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Mirrors > Home > MPE Home > Th. List > ist1-5lem | Structured version Visualization version GIF version |
Description: Lemma for ist1-5 22427 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 22424 | . . . . 5 ⊢ (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽)) | |
3 | 1, 2 | sylib 221 | . . . 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 515 | . 2 ⊢ (𝐽 ∈ 𝐴 → (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴)) |
7 | hmphsym 22387 | . . . . 5 ⊢ (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽) | |
8 | 2, 7 | sylbi 220 | . . . 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 410 | . 2 ⊢ ((𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴) → 𝐽 ∈ 𝐴) |
12 | 6, 11 | impbii 212 | 1 ⊢ (𝐽 ∈ 𝐴 ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 Kol2ct0 21911 KQckq 22298 ≃ chmph 22359 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-1o 8085 df-map 8391 df-qtop 16772 df-top 21499 df-topon 21516 df-cn 21832 df-t0 21918 df-kq 22299 df-hmeo 22360 df-hmph 22361 |
This theorem is referenced by: ist1-5 22427 ishaus3 22428 |
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