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| Mirrors > Home > MPE Home > Th. List > restcnrm | Structured version Visualization version GIF version | ||
| Description: A subspace of a completely normal space is completely normal. (Contributed by Mario Carneiro, 26-Aug-2015.) |
| Ref | Expression |
|---|---|
| restcnrm | ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ CNrm) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 2 | 1 | restin 23174 | . 2 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽))) |
| 3 | simpll 767 | . . . . . 6 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → 𝐽 ∈ CNrm) | |
| 4 | elpwi 4607 | . . . . . . 7 ⊢ (𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽) → 𝑥 ⊆ (𝐴 ∩ ∪ 𝐽)) | |
| 5 | 4 | adantl 481 | . . . . . 6 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → 𝑥 ⊆ (𝐴 ∩ ∪ 𝐽)) |
| 6 | inex1g 5319 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ ∪ 𝐽) ∈ V) | |
| 7 | 6 | ad2antlr 727 | . . . . . 6 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → (𝐴 ∩ ∪ 𝐽) ∈ V) |
| 8 | restabs 23173 | . . . . . 6 ⊢ ((𝐽 ∈ CNrm ∧ 𝑥 ⊆ (𝐴 ∩ ∪ 𝐽) ∧ (𝐴 ∩ ∪ 𝐽) ∈ V) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) = (𝐽 ↾t 𝑥)) | |
| 9 | 3, 5, 7, 8 | syl3anc 1373 | . . . . 5 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) = (𝐽 ↾t 𝑥)) |
| 10 | cnrmi 23368 | . . . . . 6 ⊢ ((𝐽 ∈ CNrm ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → (𝐽 ↾t 𝑥) ∈ Nrm) | |
| 11 | 10 | adantlr 715 | . . . . 5 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → (𝐽 ↾t 𝑥) ∈ Nrm) |
| 12 | 9, 11 | eqeltrd 2841 | . . . 4 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) ∈ Nrm) |
| 13 | 12 | ralrimiva 3146 | . . 3 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → ∀𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) ∈ Nrm) |
| 14 | cnrmtop 23345 | . . . . . . 7 ⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Top) | |
| 15 | 14 | adantr 480 | . . . . . 6 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → 𝐽 ∈ Top) |
| 16 | toptopon2 22924 | . . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
| 17 | 15, 16 | sylib 218 | . . . . 5 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
| 18 | inss2 4238 | . . . . 5 ⊢ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽 | |
| 19 | resttopon 23169 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽) → (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ (TopOn‘(𝐴 ∩ ∪ 𝐽))) | |
| 20 | 17, 18, 19 | sylancl 586 | . . . 4 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ (TopOn‘(𝐴 ∩ ∪ 𝐽))) |
| 21 | iscnrm2 23346 | . . . 4 ⊢ ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ (TopOn‘(𝐴 ∩ ∪ 𝐽)) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) ∈ Nrm)) | |
| 22 | 20, 21 | syl 17 | . . 3 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ CNrm ↔ ∀𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) ∈ Nrm)) |
| 23 | 13, 22 | mpbird 257 | . 2 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ CNrm) |
| 24 | 2, 23 | eqeltrd 2841 | 1 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ CNrm) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 ‘cfv 6561 (class class class)co 7431 ↾t crest 17465 Topctop 22899 TopOnctopon 22916 Nrmcnrm 23318 CNrmccnrm 23319 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-en 8986 df-fin 8989 df-fi 9451 df-rest 17467 df-topgen 17488 df-top 22900 df-topon 22917 df-bases 22953 df-cnrm 23326 |
| This theorem is referenced by: (None) |
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