Step | Hyp | Ref
| Expression |
1 | | igenval2.1 |
. . . . 5
⊢ 𝐺 = (1st ‘𝑅) |
2 | | igenval2.2 |
. . . . 5
⊢ 𝑋 = ran 𝐺 |
3 | 1, 2 | igenidl 36221 |
. . . 4
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅)) |
4 | 1, 2 | igenss 36220 |
. . . 4
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆)) |
5 | | igenmin 36222 |
. . . . . . 7
⊢ ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝑗) → (𝑅 IdlGen 𝑆) ⊆ 𝑗) |
6 | 5 | 3expia 1120 |
. . . . . 6
⊢ ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) → (𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗)) |
7 | 6 | ralrimiva 3103 |
. . . . 5
⊢ (𝑅 ∈ RingOps →
∀𝑗 ∈
(Idl‘𝑅)(𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗)) |
8 | 7 | adantr 481 |
. . . 4
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗)) |
9 | 3, 4, 8 | 3jca 1127 |
. . 3
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ (𝑅 IdlGen 𝑆) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))) |
10 | | eleq1 2826 |
. . . 4
⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ↔ 𝐼 ∈ (Idl‘𝑅))) |
11 | | sseq2 3947 |
. . . 4
⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → (𝑆 ⊆ (𝑅 IdlGen 𝑆) ↔ 𝑆 ⊆ 𝐼)) |
12 | | sseq1 3946 |
. . . . . 6
⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑅 IdlGen 𝑆) ⊆ 𝑗 ↔ 𝐼 ⊆ 𝑗)) |
13 | 12 | imbi2d 341 |
. . . . 5
⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗) ↔ (𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) |
14 | 13 | ralbidv 3112 |
. . . 4
⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → (∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗) ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) |
15 | 10, 11, 14 | 3anbi123d 1435 |
. . 3
⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → (((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ (𝑅 IdlGen 𝑆) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗)) ↔ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗)))) |
16 | 9, 15 | syl5ibcom 244 |
. 2
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 → (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗)))) |
17 | | igenmin 36222 |
. . . . . 6
⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼) |
18 | 17 | 3adant3r3 1183 |
. . . . 5
⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼) |
19 | 18 | adantlr 712 |
. . . 4
⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼) |
20 | | ssint 4895 |
. . . . . . . 8
⊢ (𝐼 ⊆ ∩ {𝑖
∈ (Idl‘𝑅)
∣ 𝑆 ⊆ 𝑖} ↔ ∀𝑗 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖}𝐼 ⊆ 𝑗) |
21 | | sseq2 3947 |
. . . . . . . . 9
⊢ (𝑖 = 𝑗 → (𝑆 ⊆ 𝑖 ↔ 𝑆 ⊆ 𝑗)) |
22 | 21 | ralrab 3630 |
. . . . . . . 8
⊢
(∀𝑗 ∈
{𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖}𝐼 ⊆ 𝑗 ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗)) |
23 | 20, 22 | sylbbr 235 |
. . . . . . 7
⊢
(∀𝑗 ∈
(Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗) → 𝐼 ⊆ ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖}) |
24 | 23 | 3ad2ant3 1134 |
. . . . . 6
⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗)) → 𝐼 ⊆ ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖}) |
25 | 24 | adantl 482 |
. . . . 5
⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → 𝐼 ⊆ ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖}) |
26 | 1, 2 | igenval 36219 |
. . . . . 6
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖}) |
27 | 26 | adantr 481 |
. . . . 5
⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → (𝑅 IdlGen 𝑆) = ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖}) |
28 | 25, 27 | sseqtrrd 3962 |
. . . 4
⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → 𝐼 ⊆ (𝑅 IdlGen 𝑆)) |
29 | 19, 28 | eqssd 3938 |
. . 3
⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → (𝑅 IdlGen 𝑆) = 𝐼) |
30 | 29 | ex 413 |
. 2
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗)) → (𝑅 IdlGen 𝑆) = 𝐼)) |
31 | 16, 30 | impbid 211 |
1
⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 ↔ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗)))) |