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Theorem kqnrm 23600
Description: The Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
kqnrm (𝐽 ∈ Nrm ↔ (KQβ€˜π½) ∈ Nrm)

Proof of Theorem kqnrm
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nrmtop 23184 . . . 4 (𝐽 ∈ Nrm β†’ 𝐽 ∈ Top)
2 toptopon2 22764 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
31, 2sylib 217 . . 3 (𝐽 ∈ Nrm β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
4 eqid 2724 . . . 4 (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦}) = (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
54kqnrmlem1 23591 . . 3 ((𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) ∧ 𝐽 ∈ Nrm) β†’ (KQβ€˜π½) ∈ Nrm)
63, 5mpancom 685 . 2 (𝐽 ∈ Nrm β†’ (KQβ€˜π½) ∈ Nrm)
7 nrmtop 23184 . . . . 5 ((KQβ€˜π½) ∈ Nrm β†’ (KQβ€˜π½) ∈ Top)
8 kqtop 23593 . . . . 5 (𝐽 ∈ Top ↔ (KQβ€˜π½) ∈ Top)
97, 8sylibr 233 . . . 4 ((KQβ€˜π½) ∈ Nrm β†’ 𝐽 ∈ Top)
109, 2sylib 217 . . 3 ((KQβ€˜π½) ∈ Nrm β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
114kqnrmlem2 23592 . . 3 ((𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) ∧ (KQβ€˜π½) ∈ Nrm) β†’ 𝐽 ∈ Nrm)
1210, 11mpancom 685 . 2 ((KQβ€˜π½) ∈ Nrm β†’ 𝐽 ∈ Nrm)
136, 12impbii 208 1 (𝐽 ∈ Nrm ↔ (KQβ€˜π½) ∈ Nrm)
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∈ wcel 2098  {crab 3424  βˆͺ cuni 4900   ↦ cmpt 5222  β€˜cfv 6534  Topctop 22739  TopOnctopon 22756  Nrmcnrm 23158  KQckq 23541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-iin 4991  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819  df-qtop 17458  df-top 22740  df-topon 22757  df-cld 22867  df-cls 22869  df-cn 23075  df-nrm 23165  df-kq 23542
This theorem is referenced by: (None)
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