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Theorem kqnrm 23010
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 22594 . . . 4 (𝐽 ∈ Nrm β†’ 𝐽 ∈ Top)
2 toptopon2 22174 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
31, 2sylib 217 . . 3 (𝐽 ∈ Nrm β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
4 eqid 2736 . . . 4 (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦}) = (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
54kqnrmlem1 23001 . . 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 22594 . . . . 5 ((KQβ€˜π½) ∈ Nrm β†’ (KQβ€˜π½) ∈ Top)
8 kqtop 23003 . . . . 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 23002 . . 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 2105  {crab 3403  βˆͺ cuni 4853   ↦ cmpt 5176  β€˜cfv 6480  Topctop 22149  TopOnctopon 22166  Nrmcnrm 22568  KQckq 22951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5230  ax-sep 5244  ax-nul 5251  ax-pow 5309  ax-pr 5373  ax-un 7651
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4271  df-if 4475  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4854  df-int 4896  df-iun 4944  df-iin 4945  df-br 5094  df-opab 5156  df-mpt 5177  df-id 5519  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6432  df-fun 6482  df-fn 6483  df-f 6484  df-f1 6485  df-fo 6486  df-f1o 6487  df-fv 6488  df-ov 7341  df-oprab 7342  df-mpo 7343  df-map 8689  df-qtop 17316  df-top 22150  df-topon 22167  df-cld 22277  df-cls 22279  df-cn 22485  df-nrm 22575  df-kq 22952
This theorem is referenced by: (None)
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