Step |
Hyp |
Ref |
Expression |
1 |
|
prmidl0.1 |
⊢ 0 = ( 0g ‘ 𝑅 ) |
2 |
|
df-3an |
⊢ ( ( { 0 } ∈ ( LIdeal ‘ 𝑅 ) ∧ { 0 } ≠ ( Base ‘ 𝑅 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ { 0 } → ( 𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 } ) ) ) ↔ ( ( { 0 } ∈ ( LIdeal ‘ 𝑅 ) ∧ { 0 } ≠ ( Base ‘ 𝑅 ) ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ { 0 } → ( 𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 } ) ) ) ) |
3 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
4 |
3
|
ad2antrr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ { 0 } ∈ ( LIdeal ‘ 𝑅 ) ) ∧ ¬ 𝑅 ∈ NzRing ) → 𝑅 ∈ Ring ) |
5 |
|
0ringnnzr |
⊢ ( 𝑅 ∈ Ring → ( ( ♯ ‘ ( Base ‘ 𝑅 ) ) = 1 ↔ ¬ 𝑅 ∈ NzRing ) ) |
6 |
5
|
biimpar |
⊢ ( ( 𝑅 ∈ Ring ∧ ¬ 𝑅 ∈ NzRing ) → ( ♯ ‘ ( Base ‘ 𝑅 ) ) = 1 ) |
7 |
4 6
|
sylancom |
⊢ ( ( ( 𝑅 ∈ CRing ∧ { 0 } ∈ ( LIdeal ‘ 𝑅 ) ) ∧ ¬ 𝑅 ∈ NzRing ) → ( ♯ ‘ ( Base ‘ 𝑅 ) ) = 1 ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
9 |
8 1
|
0ring |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ ( Base ‘ 𝑅 ) ) = 1 ) → ( Base ‘ 𝑅 ) = { 0 } ) |
10 |
4 7 9
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ CRing ∧ { 0 } ∈ ( LIdeal ‘ 𝑅 ) ) ∧ ¬ 𝑅 ∈ NzRing ) → ( Base ‘ 𝑅 ) = { 0 } ) |
11 |
10
|
eqcomd |
⊢ ( ( ( 𝑅 ∈ CRing ∧ { 0 } ∈ ( LIdeal ‘ 𝑅 ) ) ∧ ¬ 𝑅 ∈ NzRing ) → { 0 } = ( Base ‘ 𝑅 ) ) |
12 |
11
|
ex |
⊢ ( ( 𝑅 ∈ CRing ∧ { 0 } ∈ ( LIdeal ‘ 𝑅 ) ) → ( ¬ 𝑅 ∈ NzRing → { 0 } = ( Base ‘ 𝑅 ) ) ) |
13 |
12
|
necon1ad |
⊢ ( ( 𝑅 ∈ CRing ∧ { 0 } ∈ ( LIdeal ‘ 𝑅 ) ) → ( { 0 } ≠ ( Base ‘ 𝑅 ) → 𝑅 ∈ NzRing ) ) |
14 |
13
|
impr |
⊢ ( ( 𝑅 ∈ CRing ∧ ( { 0 } ∈ ( LIdeal ‘ 𝑅 ) ∧ { 0 } ≠ ( Base ‘ 𝑅 ) ) ) → 𝑅 ∈ NzRing ) |
15 |
|
nzrring |
⊢ ( 𝑅 ∈ NzRing → 𝑅 ∈ Ring ) |
16 |
|
eqid |
⊢ ( LIdeal ‘ 𝑅 ) = ( LIdeal ‘ 𝑅 ) |
17 |
16 1
|
lidl0 |
⊢ ( 𝑅 ∈ Ring → { 0 } ∈ ( LIdeal ‘ 𝑅 ) ) |
18 |
15 17
|
syl |
⊢ ( 𝑅 ∈ NzRing → { 0 } ∈ ( LIdeal ‘ 𝑅 ) ) |
19 |
1
|
fvexi |
⊢ 0 ∈ V |
20 |
|
hashsng |
⊢ ( 0 ∈ V → ( ♯ ‘ { 0 } ) = 1 ) |
21 |
19 20
|
ax-mp |
⊢ ( ♯ ‘ { 0 } ) = 1 |
22 |
|
1re |
⊢ 1 ∈ ℝ |
23 |
21 22
|
eqeltri |
⊢ ( ♯ ‘ { 0 } ) ∈ ℝ |
24 |
23
|
a1i |
⊢ ( 𝑅 ∈ NzRing → ( ♯ ‘ { 0 } ) ∈ ℝ ) |
25 |
8
|
isnzr2hash |
⊢ ( 𝑅 ∈ NzRing ↔ ( 𝑅 ∈ Ring ∧ 1 < ( ♯ ‘ ( Base ‘ 𝑅 ) ) ) ) |
26 |
25
|
simprbi |
⊢ ( 𝑅 ∈ NzRing → 1 < ( ♯ ‘ ( Base ‘ 𝑅 ) ) ) |
27 |
21 26
|
eqbrtrid |
⊢ ( 𝑅 ∈ NzRing → ( ♯ ‘ { 0 } ) < ( ♯ ‘ ( Base ‘ 𝑅 ) ) ) |
28 |
24 27
|
ltned |
⊢ ( 𝑅 ∈ NzRing → ( ♯ ‘ { 0 } ) ≠ ( ♯ ‘ ( Base ‘ 𝑅 ) ) ) |
29 |
|
fveq2 |
⊢ ( { 0 } = ( Base ‘ 𝑅 ) → ( ♯ ‘ { 0 } ) = ( ♯ ‘ ( Base ‘ 𝑅 ) ) ) |
30 |
29
|
necon3i |
⊢ ( ( ♯ ‘ { 0 } ) ≠ ( ♯ ‘ ( Base ‘ 𝑅 ) ) → { 0 } ≠ ( Base ‘ 𝑅 ) ) |
31 |
28 30
|
syl |
⊢ ( 𝑅 ∈ NzRing → { 0 } ≠ ( Base ‘ 𝑅 ) ) |
32 |
18 31
|
jca |
⊢ ( 𝑅 ∈ NzRing → ( { 0 } ∈ ( LIdeal ‘ 𝑅 ) ∧ { 0 } ≠ ( Base ‘ 𝑅 ) ) ) |
33 |
32
|
adantl |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑅 ∈ NzRing ) → ( { 0 } ∈ ( LIdeal ‘ 𝑅 ) ∧ { 0 } ≠ ( Base ‘ 𝑅 ) ) ) |
34 |
14 33
|
impbida |
⊢ ( 𝑅 ∈ CRing → ( ( { 0 } ∈ ( LIdeal ‘ 𝑅 ) ∧ { 0 } ≠ ( Base ‘ 𝑅 ) ) ↔ 𝑅 ∈ NzRing ) ) |
35 |
19
|
elsn2 |
