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	⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Kol2)
ist1-5lem.2	⊢ (𝐽 ≃ (KQ‘𝐽) → (𝐽 ∈ 𝐴 → (KQ‘𝐽) ∈ 𝐴))
ist1-5lem.3	⊢ ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ 𝐴 → 𝐽 ∈ 𝐴))

Assertion

Ref	Expression
ist1-5lem	⊢ (𝐽 ∈ 𝐴 ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))

Proof of Theorem ist1-5lem

Step	Hyp	Ref	Expression
1		ist1-5lem.1	. . 3 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Kol2)
2		kqhmph 23827	. . . . 5 ⊢ (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽))
3	1, 2	sylib 218	. . . 4 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ≃ (KQ‘𝐽))
4		ist1-5lem.2	. . . 4 ⊢ (𝐽 ≃ (KQ‘𝐽) → (𝐽 ∈ 𝐴 → (KQ‘𝐽) ∈ 𝐴))
5	3, 4	mpcom 38	. . 3 ⊢ (𝐽 ∈ 𝐴 → (KQ‘𝐽) ∈ 𝐴)
6	1, 5	jca 511	. 2 ⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))
7		hmphsym 23790	. . . . 5 ⊢ (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽)
8	2, 7	sylbi 217	. . . 4 ⊢ (𝐽 ∈ Kol2 → (KQ‘𝐽) ≃ 𝐽)
9		ist1-5lem.3	. . . 4 ⊢ ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ 𝐴 → 𝐽 ∈ 𝐴))
10	8, 9	syl 17	. . 3 ⊢ (𝐽 ∈ Kol2 → ((KQ‘𝐽) ∈ 𝐴 → 𝐽 ∈ 𝐴))
11	10	imp 406	. 2 ⊢ ((𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴) → 𝐽 ∈ 𝐴)
12	6, 11	impbii 209	1 ⊢ (𝐽 ∈ 𝐴 ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))

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