Step |
Hyp |
Ref |
Expression |
1 |
|
lidlabl.l |
⊢ 𝐿 = ( LIdeal ‘ 𝑅 ) |
2 |
|
lidlabl.i |
⊢ 𝐼 = ( 𝑅 ↾s 𝑈 ) |
3 |
|
zlidlring.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
4 |
|
zlidlring.0 |
⊢ 0 = ( 0g ‘ 𝑅 ) |
5 |
|
eqid |
⊢ ( Base ‘ 𝐼 ) = ( Base ‘ 𝐼 ) |
6 |
|
eqid |
⊢ ( .r ‘ 𝐼 ) = ( .r ‘ 𝐼 ) |
7 |
5 6
|
isringrng |
⊢ ( 𝐼 ∈ Ring ↔ ( 𝐼 ∈ Rng ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) ) |
8 |
|
domnring |
⊢ ( 𝑅 ∈ Domn → 𝑅 ∈ Ring ) |
9 |
8
|
anim1i |
⊢ ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) → ( 𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿 ) ) |
10 |
1 2
|
lidlrng |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿 ) → 𝐼 ∈ Rng ) |
11 |
9 10
|
syl |
⊢ ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) → 𝐼 ∈ Rng ) |
12 |
|
ibar |
⊢ ( 𝐼 ∈ Rng → ( ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ↔ ( 𝐼 ∈ Rng ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) ) ) |
13 |
12
|
bicomd |
⊢ ( 𝐼 ∈ Rng → ( ( 𝐼 ∈ Rng ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) ↔ ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) ) |
14 |
13
|
adantl |
⊢ ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) → ( ( 𝐼 ∈ Rng ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) ↔ ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) ) |
15 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
16 |
2 15
|
ressmulr |
⊢ ( 𝑈 ∈ 𝐿 → ( .r ‘ 𝑅 ) = ( .r ‘ 𝐼 ) ) |
17 |
16
|
eqcomd |
⊢ ( 𝑈 ∈ 𝐿 → ( .r ‘ 𝐼 ) = ( .r ‘ 𝑅 ) ) |
18 |
17
|
oveqd |
⊢ ( 𝑈 ∈ 𝐿 → ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ) |
19 |
18
|
eqeq1d |
⊢ ( 𝑈 ∈ 𝐿 → ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ↔ ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 𝑦 ) ) |
20 |
17
|
oveqd |
⊢ ( 𝑈 ∈ 𝐿 → ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) ) |
21 |
20
|
eqeq1d |
⊢ ( 𝑈 ∈ 𝐿 → ( ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ↔ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) = 𝑦 ) ) |
22 |
19 21
|
anbi12d |
⊢ ( 𝑈 ∈ 𝐿 → ( ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ↔ ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) = 𝑦 ) ) ) |
23 |
22
|
ad2antlr |
⊢ ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) → ( ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ↔ ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) = 𝑦 ) ) ) |
24 |
23
|
ad2antrr |
⊢ ( ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ¬ 𝑈 = { 0 } ) ∧ 𝑥 ∈ ( Base ‘ 𝐼 ) ) → ( ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ↔ ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) = 𝑦 ) ) ) |
25 |
24
|
ralbidv |
⊢ ( ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ¬ 𝑈 = { 0 } ) ∧ 𝑥 ∈ ( Base ‘ 𝐼 ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) = 𝑦 ) ) ) |
26 |
|
simp-4l |
⊢ ( ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ¬ 𝑈 = { 0 } ) ∧ 𝑥 ∈ ( Base ‘ 𝐼 ) ) → 𝑅 ∈ Domn ) |
27 |
1 2
|
lidlbas |
⊢ ( 𝑈 ∈ 𝐿 → ( Base ‘ 𝐼 ) = 𝑈 ) |
28 |
27
|
eleq1d |
⊢ ( 𝑈 ∈ 𝐿 → ( ( Base ‘ 𝐼 ) ∈ 𝐿 ↔ 𝑈 ∈ 𝐿 ) ) |
29 |
28
|
ibir |
⊢ ( 𝑈 ∈ 𝐿 → ( Base ‘ 𝐼 ) ∈ 𝐿 ) |
30 |
29
|
ad3antlr |
⊢ ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ¬ 𝑈 = { 0 } ) → ( Base ‘ 𝐼 ) ∈ 𝐿 ) |
31 |
27
|
ad2antlr |
⊢ ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) → ( Base ‘ 𝐼 ) = 𝑈 ) |
32 |
31
|
eqeq1d |
⊢ ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) → ( ( Base ‘ 𝐼 ) = { 0 } ↔ 𝑈 = { 0 } ) ) |
33 |
32
|
biimpd |
⊢ ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) → ( ( Base ‘ 𝐼 ) = { 0 } → 𝑈 = { 0 } ) ) |
34 |
33
|
necon3bd |
⊢ ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) → ( ¬ 𝑈 = { 0 } → ( Base ‘ 𝐼 ) ≠ { 0 } ) ) |
35 |
34
|
imp |
⊢ ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ¬ 𝑈 = { 0 } ) → ( Base ‘ 𝐼 ) ≠ { 0 } ) |
