ) with typecode |- |
86 |
77 78
|
lspss |
|
87 |
76 83 85 86
|
syl3anc |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( ( RSpan ` R ) ` M ) C_ ( ( RSpan ` R ) ` ( M u. ( ( Base ` R ) .X. { x } ) ) ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( ( RSpan ` R ) ` M ) C_ ( ( RSpan ` R ) ` ( M u. ( ( Base ` R ) .X. { x } ) ) ) ) with typecode |- |
88 |
80 87
|
sstrd |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> M C_ ( ( RSpan ` R ) ` ( M u. ( ( Base ` R ) .X. { x } ) ) ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> M C_ ( ( RSpan ` R ) ` ( M u. ( ( Base ` R ) .X. { x } ) ) ) ) with typecode |- |
89 |
4 67 68 14 38 73
|
lsmidllsp |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) = ( ( RSpan ` R ) ` ( M u. ( ( Base ` R ) .X. { x } ) ) ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) = ( ( RSpan ` R ) ` ( M u. ( ( Base ` R ) .X. { x } ) ) ) ) with typecode |- |
90 |
88 89
|
sseqtrrd |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> M C_ ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> M C_ ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) ) with typecode |- |
91 |
4
|
mxidlmax |
Could not format ( ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) /\ ( ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) e. ( LIdeal ` R ) /\ M C_ ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) ) ) -> ( ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) = M \/ ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) = ( Base ` R ) ) ) : No typesetting found for |- ( ( ( R e. Ring /\ M e. ( MaxIdeal ` R ) ) /\ ( ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) e. ( LIdeal ` R ) /\ M C_ ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) ) ) -> ( ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) = M \/ ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) = ( Base ` R ) ) ) with typecode |- |
92 |
14 37 74 90 91
|
syl22anc |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) = M \/ ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) = ( Base ` R ) ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) = M \/ ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) = ( Base ` R ) ) ) with typecode |- |
93 |
|
eqid |
|
94 |
21 93
|
lidl0cl |
|
95 |
14 38 94
|
syl2anc |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( 0g ` R ) e. M ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( 0g ` R ) e. M ) with typecode |- |
96 |
|
oveq1 |
|
97 |
96
|
eqeq2d |
|
98 |
97
|
rexbidv |
|
99 |
98
|
adantl |
Could not format ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ a = ( 0g ` R ) ) -> ( E. b e. ( ( Base ` R ) .X. { x } ) x = ( a ( +g ` R ) b ) <-> E. b e. ( ( Base ` R ) .X. { x } ) x = ( ( 0g ` R ) ( +g ` R ) b ) ) ) : No typesetting found for |- ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ a = ( 0g ` R ) ) -> ( E. b e. ( ( Base ` R ) .X. { x } ) x = ( a ( +g ` R ) b ) <-> E. b e. ( ( Base ` R ) .X. { x } ) x = ( ( 0g ` R ) ( +g ` R ) b ) ) ) with typecode |- |
100 |
|
oveq1 |
|
101 |
100
|
eqeq2d |
|
102 |
101
|
adantl |
Could not format ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ a = ( 1r ` R ) ) -> ( x = ( a ( .r ` R ) x ) <-> x = ( ( 1r ` R ) ( .r ` R ) x ) ) ) : No typesetting found for |- ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ a = ( 1r ` R ) ) -> ( x = ( a ( .r ` R ) x ) <-> x = ( ( 1r ` R ) ( .r ` R ) x ) ) ) with typecode |- |
103 |
4 17 18
|
ringlidm |
|
104 |
14 29 103
|
syl2anc |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( ( 1r ` R ) ( .r ` R ) x ) = x ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( ( 1r ` R ) ( .r ` R ) x ) = x ) with typecode |- |
105 |
104
|
eqcomd |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> x = ( ( 1r ` R ) ( .r ` R ) x ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> x = ( ( 1r ` R ) ( .r ` R ) x ) ) with typecode |- |
106 |
66 102 105
|
rspcedvd |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> E. a e. ( Base ` R ) x = ( a ( .r ` R ) x ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> E. a e. ( Base ` R ) x = ( a ( .r ` R ) x ) ) with typecode |- |
107 |
57 58 1 59 60 29
|
elgrplsmsn |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( x e. ( ( Base ` R ) .X. { x } ) <-> E. a e. ( Base ` R ) x = ( a ( .r ` R ) x ) ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( x e. ( ( Base ` R ) .X. { x } ) <-> E. a e. ( Base ` R ) x = ( a ( .r ` R ) x ) ) ) with typecode |- |
108 |
106 107
|
mpbird |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> x e. ( ( Base ` R ) .X. { x } ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> x e. ( ( Base ` R ) .X. { x } ) ) with typecode |- |
109 |
|
oveq2 |
|
110 |
109
|
eqeq2d |
|
111 |
110
|
adantl |
Could not format ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ b = x ) -> ( x = ( ( 0g ` R ) ( +g ` R ) b ) <-> x = ( ( 0g ` R ) ( +g ` R ) x ) ) ) : No typesetting found for |- ( ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) /\ b = x ) -> ( x = ( ( 0g ` R ) ( +g ` R ) b ) <-> x = ( ( 0g ` R ) ( +g ` R ) x ) ) ) with typecode |- |
112 |
|
ringgrp |
|
113 |
14 112
|
syl |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> R e. Grp ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> R e. Grp ) with typecode |- |
114 |
4 33 93
|
grplid |
|
115 |
113 29 114
|
syl2anc |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( ( 0g ` R ) ( +g ` R ) x ) = x ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( ( 0g ` R ) ( +g ` R ) x ) = x ) with typecode |- |
116 |
115
|