⊢ ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ { 0 } ↔ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 ) |
36 |
|
velsn |
⊢ ( 𝑥 ∈ { 0 } ↔ 𝑥 = 0 ) |
37 |
|
velsn |
⊢ ( 𝑦 ∈ { 0 } ↔ 𝑦 = 0 ) |
38 |
36 37
|
orbi12i |
⊢ ( ( 𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 } ) ↔ ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) |
39 |
35 38
|
imbi12i |
⊢ ( ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ { 0 } → ( 𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 } ) ) ↔ ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ) |
40 |
39
|
2ralbii |
⊢ ( ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ { 0 } → ( 𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 } ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ) |
41 |
40
|
a1i |
⊢ ( 𝑅 ∈ CRing → ( ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ { 0 } → ( 𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 } ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ) ) |
42 |
34 41
|
anbi12d |
⊢ ( 𝑅 ∈ CRing → ( ( ( { 0 } ∈ ( LIdeal ‘ 𝑅 ) ∧ { 0 } ≠ ( Base ‘ 𝑅 ) ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ { 0 } → ( 𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 } ) ) ) ↔ ( 𝑅 ∈ NzRing ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ) ) ) |
43 |
2 42
|
syl5bb |
⊢ ( 𝑅 ∈ CRing → ( ( { 0 } ∈ ( LIdeal ‘ 𝑅 ) ∧ { 0 } ≠ ( Base ‘ 𝑅 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ { 0 } → ( 𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 } ) ) ) ↔ ( 𝑅 ∈ NzRing ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ) ) ) |
44 |
43
|
pm5.32i |
⊢ ( ( 𝑅 ∈ CRing ∧ ( { 0 } ∈ ( LIdeal ‘ 𝑅 ) ∧ { 0 } ≠ ( Base ‘ 𝑅 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ { 0 } → ( 𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 } ) ) ) ) ↔ ( 𝑅 ∈ CRing ∧ ( 𝑅 ∈ NzRing ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ) ) ) |
45 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
46 |
8 45
|
isprmidlc |
⊢ ( 𝑅 ∈ CRing → ( { 0 } ∈ ( PrmIdeal ‘ 𝑅 ) ↔ ( { 0 } ∈ ( LIdeal ‘ 𝑅 ) ∧ { 0 } ≠ ( Base ‘ 𝑅 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ { 0 } → ( 𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 } ) ) ) ) ) |
47 |
46
|
pm5.32i |
⊢ ( ( 𝑅 ∈ CRing ∧ { 0 } ∈ ( PrmIdeal ‘ 𝑅 ) ) ↔ ( 𝑅 ∈ CRing ∧ ( { 0 } ∈ ( LIdeal ‘ 𝑅 ) ∧ { 0 } ≠ ( Base ‘ 𝑅 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ { 0 } → ( 𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 } ) ) ) ) ) |
48 |
|
df-idom |
⊢ IDomn = ( CRing ∩ Domn ) |
49 |
48
|
eleq2i |
⊢ ( 𝑅 ∈ IDomn ↔ 𝑅 ∈ ( CRing ∩ Domn ) ) |
50 |
|
elin |
⊢ ( 𝑅 ∈ ( CRing ∩ Domn ) ↔ ( 𝑅 ∈ CRing ∧ 𝑅 ∈ Domn ) ) |
51 |
8 45 1
|
isdomn |
⊢ ( 𝑅 ∈ Domn ↔ ( 𝑅 ∈ NzRing ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ) ) |
52 |
51
|
anbi2i |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑅 ∈ Domn ) ↔ ( 𝑅 ∈ CRing ∧ ( 𝑅 ∈ NzRing ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ) ) ) |
53 |
49 50 52
|
3bitri |
⊢ ( 𝑅 ∈ IDomn ↔ ( 𝑅 ∈ CRing ∧ ( 𝑅 ∈ NzRing ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 0 → ( 𝑥 = 0 ∨ 𝑦 = 0 ) ) ) ) ) |
54 |
44 47 53
|
3bitr4i |
⊢ ( ( 𝑅 ∈ CRing ∧ { 0 } ∈ ( PrmIdeal ‘ 𝑅 ) ) ↔ 𝑅 ∈ IDomn ) |