36 |
30 35
|
jca |
⊢ ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ¬ 𝑈 = { 0 } ) → ( ( Base ‘ 𝐼 ) ∈ 𝐿 ∧ ( Base ‘ 𝐼 ) ≠ { 0 } ) ) |
37 |
36
|
adantr |
⊢ ( ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ¬ 𝑈 = { 0 } ) ∧ 𝑥 ∈ ( Base ‘ 𝐼 ) ) → ( ( Base ‘ 𝐼 ) ∈ 𝐿 ∧ ( Base ‘ 𝐼 ) ≠ { 0 } ) ) |
38 |
|
simpr |
⊢ ( ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ¬ 𝑈 = { 0 } ) ∧ 𝑥 ∈ ( Base ‘ 𝐼 ) ) → 𝑥 ∈ ( Base ‘ 𝐼 ) ) |
39 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
40 |
1 15 39 4
|
lidldomn1 |
⊢ ( ( 𝑅 ∈ Domn ∧ ( ( Base ‘ 𝐼 ) ∈ 𝐿 ∧ ( Base ‘ 𝐼 ) ≠ { 0 } ) ∧ 𝑥 ∈ ( Base ‘ 𝐼 ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) = 𝑦 ) → 𝑥 = ( 1r ‘ 𝑅 ) ) ) |
41 |
26 37 38 40
|
syl3anc |
⊢ ( ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ¬ 𝑈 = { 0 } ) ∧ 𝑥 ∈ ( Base ‘ 𝐼 ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) = 𝑦 ) → 𝑥 = ( 1r ‘ 𝑅 ) ) ) |
42 |
25 41
|
sylbid |
⊢ ( ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ¬ 𝑈 = { 0 } ) ∧ 𝑥 ∈ ( Base ‘ 𝐼 ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) → 𝑥 = ( 1r ‘ 𝑅 ) ) ) |
43 |
42
|
imp |
⊢ ( ( ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ¬ 𝑈 = { 0 } ) ∧ 𝑥 ∈ ( Base ‘ 𝐼 ) ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) → 𝑥 = ( 1r ‘ 𝑅 ) ) |
44 |
27
|
ad3antlr |
⊢ ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ¬ 𝑈 = { 0 } ) → ( Base ‘ 𝐼 ) = 𝑈 ) |
45 |
44
|
eleq2d |
⊢ ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ¬ 𝑈 = { 0 } ) → ( 𝑥 ∈ ( Base ‘ 𝐼 ) ↔ 𝑥 ∈ 𝑈 ) ) |
46 |
45
|
biimpd |
⊢ ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ¬ 𝑈 = { 0 } ) → ( 𝑥 ∈ ( Base ‘ 𝐼 ) → 𝑥 ∈ 𝑈 ) ) |
47 |
46
|
imp |
⊢ ( ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ¬ 𝑈 = { 0 } ) ∧ 𝑥 ∈ ( Base ‘ 𝐼 ) ) → 𝑥 ∈ 𝑈 ) |
48 |
47
|
adantr |
⊢ ( ( ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ¬ 𝑈 = { 0 } ) ∧ 𝑥 ∈ ( Base ‘ 𝐼 ) ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) → 𝑥 ∈ 𝑈 ) |
49 |
43 48
|
eqeltrrd |
⊢ ( ( ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ¬ 𝑈 = { 0 } ) ∧ 𝑥 ∈ ( Base ‘ 𝐼 ) ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) → ( 1r ‘ 𝑅 ) ∈ 𝑈 ) |
50 |
49
|
rexlimdva2 |
⊢ ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ¬ 𝑈 = { 0 } ) → ( ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) → ( 1r ‘ 𝑅 ) ∈ 𝑈 ) ) |
51 |
50
|
impancom |
⊢ ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) → ( ¬ 𝑈 = { 0 } → ( 1r ‘ 𝑅 ) ∈ 𝑈 ) ) |
52 |
9
|
adantr |
⊢ ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) → ( 𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿 ) ) |
53 |
1 3 39
|
lidl1el |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿 ) → ( ( 1r ‘ 𝑅 ) ∈ 𝑈 ↔ 𝑈 = 𝐵 ) ) |
54 |
52 53
|
syl |
⊢ ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) → ( ( 1r ‘ 𝑅 ) ∈ 𝑈 ↔ 𝑈 = 𝐵 ) ) |
55 |
54
|
adantr |
⊢ ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) → ( ( 1r ‘ 𝑅 ) ∈ 𝑈 ↔ 𝑈 = 𝐵 ) ) |
56 |
51 55
|
sylibd |
⊢ ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) → ( ¬ 𝑈 = { 0 } → 𝑈 = 𝐵 ) ) |
57 |
56
|
orrd |
⊢ ( ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) → ( 𝑈 = { 0 } ∨ 𝑈 = 𝐵 ) ) |
58 |
57
|
ex |
⊢ ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) → ( ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) → ( 𝑈 = { 0 } ∨ 𝑈 = 𝐵 ) ) ) |
59 |
1 2 3 4
|
zlidlring |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑈 = { 0 } ) → 𝐼 ∈ Ring ) |
60 |
7
|
simprbi |
⊢ ( 𝐼 ∈ Ring → ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) |
61 |
59 60
|
syl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑈 = { 0 } ) → ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) |
62 |
61
|
ex |
⊢ ( 𝑅 ∈ Ring → ( 𝑈 = { 0 } → ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) ) |
