Step | Hyp | Ref
| Expression |
1 | | simp1 1061 |
. 2
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → 𝑅 ∈ RingOps) |
2 | | smprngpr.1 |
. . . . 5
⊢ 𝐺 = (1st ‘𝑅) |
3 | | smprngpr.4 |
. . . . 5
⊢ 𝑍 = (GId‘𝐺) |
4 | 2, 3 | 0idl 33824 |
. . . 4
⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅)) |
5 | 4 | 3ad2ant1 1082 |
. . 3
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → {𝑍} ∈ (Idl‘𝑅)) |
6 | | smprngpr.2 |
. . . . . . . 8
⊢ 𝐻 = (2nd ‘𝑅) |
7 | | smprngpr.3 |
. . . . . . . 8
⊢ 𝑋 = ran 𝐺 |
8 | | smprngpr.5 |
. . . . . . . 8
⊢ 𝑈 = (GId‘𝐻) |
9 | 2, 6, 7, 3, 8 | 0rngo 33826 |
. . . . . . 7
⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍})) |
10 | | eqcom 2629 |
. . . . . . 7
⊢ (𝑈 = 𝑍 ↔ 𝑍 = 𝑈) |
11 | | eqcom 2629 |
. . . . . . 7
⊢ ({𝑍} = 𝑋 ↔ 𝑋 = {𝑍}) |
12 | 9, 10, 11 | 3bitr4g 303 |
. . . . . 6
⊢ (𝑅 ∈ RingOps → (𝑈 = 𝑍 ↔ {𝑍} = 𝑋)) |
13 | 12 | necon3bid 2838 |
. . . . 5
⊢ (𝑅 ∈ RingOps → (𝑈 ≠ 𝑍 ↔ {𝑍} ≠ 𝑋)) |
14 | 13 | biimpa 501 |
. . . 4
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → {𝑍} ≠ 𝑋) |
15 | 14 | 3adant3 1081 |
. . 3
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → {𝑍} ≠ 𝑋) |
16 | | df-pr 4180 |
. . . . . . . 8
⊢ {{𝑍}, 𝑋} = ({{𝑍}} ∪ {𝑋}) |
17 | 16 | eqeq2i 2634 |
. . . . . . 7
⊢
((Idl‘𝑅) =
{{𝑍}, 𝑋} ↔ (Idl‘𝑅) = ({{𝑍}} ∪ {𝑋})) |
18 | | eleq2 2690 |
. . . . . . . . 9
⊢
((Idl‘𝑅) =
({{𝑍}} ∪ {𝑋}) → (𝑖 ∈ (Idl‘𝑅) ↔ 𝑖 ∈ ({{𝑍}} ∪ {𝑋}))) |
19 | | eleq2 2690 |
. . . . . . . . 9
⊢
((Idl‘𝑅) =
({{𝑍}} ∪ {𝑋}) → (𝑗 ∈ (Idl‘𝑅) ↔ 𝑗 ∈ ({{𝑍}} ∪ {𝑋}))) |
20 | 18, 19 | anbi12d 747 |
. . . . . . . 8
⊢
((Idl‘𝑅) =
({{𝑍}} ∪ {𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ∧ 𝑗 ∈ ({{𝑍}} ∪ {𝑋})))) |
21 | | elun 3753 |
. . . . . . . . . 10
⊢ (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑖 ∈ {{𝑍}} ∨ 𝑖 ∈ {𝑋})) |
22 | | velsn 4193 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ {{𝑍}} ↔ 𝑖 = {𝑍}) |
23 | | velsn 4193 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ {𝑋} ↔ 𝑖 = 𝑋) |
24 | 22, 23 | orbi12i 543 |
. . . . . . . . . 10
⊢ ((𝑖 ∈ {{𝑍}} ∨ 𝑖 ∈ {𝑋}) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋)) |
25 | 21, 24 | bitri 264 |
. . . . . . . . 9
⊢ (𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑖 = {𝑍} ∨ 𝑖 = 𝑋)) |
26 | | elun 3753 |
. . . . . . . . . 10
⊢ (𝑗 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑗 ∈ {{𝑍}} ∨ 𝑗 ∈ {𝑋})) |
27 | | velsn 4193 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ {{𝑍}} ↔ 𝑗 = {𝑍}) |
28 | | velsn 4193 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ {𝑋} ↔ 𝑗 = 𝑋) |
29 | 27, 28 | orbi12i 543 |
. . . . . . . . . 10
⊢ ((𝑗 ∈ {{𝑍}} ∨ 𝑗 ∈ {𝑋}) ↔ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)) |
30 | 26, 29 | bitri 264 |
. . . . . . . . 9
⊢ (𝑗 ∈ ({{𝑍}} ∪ {𝑋}) ↔ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)) |
31 | 25, 30 | anbi12i 733 |
. . . . . . . 8
⊢ ((𝑖 ∈ ({{𝑍}} ∪ {𝑋}) ∧ 𝑗 ∈ ({{𝑍}} ∪ {𝑋})) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋))) |
32 | 20, 31 | syl6bb 276 |
. . . . . . 7
⊢
((Idl‘𝑅) =
({{𝑍}} ∪ {𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)))) |
33 | 17, 32 | sylbi 207 |
. . . . . 6
⊢
((Idl‘𝑅) =
{{𝑍}, 𝑋} → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)))) |
34 | 33 | 3ad2ant3 1084 |
. . . . 5
