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Theorem kqnrm 22352
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 21936 . . . 4 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
2 toptopon2 21518 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
31, 2sylib 220 . . 3 (𝐽 ∈ Nrm → 𝐽 ∈ (TopOn‘ 𝐽))
4 eqid 2819 . . . 4 (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦}) = (𝑥 𝐽 ↦ {𝑦𝐽𝑥𝑦})
54kqnrmlem1 22343 . . 3 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm)
63, 5mpancom 686 . 2 (𝐽 ∈ Nrm → (KQ‘𝐽) ∈ Nrm)
7 nrmtop 21936 . . . . 5 ((KQ‘𝐽) ∈ Nrm → (KQ‘𝐽) ∈ Top)
8 kqtop 22345 . . . . 5 (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Top)
97, 8sylibr 236 . . . 4 ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ Top)
109, 2sylib 220 . . 3 ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ (TopOn‘ 𝐽))
114kqnrmlem2 22344 . . 3 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)
1210, 11mpancom 686 . 2 ((KQ‘𝐽) ∈ Nrm → 𝐽 ∈ Nrm)
136, 12impbii 211 1 (𝐽 ∈ Nrm ↔ (KQ‘𝐽) ∈ Nrm)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wcel 2108  {crab 3140   cuni 4830  cmpt 5137  cfv 6348  Topctop 21493  TopOnctopon 21510  Nrmcnrm 21910  KQckq 22293
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-map 8400  df-qtop 16772  df-top 21494  df-topon 21511  df-cld 21619  df-cls 21621  df-cn 21827  df-nrm 21917  df-kq 22294
This theorem is referenced by: (None)
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