Step |
Hyp |
Ref |
Expression |
1 |
|
ssmxidl.1 |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
ssmxidllem.1 |
⊢ 𝑃 = { 𝑝 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝 ) } |
3 |
|
ssmxidllem.2 |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
4 |
|
ssmxidllem.3 |
⊢ ( 𝜑 → 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) |
5 |
|
ssmxidllem.4 |
⊢ ( 𝜑 → 𝐼 ≠ 𝐵 ) |
6 |
|
ssmxidllem2.1 |
⊢ ( 𝜑 → 𝑍 ⊆ 𝑃 ) |
7 |
|
ssmxidllem2.2 |
⊢ ( 𝜑 → 𝑍 ≠ ∅ ) |
8 |
|
ssmxidllem2.3 |
⊢ ( 𝜑 → [⊊] Or 𝑍 ) |
9 |
|
neeq1 |
⊢ ( 𝑝 = ∪ 𝑍 → ( 𝑝 ≠ 𝐵 ↔ ∪ 𝑍 ≠ 𝐵 ) ) |
10 |
|
sseq2 |
⊢ ( 𝑝 = ∪ 𝑍 → ( 𝐼 ⊆ 𝑝 ↔ 𝐼 ⊆ ∪ 𝑍 ) ) |
11 |
9 10
|
anbi12d |
⊢ ( 𝑝 = ∪ 𝑍 → ( ( 𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝 ) ↔ ( ∪ 𝑍 ≠ 𝐵 ∧ 𝐼 ⊆ ∪ 𝑍 ) ) ) |
12 |
2
|
ssrab3 |
⊢ 𝑃 ⊆ ( LIdeal ‘ 𝑅 ) |
13 |
6 12
|
sstrdi |
⊢ ( 𝜑 → 𝑍 ⊆ ( LIdeal ‘ 𝑅 ) ) |
14 |
13
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) |
15 |
|
eqid |
⊢ ( LIdeal ‘ 𝑅 ) = ( LIdeal ‘ 𝑅 ) |
16 |
1 15
|
lidlss |
⊢ ( 𝑗 ∈ ( LIdeal ‘ 𝑅 ) → 𝑗 ⊆ 𝐵 ) |
17 |
14 16
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑗 ⊆ 𝐵 ) |
18 |
17
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑗 ∈ 𝑍 𝑗 ⊆ 𝐵 ) |
19 |
|
unissb |
⊢ ( ∪ 𝑍 ⊆ 𝐵 ↔ ∀ 𝑗 ∈ 𝑍 𝑗 ⊆ 𝐵 ) |
20 |
18 19
|
sylibr |
⊢ ( 𝜑 → ∪ 𝑍 ⊆ 𝐵 ) |
21 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑅 ∈ Ring ) |
22 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
23 |
15 22
|
lidl0cl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) → ( 0g ‘ 𝑅 ) ∈ 𝑗 ) |
24 |
21 14 23
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 0g ‘ 𝑅 ) ∈ 𝑗 ) |
25 |
|
n0i |
⊢ ( ( 0g ‘ 𝑅 ) ∈ 𝑗 → ¬ 𝑗 = ∅ ) |
26 |
24 25
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ¬ 𝑗 = ∅ ) |
27 |
26
|
reximdva0 |
⊢ ( ( 𝜑 ∧ 𝑍 ≠ ∅ ) → ∃ 𝑗 ∈ 𝑍 ¬ 𝑗 = ∅ ) |
28 |
7 27
|
mpdan |
⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ¬ 𝑗 = ∅ ) |
29 |
|
rexnal |
⊢ ( ∃ 𝑗 ∈ 𝑍 ¬ 𝑗 = ∅ ↔ ¬ ∀ 𝑗 ∈ 𝑍 𝑗 = ∅ ) |
30 |
28 29
|
sylib |
⊢ ( 𝜑 → ¬ ∀ 𝑗 ∈ 𝑍 𝑗 = ∅ ) |
31 |
|
uni0c |
⊢ ( ∪ 𝑍 = ∅ ↔ ∀ 𝑗 ∈ 𝑍 𝑗 = ∅ ) |
32 |
31
|
necon3abii |
⊢ ( ∪ 𝑍 ≠ ∅ ↔ ¬ ∀ 𝑗 ∈ 𝑍 𝑗 = ∅ ) |
33 |
30 32
|
sylibr |
⊢ ( 𝜑 → ∪ 𝑍 ≠ ∅ ) |
34 |
|
eluni2 |
⊢ ( 𝑎 ∈ ∪ 𝑍 ↔ ∃ 𝑖 ∈ 𝑍 𝑎 ∈ 𝑖 ) |
35 |
|
eluni2 |
⊢ ( 𝑏 ∈ ∪ 𝑍 ↔ ∃ 𝑗 ∈ 𝑍 𝑏 ∈ 𝑗 ) |
36 |
34 35
|
anbi12i |
⊢ ( ( 𝑎 ∈ ∪ 𝑍 ∧ 𝑏 ∈ ∪ 𝑍 ) ↔ ( ∃ 𝑖 ∈ 𝑍 𝑎 ∈ 𝑖 ∧ ∃ 𝑗 ∈ 𝑍 𝑏 ∈ 𝑗 ) ) |
37 |
|
an32 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ∃ 𝑖 ∈ 𝑍 𝑎 ∈ 𝑖 ) ∧ 𝑗 ∈ 𝑍 ) ↔ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ ∃ 𝑖 ∈ 𝑍 𝑎 ∈ 𝑖 ) ) |
38 |
3
|
ad6antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑖 ⊆ 𝑗 ) → 𝑅 ∈ Ring ) |
39 |
13
|
ad5antr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) → 𝑍 ⊆ ( LIdeal ‘ 𝑅 ) ) |