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) ↔ ((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)))) |
35 | | eqimss 3657 |
. . . . . . . . . . 11
⊢ (𝑖 = {𝑍} → 𝑖 ⊆ {𝑍}) |
36 | 35 | orcd 407 |
. . . . . . . . . 10
⊢ (𝑖 = {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})) |
37 | 36 | adantr 481 |
. . . . . . . . 9
⊢ ((𝑖 = {𝑍} ∧ 𝑗 = {𝑍}) → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})) |
38 | 37 | a1d 25 |
. . . . . . . 8
⊢ ((𝑖 = {𝑍} ∧ 𝑗 = {𝑍}) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))) |
39 | 38 | a1i 11 |
. . . . . . 7
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → ((𝑖 = {𝑍} ∧ 𝑗 = {𝑍}) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))) |
40 | | eqimss 3657 |
. . . . . . . . . . 11
⊢ (𝑗 = {𝑍} → 𝑗 ⊆ {𝑍}) |
41 | 40 | olcd 408 |
. . . . . . . . . 10
⊢ (𝑗 = {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})) |
42 | 41 | adantl 482 |
. . . . . . . . 9
⊢ ((𝑖 = 𝑋 ∧ 𝑗 = {𝑍}) → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})) |
43 | 42 | a1d 25 |
. . . . . . . 8
⊢ ((𝑖 = 𝑋 ∧ 𝑗 = {𝑍}) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))) |
44 | 43 | a1i 11 |
. . . . . . 7
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → ((𝑖 = 𝑋 ∧ 𝑗 = {𝑍}) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))) |
45 | 36 | adantr 481 |
. . . . . . . . 9
⊢ ((𝑖 = {𝑍} ∧ 𝑗 = 𝑋) → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})) |
46 | 45 | a1d 25 |
. . . . . . . 8
⊢ ((𝑖 = {𝑍} ∧ 𝑗 = 𝑋) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))) |
47 | 46 | a1i 11 |
. . . . . . 7
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → ((𝑖 = {𝑍} ∧ 𝑗 = 𝑋) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))) |
48 | 2 | rneqi 5352 |
. . . . . . . . . . . . . 14
⊢ ran 𝐺 = ran (1st
‘𝑅) |
49 | 7, 48 | eqtri 2644 |
. . . . . . . . . . . . 13
⊢ 𝑋 = ran (1st
‘𝑅) |
50 | 49, 6, 8 | rngo1cl 33738 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋) |
51 | 50 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → 𝑈 ∈ 𝑋) |
52 | 6, 49, 8 | rngolidm 33736 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ∈ 𝑋) → (𝑈𝐻𝑈) = 𝑈) |
53 | 50, 52 | mpdan 702 |
. . . . . . . . . . . . . . 15
⊢ (𝑅 ∈ RingOps → (𝑈𝐻𝑈) = 𝑈) |
54 | 53 | eleq1d 2686 |
. . . . . . . . . . . . . 14
⊢ (𝑅 ∈ RingOps → ((𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈 ∈ {𝑍})) |
55 | | fvex 6201 |
. . . . . . . . . . . . . . . 16
⊢
(GId‘𝐻) ∈
V |
56 | 8, 55 | eqeltri 2697 |
. . . . . . . . . . . . . . 15
⊢ 𝑈 ∈ V |
57 | 56 | elsn 4192 |
. . . . . . . . . . . . . 14
⊢ (𝑈 ∈ {𝑍} ↔ 𝑈 = 𝑍) |
58 | 54, 57 | syl6bb 276 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ RingOps → ((𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈 = 𝑍)) |
59 | 58 | necon3bbid 2831 |
. . . . . . . . . . . 12
⊢ (𝑅 ∈ RingOps → (¬
(𝑈𝐻𝑈) ∈ {𝑍} ↔ 𝑈 ≠ 𝑍)) |
60 | 59 | biimpar 502 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → ¬ (𝑈𝐻𝑈) ∈ {𝑍}) |
61 | | oveq1 6657 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑈 → (𝑥𝐻𝑦) = (𝑈𝐻𝑦)) |
62 | 61 | eleq1d 2686 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑈 → ((𝑥𝐻𝑦) ∈ {𝑍} ↔ (𝑈𝐻𝑦) ∈ {𝑍})) |
63 | 62 | notbid 308 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑈 → (¬ (𝑥𝐻𝑦) ∈ {𝑍} ↔ ¬ (𝑈𝐻𝑦) ∈ {𝑍})) |
64 | | oveq2 6658 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑈 → (𝑈𝐻𝑦) = (𝑈𝐻𝑈)) |
