Proof of Theorem isnrm3
Step | Hyp | Ref
| Expression |
1 | | nrmtop 21140 |
. . 3
⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top) |
2 | | nrmsep 21161 |
. . . . . 6
⊢ ((𝐽 ∈ Nrm ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽) ∧ (𝑐 ∩ 𝑑) = ∅)) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) |
3 | 2 | 3exp2 1285 |
. . . . 5
⊢ (𝐽 ∈ Nrm → (𝑐 ∈ (Clsd‘𝐽) → (𝑑 ∈ (Clsd‘𝐽) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))))) |
4 | 3 | impd 447 |
. . . 4
⊢ (𝐽 ∈ Nrm → ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))) |
5 | 4 | ralrimivv 2970 |
. . 3
⊢ (𝐽 ∈ Nrm → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))) |
6 | 1, 5 | jca 554 |
. 2
⊢ (𝐽 ∈ Nrm → (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))) |
7 | | simpl 473 |
. . 3
⊢ ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))) → 𝐽 ∈ Top) |
8 | | simpr1 1067 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑐 ⊆ 𝑥) |
9 | | simpr2 1068 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑑 ⊆ 𝑦) |
10 | | sslin 3839 |
. . . . . . . . . . . . 13
⊢ (𝑑 ⊆ 𝑦 → (((cls‘𝐽)‘𝑥) ∩ 𝑑) ⊆ (((cls‘𝐽)‘𝑥) ∩ 𝑦)) |
11 | 9, 10 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) ⊆ (((cls‘𝐽)‘𝑥) ∩ 𝑦)) |
12 | | simplll 798 |
. . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝐽 ∈ Top) |
13 | | simplr 792 |
. . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑦 ∈ 𝐽) |
14 | | eqid 2622 |
. . . . . . . . . . . . . . 15
⊢ ∪ 𝐽 =
∪ 𝐽 |
15 | 14 | opncld 20837 |
. . . . . . . . . . . . . 14
⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽)) |
16 | 12, 13, 15 | syl2anc 693 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (∪ 𝐽
∖ 𝑦) ∈
(Clsd‘𝐽)) |
17 | | simpr3 1069 |
. . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (𝑥 ∩ 𝑦) = ∅) |
18 | | simpllr 799 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑥 ∈ 𝐽) |
19 | | elssuni 4467 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽) |
20 | | reldisj 4020 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ⊆ ∪ 𝐽
→ ((𝑥 ∩ 𝑦) = ∅ ↔ 𝑥 ⊆ (∪ 𝐽
∖ 𝑦))) |
21 | 18, 19, 20 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → ((𝑥 ∩ 𝑦) = ∅ ↔ 𝑥 ⊆ (∪ 𝐽 ∖ 𝑦))) |
22 | 17, 21 | mpbid 222 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑥 ⊆ (∪ 𝐽 ∖ 𝑦)) |
23 | 14 | clsss2 20876 |
. . . . . . . . . . . . . 14
⊢ (((∪ 𝐽
∖ 𝑦) ∈
(Clsd‘𝐽) ∧ 𝑥 ⊆ (∪ 𝐽
∖ 𝑦)) →
((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽
∖ 𝑦)) |
24 | | ssdifin0 4050 |
. . . . . . . . . . . . . 14
⊢
(((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝑦) → (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅) |
25 | 23, 24 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((∪ 𝐽
∖ 𝑦) ∈
(Clsd‘𝐽) ∧ 𝑥 ⊆ (∪ 𝐽
∖ 𝑦)) →
(((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅) |
26 | 16, 22, 25 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅) |
27 | | sseq0 3975 |
. . . . . . . . . . . 12
⊢
(((((cls‘𝐽)‘𝑥) ∩ 𝑑) ⊆ (((cls‘𝐽)‘𝑥) ∩ 𝑦) ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅) |
28 | 11, 26, 27 | syl2anc 693 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅) |
29 | 8, 28 | jca 554 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)) |
30 | 29 | ex 450 |
. . . . . . . . 9
⊢ (((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) → ((𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))) |
31 | 30 | rexlimdva 3031 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))) |
32 | 31 | reximdva 3017 |
. . . . . . 7
⊢ (𝐽 ∈ Top → (∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅) → ∃𝑥 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))) |
33 | 32 | imim2d 57 |
. . . . . 6
⊢ (𝐽 ∈ Top → (((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))) |
34 | 33 | ralimdv 2963 |
. . . . 5
⊢ (𝐽 ∈ Top →
(∀𝑑 ∈
(Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → ∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))) |
35 | 34 | ralimdv 2963 |
. . . 4
⊢ (𝐽 ∈ Top →
(∀𝑐 ∈
(Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))) |
36 | 35 | imp 445 |
. . 3
⊢ ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))) → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))) |
37 | | isnrm2 21162 |
. . 3
⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))) |
38 | 7, 36, 37 | sylanbrc 698 |
. 2
⊢ ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))) → 𝐽 ∈ Nrm) |
39 | 6, 38 | impbii 199 |
1
⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))) |