Step | Hyp | Ref
| Expression |
1 | | simp1 1170 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → 𝑅 ∈ Ring) |
2 | | simp3 1172 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵) |
3 | 2 | snssd 4558 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → {𝐺} ⊆ 𝐵) |
4 | | lidldvgen.k |
. . . . . . 7
⊢ 𝐾 = (RSpan‘𝑅) |
5 | | lidldvgen.b |
. . . . . . 7
⊢ 𝐵 = (Base‘𝑅) |
6 | 4, 5 | rspssid 19584 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ {𝐺} ⊆ 𝐵) → {𝐺} ⊆ (𝐾‘{𝐺})) |
7 | 1, 3, 6 | syl2anc 579 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → {𝐺} ⊆ (𝐾‘{𝐺})) |
8 | | snssg 4534 |
. . . . . 6
⊢ (𝐺 ∈ 𝐵 → (𝐺 ∈ (𝐾‘{𝐺}) ↔ {𝐺} ⊆ (𝐾‘{𝐺}))) |
9 | 8 | 3ad2ant3 1169 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → (𝐺 ∈ (𝐾‘{𝐺}) ↔ {𝐺} ⊆ (𝐾‘{𝐺}))) |
10 | 7, 9 | mpbird 249 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ (𝐾‘{𝐺})) |
11 | | lidldvgen.d |
. . . . . . . . . 10
⊢ ∥ =
(∥r‘𝑅) |
12 | 5, 4, 11 | rspsn 19615 |
. . . . . . . . 9
⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐾‘{𝐺}) = {𝑦 ∣ 𝐺 ∥ 𝑦}) |
13 | 12 | 3adant2 1165 |
. . . . . . . 8
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → (𝐾‘{𝐺}) = {𝑦 ∣ 𝐺 ∥ 𝑦}) |
14 | 13 | eleq2d 2892 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝐺}) ↔ 𝑥 ∈ {𝑦 ∣ 𝐺 ∥ 𝑦})) |
15 | | vex 3417 |
. . . . . . . 8
⊢ 𝑥 ∈ V |
16 | | breq2 4877 |
. . . . . . . 8
⊢ (𝑦 = 𝑥 → (𝐺 ∥ 𝑦 ↔ 𝐺 ∥ 𝑥)) |
17 | 15, 16 | elab 3571 |
. . . . . . 7
⊢ (𝑥 ∈ {𝑦 ∣ 𝐺 ∥ 𝑦} ↔ 𝐺 ∥ 𝑥) |
18 | 14, 17 | syl6bb 279 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝐺}) ↔ 𝐺 ∥ 𝑥)) |
19 | 18 | biimpd 221 |
. . . . 5
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝐺}) → 𝐺 ∥ 𝑥)) |
20 | 19 | ralrimiv 3174 |
. . . 4
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → ∀𝑥 ∈ (𝐾‘{𝐺})𝐺 ∥ 𝑥) |
21 | 10, 20 | jca 507 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → (𝐺 ∈ (𝐾‘{𝐺}) ∧ ∀𝑥 ∈ (𝐾‘{𝐺})𝐺 ∥ 𝑥)) |
22 | | eleq2 2895 |
. . . 4
⊢ (𝐼 = (𝐾‘{𝐺}) → (𝐺 ∈ 𝐼 ↔ 𝐺 ∈ (𝐾‘{𝐺}))) |
23 | | raleq 3350 |
. . . 4
⊢ (𝐼 = (𝐾‘{𝐺}) → (∀𝑥 ∈ 𝐼 𝐺 ∥ 𝑥 ↔ ∀𝑥 ∈ (𝐾‘{𝐺})𝐺 ∥ 𝑥)) |
24 | 22, 23 | anbi12d 624 |
. . 3
⊢ (𝐼 = (𝐾‘{𝐺}) → ((𝐺 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 𝐺 ∥ 𝑥) ↔ (𝐺 ∈ (𝐾‘{𝐺}) ∧ ∀𝑥 ∈ (𝐾‘{𝐺})𝐺 ∥ 𝑥))) |
25 | 21, 24 | syl5ibrcom 239 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → (𝐼 = (𝐾‘{𝐺}) → (𝐺 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 𝐺 ∥ 𝑥))) |
26 | | df-ral 3122 |