63 |
8 62
|
syl |
⊢ ( 𝑅 ∈ Domn → ( 𝑈 = { 0 } → ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) ) |
64 |
63
|
ad2antrr |
⊢ ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) → ( 𝑈 = { 0 } → ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) ) |
65 |
9
|
anim1i |
⊢ ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) → ( ( 𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ) |
66 |
3 15
|
ringideu |
⊢ ( 𝑅 ∈ Ring → ∃! 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) = 𝑦 ) ) |
67 |
|
reurex |
⊢ ( ∃! 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) = 𝑦 ) → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) = 𝑦 ) ) |
68 |
66 67
|
syl |
⊢ ( 𝑅 ∈ Ring → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) = 𝑦 ) ) |
69 |
68
|
adantr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿 ) → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) = 𝑦 ) ) |
70 |
69
|
ad2antrr |
⊢ ( ( ( ( 𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ 𝑈 = 𝐵 ) → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) = 𝑦 ) ) |
71 |
2 3
|
ressbas |
⊢ ( 𝑈 ∈ 𝐿 → ( 𝑈 ∩ 𝐵 ) = ( Base ‘ 𝐼 ) ) |
72 |
71
|
ad3antlr |
⊢ ( ( ( ( 𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ 𝑈 = 𝐵 ) → ( 𝑈 ∩ 𝐵 ) = ( Base ‘ 𝐼 ) ) |
73 |
|
ineq1 |
⊢ ( 𝑈 = 𝐵 → ( 𝑈 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐵 ) ) |
74 |
|
inidm |
⊢ ( 𝐵 ∩ 𝐵 ) = 𝐵 |
75 |
73 74
|
eqtrdi |
⊢ ( 𝑈 = 𝐵 → ( 𝑈 ∩ 𝐵 ) = 𝐵 ) |
76 |
75
|
adantl |
⊢ ( ( ( ( 𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ 𝑈 = 𝐵 ) → ( 𝑈 ∩ 𝐵 ) = 𝐵 ) |
77 |
72 76
|
eqtr3d |
⊢ ( ( ( ( 𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ 𝑈 = 𝐵 ) → ( Base ‘ 𝐼 ) = 𝐵 ) |
78 |
22
|
ad3antlr |
⊢ ( ( ( ( 𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ 𝑈 = 𝐵 ) → ( ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ↔ ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) = 𝑦 ) ) ) |
79 |
77 78
|
raleqbidv |
⊢ ( ( ( ( 𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ 𝑈 = 𝐵 ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ↔ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) = 𝑦 ) ) ) |
80 |
77 79
|
rexeqbidv |
⊢ ( ( ( ( 𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ 𝑈 = 𝐵 ) → ( ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ↔ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) = 𝑦 ) ) ) |
81 |
70 80
|
mpbird |
⊢ ( ( ( ( 𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) ∧ 𝑈 = 𝐵 ) → ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) |
82 |
81
|
ex |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) → ( 𝑈 = 𝐵 → ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) ) |
83 |
65 82
|
syl |
⊢ ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) → ( 𝑈 = 𝐵 → ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) ) |
84 |
64 83
|
jaod |
⊢ ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) → ( ( 𝑈 = { 0 } ∨ 𝑈 = 𝐵 ) → ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) ) |
85 |
58 84
|
impbid |
⊢ ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) → ( ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ↔ ( 𝑈 = { 0 } ∨ 𝑈 = 𝐵 ) ) ) |
86 |
14 85
|
bitrd |
⊢ ( ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) ∧ 𝐼 ∈ Rng ) → ( ( 𝐼 ∈ Rng ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) ↔ ( 𝑈 = { 0 } ∨ 𝑈 = 𝐵 ) ) ) |
87 |
11 86
|
mpdan |
⊢ ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) → ( ( 𝐼 ∈ Rng ∧ ∃ 𝑥 ∈ ( Base ‘ 𝐼 ) ∀ 𝑦 ∈ ( Base ‘ 𝐼 ) ( ( 𝑥 ( .r ‘ 𝐼 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝐼 ) 𝑥 ) = 𝑦 ) ) ↔ ( 𝑈 = { 0 } ∨ 𝑈 = 𝐵 ) ) ) |
88 |
7 87
|
syl5bb |
⊢ ( ( 𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿 ) → ( 𝐼 ∈ Ring ↔ ( 𝑈 = { 0 } ∨ 𝑈 = 𝐵 ) ) ) |