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Theorem kqnrm 21307
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 20892 . . . 4 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2 eqid 2609 . . . . 5 𝐽 = 𝐽
32toptopon 20490 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
41, 3sylib 206 . . 3 (𝐽 ∈ Nrm → 𝐽 ∈ (TopOn‘ 𝐽))
5 eqid 2609 . . . 4 (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})
65kqnrmlem1 21298 . . 3 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm)
74, 6mpancom 699 . 2 (𝐽 ∈ Nrm → (KQ‘𝐽) ∈ Nrm)
8 nrmtop 20892 . . . . 5 ((KQ‘𝐽) ∈ Nrm → (KQ‘𝐽) ∈ Top)
9 kqtop 21300 . . . . 5 (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
108, 9sylibr 222 . . . 4 ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ Top)
1110, 3sylib 206 . . 3 ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ (TopOn‘ 𝐽))
125kqnrmlem2 21299 . . 3 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)
1311, 12mpancom 699 . 2 ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ Nrm)
147, 13impbii 197 1 (𝐽 ∈ Nrm ↔ (KQ‘𝐽) ∈ Nrm)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wcel 1976  {crab 2899   cuni 4366  cmpt 4637  cfv 5790  Topctop 20459  TopOnctopon 20460  Nrmcnrm 20866  KQckq 21248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-map 7723  df-qtop 15936  df-top 20463  df-topon 20465  df-cld 20575  df-cls 20577  df-cn 20783  df-nrm 20873  df-kq 21249
This theorem is referenced by: (None)
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