Proof of Theorem isnrm3
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | nrmtop 23345 | . . 3
⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top) | 
| 2 |  | nrmsep 23366 | . . . . . 6
⊢ ((𝐽 ∈ Nrm ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽) ∧ (𝑐 ∩ 𝑑) = ∅)) → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) | 
| 3 | 2 | 3exp2 1354 | . . . . 5
⊢ (𝐽 ∈ Nrm → (𝑐 ∈ (Clsd‘𝐽) → (𝑑 ∈ (Clsd‘𝐽) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))))) | 
| 4 | 3 | impd 410 | . . . 4
⊢ (𝐽 ∈ Nrm → ((𝑐 ∈ (Clsd‘𝐽) ∧ 𝑑 ∈ (Clsd‘𝐽)) → ((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))) | 
| 5 | 4 | ralrimivv 3199 | . . 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 511 | . 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 482 | . . 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 1194 | . . . . . . . . . 10
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑐 ⊆ 𝑥) | 
| 9 |  | simpr2 1195 | . . . . . . . . . . . 12
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑑 ⊆ 𝑦) | 
| 10 |  | sslin 4242 | . . . . . . . . . . . 12
⊢ (𝑑 ⊆ 𝑦 → (((cls‘𝐽)‘𝑥) ∩ 𝑑) ⊆ (((cls‘𝐽)‘𝑥) ∩ 𝑦)) | 
| 11 | 9, 10 | syl 17 | . . . . . . . . . . 11
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) ⊆ (((cls‘𝐽)‘𝑥) ∩ 𝑦)) | 
| 12 |  | eqid 2736 | . . . . . . . . . . . . . 14
⊢ ∪ 𝐽 =
∪ 𝐽 | 
| 13 | 12 | opncld 23042 | . . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽)) | 
| 14 | 13 | ad4ant13 751 | . . . . . . . . . . . 12
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (∪ 𝐽
∖ 𝑦) ∈
(Clsd‘𝐽)) | 
| 15 |  | simpr3 1196 | . . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (𝑥 ∩ 𝑦) = ∅) | 
| 16 |  | simpllr 775 | . . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑥 ∈ 𝐽) | 
| 17 |  | elssuni 4936 | . . . . . . . . . . . . . 14
⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽) | 
| 18 |  | reldisj 4452 | . . . . . . . . . . . . . 14
⊢ (𝑥 ⊆ ∪ 𝐽
→ ((𝑥 ∩ 𝑦) = ∅ ↔ 𝑥 ⊆ (∪ 𝐽
∖ 𝑦))) | 
| 19 | 16, 17, 18 | 3syl 18 | . . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → ((𝑥 ∩ 𝑦) = ∅ ↔ 𝑥 ⊆ (∪ 𝐽 ∖ 𝑦))) | 
| 20 | 15, 19 | mpbid 232 | . . . . . . . . . . . 12
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → 𝑥 ⊆ (∪ 𝐽 ∖ 𝑦)) | 
| 21 | 12 | clsss2 23081 | . . . . . . . . . . . . 13
⊢ (((∪ 𝐽
∖ 𝑦) ∈
(Clsd‘𝐽) ∧ 𝑥 ⊆ (∪ 𝐽
∖ 𝑦)) →
((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽
∖ 𝑦)) | 
| 22 |  | ssdifin0 4485 | . . . . . . . . . . . . 13
⊢
(((cls‘𝐽)‘𝑥) ⊆ (∪ 𝐽 ∖ 𝑦) → (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅) | 
| 23 | 21, 22 | syl 17 | . . . . . . . . . . . 12
⊢ (((∪ 𝐽
∖ 𝑦) ∈
(Clsd‘𝐽) ∧ 𝑥 ⊆ (∪ 𝐽
∖ 𝑦)) →
(((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅) | 
| 24 | 14, 20, 23 | syl2anc 584 | . . . . . . . . . . 11
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅) | 
| 25 |  | sseq0 4402 | . . . . . . . . . . 11
⊢
(((((cls‘𝐽)‘𝑥) ∩ 𝑑) ⊆ (((cls‘𝐽)‘𝑥) ∩ 𝑦) ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑦) = ∅) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅) | 
| 26 | 11, 24, 25 | syl2anc 584 | . . . . . . . . . 10
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅) | 
| 27 | 8, 26 | jca 511 | . . . . . . . . 9
⊢ ((((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝐽) ∧ (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)) → (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)) | 
| 28 | 27 | rexlimdva2 3156 | . . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))) | 
| 29 | 28 | reximdva 3167 | . . . . . . 7
⊢ (𝐽 ∈ Top → (∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅) → ∃𝑥 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅))) | 
| 30 | 29 | 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) = ∅)))) | 
| 31 | 30 | ralimdv 3168 | . . . . 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) = ∅)))) | 
| 32 | 31 | ralimdv 3168 | . . . 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) = ∅)))) | 
| 33 | 32 | imp 406 | . . 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) = ∅))) | 
| 34 |  | isnrm2 23367 | . . 3
⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ (((cls‘𝐽)‘𝑥) ∩ 𝑑) = ∅)))) | 
| 35 | 7, 33, 34 | sylanbrc 583 | . 2
⊢ ((𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))) → 𝐽 ∈ Nrm) | 
| 36 | 6, 35 | impbii 209 | 1
⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))) |