Step |
Hyp |
Ref |
Expression |
1 |
|
prmidl0.1 |
|
2 |
|
df-3an |
|
3 |
|
crngring |
|
4 |
3
|
ad2antrr |
|
5 |
|
0ringnnzr |
|
6 |
5
|
biimpar |
|
7 |
4 6
|
sylancom |
|
8 |
|
eqid |
|
9 |
8 1
|
0ring |
|
10 |
4 7 9
|
syl2anc |
|
11 |
10
|
eqcomd |
|
12 |
11
|
ex |
|
13 |
12
|
necon1ad |
|
14 |
13
|
impr |
|
15 |
|
nzrring |
|
16 |
|
eqid |
|
17 |
16 1
|
lidl0 |
|
18 |
15 17
|
syl |
|
19 |
1
|
fvexi |
|
20 |
|
hashsng |
|
21 |
19 20
|
ax-mp |
|
22 |
|
1re |
|
23 |
21 22
|
eqeltri |
|
24 |
23
|
a1i |
|
25 |
8
|
isnzr2hash |
|
26 |
25
|
simprbi |
|
27 |
21 26
|
eqbrtrid |
|
28 |
24 27
|
ltned |
|
29 |
|
fveq2 |
|
30 |
29
|
necon3i |
|
31 |
28 30
|
syl |
|
32 |
18 31
|
jca |
|
33 |
32
|
adantl |
|
34 |
14 33
|
impbida |
|
35 |
19
|
elsn2 |
|
36 |
|
velsn |
|
37 |
|
velsn |
|
38 |
36 37
|
orbi12i |
|
39 |
35 38
|
imbi12i |
|
40 |
39
|
2ralbii |
|
41 |
40
|
a1i |
|
42 |
34 41
|
anbi12d |
|
43 |
2 42
|
syl5bb |
|
44 |
43
|
pm5.32i |
|
45 |
|
eqid |
|
46 |
8 45
|
isprmidlc |
Could not format ( R e. CRing -> ( { .0. } e. ( PrmIdeal ` R ) <-> ( { .0. } e. ( LIdeal ` R ) /\ { .0. } =/= ( Base ` R ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) e. { .0. } -> ( x e. { .0. } \/ y e. { .0. } ) ) ) ) ) : No typesetting found for |- ( R e. CRing -> ( { .0. } e. ( PrmIdeal ` R ) <-> ( { .0. } e. ( LIdeal ` R ) /\ { .0. } =/= ( Base ` R ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) e. { .0. } -> ( x e. { .0. } \/ y e. { .0. } ) ) ) ) ) with typecode |- |
47 |
46
|
pm5.32i |
Could not format ( ( R e. CRing /\ { .0. } e. ( PrmIdeal ` R ) ) <-> ( R e. CRing /\ ( { .0. } e. ( LIdeal ` R ) /\ { .0. } =/= ( Base ` R ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) e. { .0. } -> ( x e. { .0. } \/ y e. { .0. } ) ) ) ) ) : No typesetting found for |- ( ( R e. CRing /\ { .0. } e. ( PrmIdeal ` R ) ) <-> ( R e. CRing /\ ( { .0. } e. ( LIdeal ` R ) /\ { .0. } =/= ( Base ` R ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) e. { .0. } -> ( x e. { .0. } \/ y e. { .0. } ) ) ) ) ) with typecode |- |
48 |
|
df-idom |
|
49 |
48
|
eleq2i |
|
50 |
|
elin |
|
51 |
8 45 1
|
isdomn |
|
52 |
51
|
anbi2i |
|
53 |
49 50 52
|
3bitri |
|
54 |
44 47 53
|
3bitr4i |
Could not format ( ( R e. CRing /\ { .0. } e. ( PrmIdeal ` R ) ) <-> R e. IDomn ) : No typesetting found for |- ( ( R e. CRing /\ { .0. } e. ( PrmIdeal ` R ) ) <-> R e. IDomn ) with typecode |- |