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Theorem kqnrm 21365
 Description: The Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
kqnrm (𝐽 ∈ Nrm ↔ (KQ‘𝐽) ∈ Nrm)

Proof of Theorem kqnrm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nrmtop 20950 . . . 4 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2 eqid 2610 . . . . 5 𝐽 = 𝐽
32toptopon 20548 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
41, 3sylib 207 . . 3 (𝐽 ∈ Nrm → 𝐽 ∈ (TopOn‘ 𝐽))
5 eqid 2610 . . . 4 (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})
65kqnrmlem1 21356 . . 3 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm)
74, 6mpancom 700 . 2 (𝐽 ∈ Nrm → (KQ‘𝐽) ∈ Nrm)
8 nrmtop 20950 . . . . 5 ((KQ‘𝐽) ∈ Nrm → (KQ‘𝐽) ∈ Top)
9 kqtop 21358 . . . . 5 (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
108, 9sylibr 223 . . . 4 ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ Top)
1110, 3sylib 207 . . 3 ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ (TopOn‘ 𝐽))
125kqnrmlem2 21357 . . 3 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)
1311, 12mpancom 700 . 2 ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ Nrm)
147, 13impbii 198 1 (𝐽 ∈ Nrm ↔ (KQ‘𝐽) ∈ Nrm)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∈ wcel 1977  {crab 2900  ∪ cuni 4372   ↦ cmpt 4643  ‘cfv 5804  Topctop 20517  TopOnctopon 20518  Nrmcnrm 20924  KQckq 21306 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-qtop 15990  df-top 20521  df-topon 20523  df-cld 20633  df-cls 20635  df-cn 20841  df-nrm 20931  df-kq 21307 This theorem is referenced by: (None)
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