65 | 64 | eleq1d 2686 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑈 → ((𝑈𝐻𝑦) ∈ {𝑍} ↔ (𝑈𝐻𝑈) ∈ {𝑍})) |
66 | 65 | notbid 308 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑈 → (¬ (𝑈𝐻𝑦) ∈ {𝑍} ↔ ¬ (𝑈𝐻𝑈) ∈ {𝑍})) |
67 | 63, 66 | rspc2ev 3324 |
. . . . . . . . . . 11
⊢ ((𝑈 ∈ 𝑋 ∧ 𝑈 ∈ 𝑋 ∧ ¬ (𝑈𝐻𝑈) ∈ {𝑍}) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍}) |
68 | 51, 51, 60, 67 | syl3anc 1326 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍}) |
69 | | rexnal2 3043 |
. . . . . . . . . 10
⊢
(∃𝑥 ∈
𝑋 ∃𝑦 ∈ 𝑋 ¬ (𝑥𝐻𝑦) ∈ {𝑍} ↔ ¬ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍}) |
70 | 68, 69 | sylib 208 |
. . . . . . . . 9
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → ¬ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍}) |
71 | 70 | pm2.21d 118 |
. . . . . . . 8
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))) |
72 | | raleq 3138 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑋 → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍})) |
73 | | raleq 3138 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑋 → (∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍})) |
74 | 73 | ralbidv 2986 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑋 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍})) |
75 | 72, 74 | sylan9bb 736 |
. . . . . . . . 9
⊢ ((𝑖 = 𝑋 ∧ 𝑗 = 𝑋) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍})) |
76 | 75 | imbi1d 331 |
. . . . . . . 8
⊢ ((𝑖 = 𝑋 ∧ 𝑗 = 𝑋) → ((∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})) ↔ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))) |
77 | 71, 76 | syl5ibrcom 237 |
. . . . . . 7
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → ((𝑖 = 𝑋 ∧ 𝑗 = 𝑋) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))) |
78 | 39, 44, 47, 77 | ccased 988 |
. . . . . 6
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍) → (((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))) |
79 | 78 | 3adant3 1081 |
. . . . 5
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → (((𝑖 = {𝑍} ∨ 𝑖 = 𝑋) ∧ (𝑗 = {𝑍} ∨ 𝑗 = 𝑋)) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))) |
80 | 34, 79 | sylbid 230 |
. . . 4
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ((𝑖 ∈ (Idl‘𝑅) ∧ 𝑗 ∈ (Idl‘𝑅)) → (∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍})))) |
81 | 80 | ralrimivv 2970 |
. . 3
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ∀𝑖 ∈ (Idl‘𝑅)∀𝑗 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))) |
82 | 2, 6, 7 | ispridl 33833 |
. . . 4
⊢ (𝑅 ∈ RingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑖 ∈ (Idl‘𝑅)∀𝑗 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))) |
83 | 82 | 3ad2ant1 1082 |
. . 3
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → ({𝑍} ∈ (PrIdl‘𝑅) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑖 ∈ (Idl‘𝑅)∀𝑗 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑖 ∀𝑦 ∈ 𝑗 (𝑥𝐻𝑦) ∈ {𝑍} → (𝑖 ⊆ {𝑍} ∨ 𝑗 ⊆ {𝑍}))))) |
84 | 5, 15, 81, 83 | mpbir3and 1245 |
. 2
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → {𝑍} ∈ (PrIdl‘𝑅)) |
85 | 2, 3 | isprrngo 33849 |
. 2
⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅))) |
86 | 1, 84, 85 | sylanbrc 698 |
1
⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → 𝑅 ∈ PrRing) |