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Mirrors > Home > MPE Home > Th. List > ist1-5lem | Structured version Visualization version GIF version |
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.) |
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)) |
Ref | Expression |
---|---|
ist1-5lem | ⊢ (𝐽 ∈ 𝐴 ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ist1-5lem.1 | . . 3 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Kol2) | |
2 | kqhmph 21432 | . . . . 5 ⊢ (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽)) | |
3 | 1, 2 | sylib 207 | . . . 4 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ≃ (KQ‘𝐽)) |
4 | ist1-5lem.2 | . . . 4 ⊢ (𝐽 ≃ (KQ‘𝐽) → (𝐽 ∈ 𝐴 → (KQ‘𝐽) ∈ 𝐴)) | |
5 | 3, 4 | mpcom 37 | . . 3 ⊢ (𝐽 ∈ 𝐴 → (KQ‘𝐽) ∈ 𝐴) |
6 | 1, 5 | jca 553 | . 2 ⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴)) |
7 | hmphsym 21395 | . . . . 5 ⊢ (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽) | |
8 | 2, 7 | sylbi 206 | . . . 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 444 | . 2 ⊢ ((𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴) → 𝐽 ∈ 𝐴) |
12 | 6, 11 | impbii 198 | 1 ⊢ (𝐽 ∈ 𝐴 ↔ (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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