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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 2735 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | restin 23190 | . 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 4612 | . . . . . . 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 5325 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ ∪ 𝐽) ∈ V) | |
7 | 6 | ad2antlr 727 | . . . . . 6 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → (𝐴 ∩ ∪ 𝐽) ∈ V) |
8 | restabs 23189 | . . . . . 6 ⊢ ((𝐽 ∈ CNrm ∧ 𝑥 ⊆ (𝐴 ∩ ∪ 𝐽) ∧ (𝐴 ∩ ∪ 𝐽) ∈ V) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) = (𝐽 ↾t 𝑥)) | |
9 | 3, 5, 7, 8 | syl3anc 1370 | . . . . 5 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) = (𝐽 ↾t 𝑥)) |
10 | cnrmi 23384 | . . . . . 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 2839 | . . . 4 ⊢ (((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) ∧ 𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)) → ((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) ∈ Nrm) |
13 | 12 | ralrimiva 3144 | . . 3 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → ∀𝑥 ∈ 𝒫 (𝐴 ∩ ∪ 𝐽)((𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ↾t 𝑥) ∈ Nrm) |
14 | cnrmtop 23361 | . . . . . . 7 ⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Top) | |
15 | 14 | adantr 480 | . . . . . 6 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → 𝐽 ∈ Top) |
16 | toptopon2 22940 | . . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
17 | 15, 16 | sylib 218 | . . . . 5 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
18 | inss2 4246 | . . . . 5 ⊢ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽 | |
19 | resttopon 23185 | . . . . 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 23362 | . . . 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 2839 | 1 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ CNrm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 𝒫 cpw 4605 ∪ cuni 4912 ‘cfv 6563 (class class class)co 7431 ↾t crest 17467 Topctop 22915 TopOnctopon 22932 Nrmcnrm 23334 CNrmccnrm 23335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-en 8985 df-fin 8988 df-fi 9449 df-rest 17469 df-topgen 17490 df-top 22916 df-topon 22933 df-bases 22969 df-cnrm 23342 |
This theorem is referenced by: (None) |
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