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Theorem ist1-5lem 23828
Description: Lemma for ist1-5 23830 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 23827 . . . . 5 (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽))
31, 2sylib 218 . . . 4 (𝐽𝐴𝐽 ≃ (KQ‘𝐽))
4 ist1-5lem.2 . . . 4 (𝐽 ≃ (KQ‘𝐽) → (𝐽𝐴 → (KQ‘𝐽) ∈ 𝐴))
53, 4mpcom 38 . . 3 (𝐽𝐴 → (KQ‘𝐽) ∈ 𝐴)
61, 5jca 511 . 2 (𝐽𝐴 → (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))
7 hmphsym 23790 . . . . 5 (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽)
82, 7sylbi 217 . . . 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 406 . 2 ((𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴) → 𝐽𝐴)
126, 11impbii 209 1 (𝐽𝐴 ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108   class class class wbr 5143  cfv 6561  Kol2ct0 23314  KQckq 23701  chmph 23762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-1o 8506  df-map 8868  df-qtop 17552  df-top 22900  df-topon 22917  df-cn 23235  df-t0 23321  df-kq 23702  df-hmeo 23763  df-hmph 23764
This theorem is referenced by:  ist1-5  23830  ishaus3  23831
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