40 |
|
simp-4r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) → 𝑗 ∈ 𝑍 ) |
41 |
39 40
|
sseldd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) → 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) |
42 |
41
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑖 ⊆ 𝑗 ) → 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) |
43 |
|
simp-6r |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑖 ⊆ 𝑗 ) → 𝑥 ∈ 𝐵 ) |
44 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑖 ⊆ 𝑗 ) → 𝑖 ⊆ 𝑗 ) |
45 |
|
simplr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑖 ⊆ 𝑗 ) → 𝑎 ∈ 𝑖 ) |
46 |
44 45
|
sseldd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑖 ⊆ 𝑗 ) → 𝑎 ∈ 𝑗 ) |
47 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
48 |
15 1 47
|
lidlmcl |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝑗 ) ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ∈ 𝑗 ) |
49 |
38 42 43 46 48
|
syl22anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑖 ⊆ 𝑗 ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ∈ 𝑗 ) |
50 |
|
simp-4r |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑖 ⊆ 𝑗 ) → 𝑏 ∈ 𝑗 ) |
51 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
52 |
15 51
|
lidlacl |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ∈ 𝑗 ∧ 𝑏 ∈ 𝑗 ) ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ 𝑗 ) |
53 |
38 42 49 50 52
|
syl22anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑖 ⊆ 𝑗 ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ 𝑗 ) |
54 |
40
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑖 ⊆ 𝑗 ) → 𝑗 ∈ 𝑍 ) |
55 |
|
elunii |
⊢ ( ( ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ 𝑗 ∧ 𝑗 ∈ 𝑍 ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ ∪ 𝑍 ) |
56 |
53 54 55
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑖 ⊆ 𝑗 ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ ∪ 𝑍 ) |
57 |
3
|
ad6antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑗 ⊆ 𝑖 ) → 𝑅 ∈ Ring ) |
58 |
39
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑗 ⊆ 𝑖 ) → 𝑍 ⊆ ( LIdeal ‘ 𝑅 ) ) |
59 |
|
simplr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) → 𝑖 ∈ 𝑍 ) |
60 |
59
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑗 ⊆ 𝑖 ) → 𝑖 ∈ 𝑍 ) |
61 |
58 60
|
sseldd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑗 ⊆ 𝑖 ) → 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ) |
62 |
|
simp-6r |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑗 ⊆ 𝑖 ) → 𝑥 ∈ 𝐵 ) |
63 |
|
simplr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑗 ⊆ 𝑖 ) → 𝑎 ∈ 𝑖 ) |
64 |
15 1 47
|
lidlmcl |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝑖 ) ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ∈ 𝑖 ) |