. . . . . . . 8
⊢
(∀𝑥 ∈
𝐼 𝐺 ∥ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝐼 → 𝐺 ∥ 𝑥)) |
27 | | ssab 3897 |
. . . . . . . 8
⊢ (𝐼 ⊆ {𝑥 ∣ 𝐺 ∥ 𝑥} ↔ ∀𝑥(𝑥 ∈ 𝐼 → 𝐺 ∥ 𝑥)) |
28 | 26, 27 | bitr4i 270 |
. . . . . . 7
⊢
(∀𝑥 ∈
𝐼 𝐺 ∥ 𝑥 ↔ 𝐼 ⊆ {𝑥 ∣ 𝐺 ∥ 𝑥}) |
29 | 28 | biimpi 208 |
. . . . . 6
⊢
(∀𝑥 ∈
𝐼 𝐺 ∥ 𝑥 → 𝐼 ⊆ {𝑥 ∣ 𝐺 ∥ 𝑥}) |
30 | 29 | ad2antll 720 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) ∧ (𝐺 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 𝐺 ∥ 𝑥)) → 𝐼 ⊆ {𝑥 ∣ 𝐺 ∥ 𝑥}) |
31 | 5, 4, 11 | rspsn 19615 |
. . . . . . 7
⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐾‘{𝐺}) = {𝑥 ∣ 𝐺 ∥ 𝑥}) |
32 | 31 | 3adant2 1165 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → (𝐾‘{𝐺}) = {𝑥 ∣ 𝐺 ∥ 𝑥}) |
33 | 32 | adantr 474 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) ∧ (𝐺 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 𝐺 ∥ 𝑥)) → (𝐾‘{𝐺}) = {𝑥 ∣ 𝐺 ∥ 𝑥}) |
34 | 30, 33 | sseqtr4d 3867 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) ∧ (𝐺 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 𝐺 ∥ 𝑥)) → 𝐼 ⊆ (𝐾‘{𝐺})) |
35 | | simpl1 1246 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) ∧ 𝐺 ∈ 𝐼) → 𝑅 ∈ Ring) |
36 | | simpl2 1248 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) ∧ 𝐺 ∈ 𝐼) → 𝐼 ∈ 𝑈) |
37 | | snssi 4557 |
. . . . . . 7
⊢ (𝐺 ∈ 𝐼 → {𝐺} ⊆ 𝐼) |
38 | 37 | adantl 475 |
. . . . . 6
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) ∧ 𝐺 ∈ 𝐼) → {𝐺} ⊆ 𝐼) |
39 | | lidldvgen.u |
. . . . . . 7
⊢ 𝑈 = (LIdeal‘𝑅) |
40 | 4, 39 | rspssp 19587 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ {𝐺} ⊆ 𝐼) → (𝐾‘{𝐺}) ⊆ 𝐼) |
41 | 35, 36, 38, 40 | syl3anc 1494 |
. . . . 5
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) ∧ 𝐺 ∈ 𝐼) → (𝐾‘{𝐺}) ⊆ 𝐼) |
42 | 41 | adantrr 708 |
. . . 4
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) ∧ (𝐺 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 𝐺 ∥ 𝑥)) → (𝐾‘{𝐺}) ⊆ 𝐼) |
43 | 34, 42 | eqssd 3844 |
. . 3
⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) ∧ (𝐺 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 𝐺 ∥ 𝑥)) → 𝐼 = (𝐾‘{𝐺})) |
44 | 43 | ex 403 |
. 2
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → ((𝐺 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 𝐺 ∥ 𝑥) → 𝐼 = (𝐾‘{𝐺}))) |
45 | 25, 44 | impbid 204 |
1
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ∈ 𝐵) → (𝐼 = (𝐾‘{𝐺}) ↔ (𝐺 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 𝐺 ∥ 𝑥))) |