Theorem ist1-5lem 23776

Description: Lemma for ist1-5 23778 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 23775	. . . . 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 23738	. . . . 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 2114   class class class wbr 5100  ‘cfv 6500  Kol2ct0 23262  KQckq 23649   ≃ chmph 23710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-1o 8407  df-map 8777  df-qtop 17440  df-top 22850  df-topon 22867  df-cn 23183  df-t0 23269  df-kq 23650  df-hmeo 23711  df-hmph 23712

This theorem is referenced by:  ist1-5  23778  ishaus3  23779