65 |
57 61 62 63 64
|
syl22anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑗 ⊆ 𝑖 ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ∈ 𝑖 ) |
66 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑗 ⊆ 𝑖 ) → 𝑗 ⊆ 𝑖 ) |
67 |
|
simp-4r |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑗 ⊆ 𝑖 ) → 𝑏 ∈ 𝑗 ) |
68 |
66 67
|
sseldd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑗 ⊆ 𝑖 ) → 𝑏 ∈ 𝑖 ) |
69 |
15 51
|
lidlacl |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ∈ 𝑖 ∧ 𝑏 ∈ 𝑖 ) ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ 𝑖 ) |
70 |
57 61 65 68 69
|
syl22anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑗 ⊆ 𝑖 ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ 𝑖 ) |
71 |
|
elunii |
⊢ ( ( ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ 𝑖 ∧ 𝑖 ∈ 𝑍 ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ ∪ 𝑍 ) |
72 |
70 60 71
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) ∧ 𝑗 ⊆ 𝑖 ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ ∪ 𝑍 ) |
73 |
8
|
ad5antr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) → [⊊] Or 𝑍 ) |
74 |
|
sorpssi |
⊢ ( ( [⊊] Or 𝑍 ∧ ( 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍 ) ) → ( 𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖 ) ) |
75 |
73 59 40 74
|
syl12anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) → ( 𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖 ) ) |
76 |
56 72 75
|
mpjaodan |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ 𝑖 ∈ 𝑍 ) ∧ 𝑎 ∈ 𝑖 ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ ∪ 𝑍 ) |
77 |
76
|
r19.29an |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) ∧ ∃ 𝑖 ∈ 𝑍 𝑎 ∈ 𝑖 ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ ∪ 𝑍 ) |
78 |
77
|
an32s |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑗 ∈ 𝑍 ) ∧ ∃ 𝑖 ∈ 𝑍 𝑎 ∈ 𝑖 ) ∧ 𝑏 ∈ 𝑗 ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ ∪ 𝑍 ) |
79 |
37 78
|
sylanb |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ∃ 𝑖 ∈ 𝑍 𝑎 ∈ 𝑖 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑏 ∈ 𝑗 ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ ∪ 𝑍 ) |
80 |
79
|
r19.29an |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ∃ 𝑖 ∈ 𝑍 𝑎 ∈ 𝑖 ) ∧ ∃ 𝑗 ∈ 𝑍 𝑏 ∈ 𝑗 ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ ∪ 𝑍 ) |
81 |
80
|
anasss |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( ∃ 𝑖 ∈ 𝑍 𝑎 ∈ 𝑖 ∧ ∃ 𝑗 ∈ 𝑍 𝑏 ∈ 𝑗 ) ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ ∪ 𝑍 ) |
82 |
36 81
|
sylan2b |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝑎 ∈ ∪ 𝑍 ∧ 𝑏 ∈ ∪ 𝑍 ) ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ ∪ 𝑍 ) |
