MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  kqnrm Structured version   Visualization version   GIF version

Theorem kqnrm 23655
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 23239 . . . 4 (𝐽 ∈ Nrm β†’ 𝐽 ∈ Top)
2 toptopon2 22819 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
31, 2sylib 217 . . 3 (𝐽 ∈ Nrm β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
4 eqid 2728 . . . 4 (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦}) = (π‘₯ ∈ βˆͺ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
54kqnrmlem1 23646 . . 3 ((𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) ∧ 𝐽 ∈ Nrm) β†’ (KQβ€˜π½) ∈ Nrm)
63, 5mpancom 687 . 2 (𝐽 ∈ Nrm β†’ (KQβ€˜π½) ∈ Nrm)
7 nrmtop 23239 . . . . 5 ((KQβ€˜π½) ∈ Nrm β†’ (KQβ€˜π½) ∈ Top)
8 kqtop 23648 . . . . 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 23647 . . 3 ((𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) ∧ (KQβ€˜π½) ∈ Nrm) β†’ 𝐽 ∈ Nrm)
1210, 11mpancom 687 . 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 2099  {crab 3429  βˆͺ cuni 4908   ↦ cmpt 5231  β€˜cfv 6548  Topctop 22794  TopOnctopon 22811  Nrmcnrm 23213  KQckq 23596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8846  df-qtop 17488  df-top 22795  df-topon 22812  df-cld 22922  df-cls 22924  df-cn 23130  df-nrm 23220  df-kq 23597
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator