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Mirrors > Home > MPE Home > Th. List > cnrmi | Structured version Visualization version GIF version |
Description: A subspace of a completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
cnrmi | ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Nrm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | restin 21768 | . 2 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽))) |
3 | oveq2 7158 | . . . 4 ⊢ (𝑥 = (𝐴 ∩ ∪ 𝐽) → (𝐽 ↾t 𝑥) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽))) | |
4 | 3 | eleq1d 2897 | . . 3 ⊢ (𝑥 = (𝐴 ∩ ∪ 𝐽) → ((𝐽 ↾t 𝑥) ∈ Nrm ↔ (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ Nrm)) |
5 | 1 | iscnrm 21925 | . . . . 5 ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm)) |
6 | 5 | simprbi 499 | . . . 4 ⊢ (𝐽 ∈ CNrm → ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm) |
7 | 6 | adantr 483 | . . 3 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm) |
8 | inss2 4205 | . . . . 5 ⊢ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽 | |
9 | inex1g 5215 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ ∪ 𝐽) ∈ V) | |
10 | elpwg 4544 | . . . . . 6 ⊢ ((𝐴 ∩ ∪ 𝐽) ∈ V → ((𝐴 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽 ↔ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽)) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝐴 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽 ↔ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽)) |
12 | 8, 11 | mpbiri 260 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽) |
13 | 12 | adantl 484 | . . 3 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽) |
14 | 4, 7, 13 | rspcdva 3624 | . 2 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ Nrm) |
15 | 2, 14 | eqeltrd 2913 | 1 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Nrm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4538 ∪ cuni 4831 (class class class)co 7150 ↾t crest 16688 Topctop 21495 Nrmcnrm 21912 CNrmccnrm 21913 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-rest 16690 df-cnrm 21920 |
This theorem is referenced by: cnrmnrm 21963 restcnrm 21964 |
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