83 |
82
|
ralrimivva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∀ 𝑎 ∈ ∪ 𝑍 ∀ 𝑏 ∈ ∪ 𝑍 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ ∪ 𝑍 ) |
84 |
83
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ∀ 𝑎 ∈ ∪ 𝑍 ∀ 𝑏 ∈ ∪ 𝑍 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ ∪ 𝑍 ) |
85 |
15 1 51 47
|
islidl |
⊢ ( ∪ 𝑍 ∈ ( LIdeal ‘ 𝑅 ) ↔ ( ∪ 𝑍 ⊆ 𝐵 ∧ ∪ 𝑍 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑎 ∈ ∪ 𝑍 ∀ 𝑏 ∈ ∪ 𝑍 ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑎 ) ( +g ‘ 𝑅 ) 𝑏 ) ∈ ∪ 𝑍 ) ) |
86 |
20 33 84 85
|
syl3anbrc |
⊢ ( 𝜑 → ∪ 𝑍 ∈ ( LIdeal ‘ 𝑅 ) ) |
87 |
6
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑗 ∈ 𝑃 ) |
88 |
|
neeq1 |
⊢ ( 𝑝 = 𝑗 → ( 𝑝 ≠ 𝐵 ↔ 𝑗 ≠ 𝐵 ) ) |
89 |
|
sseq2 |
⊢ ( 𝑝 = 𝑗 → ( 𝐼 ⊆ 𝑝 ↔ 𝐼 ⊆ 𝑗 ) ) |
90 |
88 89
|
anbi12d |
⊢ ( 𝑝 = 𝑗 → ( ( 𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝 ) ↔ ( 𝑗 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑗 ) ) ) |
91 |
90 2
|
elrab2 |
⊢ ( 𝑗 ∈ 𝑃 ↔ ( 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ∧ ( 𝑗 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑗 ) ) ) |
92 |
87 91
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ∧ ( 𝑗 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑗 ) ) ) |
93 |
92
|
simprld |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑗 ≠ 𝐵 ) |
94 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
95 |
1 94
|
pridln1 |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑗 ≠ 𝐵 ) → ¬ ( 1r ‘ 𝑅 ) ∈ 𝑗 ) |
96 |
21 14 93 95
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ¬ ( 1r ‘ 𝑅 ) ∈ 𝑗 ) |
97 |
96
|
nrexdv |
⊢ ( 𝜑 → ¬ ∃ 𝑗 ∈ 𝑍 ( 1r ‘ 𝑅 ) ∈ 𝑗 ) |
98 |
|
eluni2 |
⊢ ( ( 1r ‘ 𝑅 ) ∈ ∪ 𝑍 ↔ ∃ 𝑗 ∈ 𝑍 ( 1r ‘ 𝑅 ) ∈ 𝑗 ) |
99 |
97 98
|
sylnibr |
⊢ ( 𝜑 → ¬ ( 1r ‘ 𝑅 ) ∈ ∪ 𝑍 ) |
100 |
15 1 94
|
lidl1el |
⊢ ( ( 𝑅 ∈ Ring ∧ ∪ 𝑍 ∈ ( LIdeal ‘ 𝑅 ) ) → ( ( 1r ‘ 𝑅 ) ∈ ∪ 𝑍 ↔ ∪ 𝑍 = 𝐵 ) ) |
101 |
3 86 100
|
syl2anc |
⊢ ( 𝜑 → ( ( 1r ‘ 𝑅 ) ∈ ∪ 𝑍 ↔ ∪ 𝑍 = 𝐵 ) ) |
102 |
101
|
necon3bbid |
⊢ ( 𝜑 → ( ¬ ( 1r ‘ 𝑅 ) ∈ ∪ 𝑍 ↔ ∪ 𝑍 ≠ 𝐵 ) ) |
103 |
99 102
|
mpbid |
⊢ ( 𝜑 → ∪ 𝑍 ≠ 𝐵 ) |
104 |
92
|
simprrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝐼 ⊆ 𝑗 ) |
105 |
104
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑗 ∈ 𝑍 𝐼 ⊆ 𝑗 ) |
106 |
|
ssint |
⊢ ( 𝐼 ⊆ ∩ 𝑍 ↔ ∀ 𝑗 ∈ 𝑍 𝐼 ⊆ 𝑗 ) |
107 |
105 106
|
sylibr |
⊢ ( 𝜑 → 𝐼 ⊆ ∩ 𝑍 ) |
108 |
|
intssuni |
⊢ ( 𝑍 ≠ ∅ → ∩ 𝑍 ⊆ ∪ 𝑍 ) |
109 |
7 108
|
syl |
⊢ ( 𝜑 → ∩ 𝑍 ⊆ ∪ 𝑍 ) |
110 |
107 109
|
sstrd |
⊢ ( 𝜑 → 𝐼 ⊆ ∪ 𝑍 ) |
111 |
103 110
|
jca |
⊢ ( 𝜑 → ( ∪ 𝑍 ≠ 𝐵 ∧ 𝐼 ⊆ ∪ 𝑍 ) ) |
112 |
11 86 111
|
elrabd |
⊢ ( 𝜑 → ∪ 𝑍 ∈ { 𝑝 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝 ) } ) |
113 |
112 2
|
eleqtrrdi |
⊢ ( 𝜑 → ∪ 𝑍 ∈ 𝑃 ) |