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Theorem ist1-5lem 21433
 Description: Lemma for ist1-5 21435 and similar theorems. If 𝐴 is a topological property which implies T0, such as T1 or T2, the property can be "decomposed" into T0 and a non-T0 version of property 𝐴 (which is defined as stating that the Kolmogorov quotient of the space has property 𝐴). For example, if 𝐴 is T1, then the theorem states that a space is T1 iff it is T0 and its Kolmogorov quotient is T1 (we call this property R0). (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypotheses
Ref Expression
ist1-5lem.1 (𝐽𝐴𝐽 ∈ Kol2)
ist1-5lem.2 (𝐽 ≃ (KQ‘𝐽) → (𝐽𝐴 → (KQ‘𝐽) ∈ 𝐴))
ist1-5lem.3 ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ 𝐴𝐽𝐴))
Assertion
Ref Expression
ist1-5lem (𝐽𝐴 ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))

Proof of Theorem ist1-5lem
StepHypRef Expression
1 ist1-5lem.1 . . 3 (𝐽𝐴𝐽 ∈ Kol2)
2 kqhmph 21432 . . . . 5 (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽))
31, 2sylib 207 . . . 4 (𝐽𝐴𝐽 ≃ (KQ‘𝐽))
4 ist1-5lem.2 . . . 4 (𝐽 ≃ (KQ‘𝐽) → (𝐽𝐴 → (KQ‘𝐽) ∈ 𝐴))
53, 4mpcom 37 . . 3 (𝐽𝐴 → (KQ‘𝐽) ∈ 𝐴)
61, 5jca 553 . 2 (𝐽𝐴 → (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))
7 hmphsym 21395 . . . . 5 (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽)
82, 7sylbi 206 . . . 4 (𝐽 ∈ Kol2 → (KQ‘𝐽) ≃ 𝐽)
9 ist1-5lem.3 . . . 4 ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ 𝐴𝐽𝐴))
108, 9syl 17 . . 3 (𝐽 ∈ Kol2 → ((KQ‘𝐽) ∈ 𝐴𝐽𝐴))
1110imp 444 . 2 ((𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴) → 𝐽𝐴)
126, 11impbii 198 1 (𝐽𝐴 ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  Kol2ct0 20920  KQckq 21306   ≃ chmph 21367 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-1o 7447  df-map 7746  df-qtop 15990  df-top 20521  df-topon 20523  df-cn 20841  df-t0 20927  df-kq 21307  df-hmeo 21368  df-hmph 21369 This theorem is referenced by:  ist1-5  21435  ishaus3  21436
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