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Mirrors > Home > MPE Home > Th. List > kqnrm | Structured version Visualization version GIF version |
Description: The Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
kqnrm | ⊢ (𝐽 ∈ Nrm ↔ (KQ‘𝐽) ∈ Nrm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nrmtop 21938 | . . . 4 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top) | |
2 | toptopon2 21520 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
3 | 1, 2 | sylib 220 | . . 3 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
4 | eqid 2821 | . . . 4 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | |
5 | 4 | kqnrmlem1 22345 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm) |
6 | 3, 5 | mpancom 686 | . 2 ⊢ (𝐽 ∈ Nrm → (KQ‘𝐽) ∈ Nrm) |
7 | nrmtop 21938 | . . . . 5 ⊢ ((KQ‘𝐽) ∈ Nrm → (KQ‘𝐽) ∈ Top) | |
8 | kqtop 22347 | . . . . 5 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top) | |
9 | 7, 8 | sylibr 236 | . . . 4 ⊢ ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ Top) |
10 | 9, 2 | sylib 220 | . . 3 ⊢ ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
11 | 4 | kqnrmlem2 22346 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm) |
12 | 10, 11 | mpancom 686 | . 2 ⊢ ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ Nrm) |
13 | 6, 12 | impbii 211 | 1 ⊢ (𝐽 ∈ Nrm ↔ (KQ‘𝐽) ∈ Nrm) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2110 {crab 3142 ∪ cuni 4832 ↦ cmpt 5139 ‘cfv 6350 Topctop 21495 TopOnctopon 21512 Nrmcnrm 21912 KQckq 22295 |
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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-qtop 16774 df-top 21496 df-topon 21513 df-cld 21621 df-cls 21623 df-cn 21829 df-nrm 21919 df-kq 22296 |
This theorem is referenced by: (None) |
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