eqcomd |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> x = ( ( 0g ` R ) ( +g ` R ) x ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> x = ( ( 0g ` R ) ( +g ` R ) x ) ) with typecode |- |
117 |
108 111 116
|
rspcedvd |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> E. b e. ( ( Base ` R ) .X. { x } ) x = ( ( 0g ` R ) ( +g ` R ) b ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> E. b e. ( ( Base ` R ) .X. { x } ) x = ( ( 0g ` R ) ( +g ` R ) b ) ) with typecode |- |
118 |
95 99 117
|
rspcedvd |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> E. a e. M E. b e. ( ( Base ` R ) .X. { x } ) x = ( a ( +g ` R ) b ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> E. a e. M E. b e. ( ( Base ` R ) .X. { x } ) x = ( a ( +g ` R ) b ) ) with typecode |- |
119 |
|
simp-5l |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> R e. CRing ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> R e. CRing ) with typecode |- |
120 |
4 33 67
|
lsmelvalx |
|
121 |
119 24 82 120
|
syl3anc |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( x e. ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) <-> E. a e. M E. b e. ( ( Base ` R ) .X. { x } ) x = ( a ( +g ` R ) b ) ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( x e. ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) <-> E. a e. M E. b e. ( ( Base ` R ) .X. { x } ) x = ( a ( +g ` R ) b ) ) ) with typecode |- |
122 |
118 121
|
mpbird |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> x e. ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> x e. ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) ) with typecode |- |
123 |
|
simpr |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> -. x e. M ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> -. x e. M ) with typecode |- |
124 |
|
nelne1 |
|
125 |
122 123 124
|
syl2anc |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) =/= M ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) =/= M ) with typecode |- |
126 |
125
|
neneqd |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> -. ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) = M ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> -. ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) = M ) with typecode |- |
127 |
92 126
|
orcnd |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) = ( Base ` R ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) = ( Base ` R ) ) with typecode |- |
128 |
66 127
|
eleqtrrd |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( 1r ` R ) e. ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( 1r ` R ) e. ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) ) with typecode |- |
129 |
4 33 67
|
lsmelvalx |
|
130 |
119 24 82 129
|
syl3anc |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( ( 1r ` R ) e. ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) <-> E. u e. M E. k e. ( ( Base ` R ) .X. { x } ) ( 1r ` R ) = ( u ( +g ` R ) k ) ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> ( ( 1r ` R ) e. ( M ( LSSum ` R ) ( ( Base ` R ) .X. { x } ) ) <-> E. u e. M E. k e. ( ( Base ` R ) .X. { x } ) ( 1r ` R ) = ( u ( +g ` R ) k ) ) ) with typecode |- |
131 |
128 130
|
mpbid |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> E. u e. M E. k e. ( ( Base ` R ) .X. { x } ) ( 1r ` R ) = ( u ( +g ` R ) k ) ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> E. u e. M E. k e. ( ( Base ` R ) .X. { x } ) ( 1r ` R ) = ( u ( +g ` R ) k ) ) with typecode |- |
132 |
64 131
|
r19.29vva |
Could not format ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> y e. M ) : No typesetting found for |- ( ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) /\ -. x e. M ) -> y e. M ) with typecode |- |
133 |
132
|
ex |
Could not format ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) -> ( -. x e. M -> y e. M ) ) : No typesetting found for |- ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) -> ( -. x e. M -> y e. M ) ) with typecode |- |
134 |
133
|
orrd |
Could not format ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) -> ( x e. M \/ y e. M ) ) : No typesetting found for |- ( ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) /\ ( x ( .r ` R ) y ) e. M ) -> ( x e. M \/ y e. M ) ) with typecode |- |
135 |
134
|
ex |
Could not format ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( ( x ( .r ` R ) y ) e. M -> ( x e. M \/ y e. M ) ) ) : No typesetting found for |- ( ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( ( x ( .r ` R ) y ) e. M -> ( x e. M \/ y e. M ) ) ) with typecode |- |
136 |
135
|
anasss |
Could not format ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( ( x ( .r ` R ) y ) e. M -> ( x e. M \/ y e. M ) ) ) : No typesetting found for |- ( ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( ( x ( .r ` R ) y ) e. M -> ( x e. M \/ y e. M ) ) ) with typecode |- |
137 |
136
|
ralrimivva |
Could not format ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) e. M -> ( x e. M \/ y e. M ) ) ) : No typesetting found for |- ( ( R e. CRing /\ M e. ( MaxIdeal ` R ) ) -> A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) e. M -> ( x e. M \/ y e. M ) ) ) with typecode |- |
138 |
4 17
|
prmidl2 |
Could not format ( ( ( R e. Ring /\ M e. ( LIdeal ` R ) ) /\ ( M =/= ( Base ` R ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) e. M -> ( x e. M \/ y e. M ) ) ) ) -> M e. ( PrmIdeal ` R ) ) : No typesetting found for |- ( ( ( R e. Ring /\ M e. ( LIdeal ` R ) ) /\ ( M =/= ( Base ` R ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) e. M -> ( x e. M \/ y e. M ) ) ) ) -> M e. ( PrmIdeal ` R ) ) with typecode |- |
139 |
3 6 8 137 138
|
syl22anc |
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 |- |