Theorem ist1-5lem 23043

Description: Lemma for ist1-5 23045 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 23042	. . . . 5 ⊢ (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽))
3	1, 2	sylib 217	. . . 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 512	. 2 ⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))
7		hmphsym 23005	. . . . 5 ⊢ (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽)
8	2, 7	sylbi 216	. . . 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 407	. 2 ⊢ ((𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴) → 𝐽 ∈ 𝐴)
12	6, 11	impbii 208	1 ⊢ (𝐽 ∈ 𝐴 ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2105   class class class wbr 5087  ‘cfv 6465  Kol2ct0 22529  KQckq 22916   ≃ chmph 22977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7628

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-iun 4939  df-br 5088  df-opab 5150  df-mpt 5171  df-id 5507  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-ov 7318  df-oprab 7319  df-mpo 7320  df-1st 7876  df-2nd 7877  df-1o 8344  df-map 8665  df-qtop 17288  df-top 22115  df-topon 22132  df-cn 22450  df-t0 22536  df-kq 22917  df-hmeo 22978  df-hmph 22979

This theorem is referenced by:  ist1-5  23045  ishaus3  23046