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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 2733 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | restin 22540 | . 2 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽))) |
3 | oveq2 7369 | . . . 4 ⊢ (𝑥 = (𝐴 ∩ ∪ 𝐽) → (𝐽 ↾t 𝑥) = (𝐽 ↾t (𝐴 ∩ ∪ 𝐽))) | |
4 | 3 | eleq1d 2819 | . . 3 ⊢ (𝑥 = (𝐴 ∩ ∪ 𝐽) → ((𝐽 ↾t 𝑥) ∈ Nrm ↔ (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ Nrm)) |
5 | 1 | iscnrm 22697 | . . . . 5 ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm)) |
6 | 5 | simprbi 498 | . . . 4 ⊢ (𝐽 ∈ CNrm → ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm) |
7 | 6 | adantr 482 | . . 3 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm) |
8 | inss2 4193 | . . . . 5 ⊢ (𝐴 ∩ ∪ 𝐽) ⊆ ∪ 𝐽 | |
9 | inex1g 5280 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ ∪ 𝐽) ∈ V) | |
10 | elpwg 4567 | . . . . . 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 258 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽) |
13 | 12 | adantl 483 | . . 3 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ ∪ 𝐽) ∈ 𝒫 ∪ 𝐽) |
14 | 4, 7, 13 | rspcdva 3584 | . 2 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t (𝐴 ∩ ∪ 𝐽)) ∈ Nrm) |
15 | 2, 14 | eqeltrd 2834 | 1 ⊢ ((𝐽 ∈ CNrm ∧ 𝐴 ∈ 𝑉) → (𝐽 ↾t 𝐴) ∈ Nrm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 Vcvv 3447 ∩ cin 3913 ⊆ wss 3914 𝒫 cpw 4564 ∪ cuni 4869 (class class class)co 7361 ↾t crest 17310 Topctop 22265 Nrmcnrm 22684 CNrmccnrm 22685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-rest 17312 df-cnrm 22692 |
This theorem is referenced by: cnrmnrm 22735 restcnrm 22736 |
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