ismntop - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ismntop		Structured version Visualization version GIF version

Theorem ismntop 34416

Description: Property of being a manifold. (Contributed by Thierry Arnoux, 5-Jan-2020.)

**Assertion**
Ref	Expression
ismntop	⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2^ndω ∧ 𝐽 ∈ Haus ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ≃ (TopOpen‘(𝔼_hil‘𝑁))))))

Distinct variable groups: 𝑢,𝐽,𝑥,𝑦 𝑢,𝑁,𝑥,𝑦 Allowed substitution hints: 𝑉(𝑥,𝑦,𝑢)

**Proof of Theorem ismntop**
Step	Hyp	Ref	Expression
1		ismntoplly 34415	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2^ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼_hil‘𝑁))] ≃ )))
2		haustop 23488	. . . . . . . . 9 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
3	2	adantl 486	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Top)
4	3	biantrurd 541	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ Haus) → (∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ∈ [(TopOpen‘(𝔼_hil‘𝑁))] ≃ ) ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ∈ [(TopOpen‘(𝔼_hil‘𝑁))] ≃ ))))
5		hmpher 23941	. . . . . . . . . . . . 13 ⊢ ≃ Er Top
6		errel 8700	. . . . . . . . . . . . 13 ⊢ ( ≃ Er Top → Rel ≃ )
7		relelec 8738	. . . . . . . . . . . . 13 ⊢ (Rel ≃ → ((𝐽 ↾_t 𝑢) ∈ [(TopOpen‘(𝔼_hil‘𝑁))] ≃ ↔ (TopOpen‘(𝔼_hil‘𝑁)) ≃ (𝐽 ↾_t 𝑢)))
8	5, 6, 7	mp2b 10	. . . . . . . . . . . 12 ⊢ ((𝐽 ↾_t 𝑢) ∈ [(TopOpen‘(𝔼_hil‘𝑁))] ≃ ↔ (TopOpen‘(𝔼_hil‘𝑁)) ≃ (𝐽 ↾_t 𝑢))
9		hmphsymb 23943	. . . . . . . . . . . 12 ⊢ ((TopOpen‘(𝔼_hil‘𝑁)) ≃ (𝐽 ↾_t 𝑢) ↔ (𝐽 ↾_t 𝑢) ≃ (TopOpen‘(𝔼_hil‘𝑁)))
10	8, 9	bitr2i 279	. . . . . . . . . . 11 ⊢ ((𝐽 ↾_t 𝑢) ≃ (TopOpen‘(𝔼_hil‘𝑁)) ↔ (𝐽 ↾_t 𝑢) ∈ [(TopOpen‘(𝔼_hil‘𝑁))] ≃ )
11	10	a1i 11	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ Haus) → ((𝐽 ↾_t 𝑢) ≃ (TopOpen‘(𝔼_hil‘𝑁)) ↔ (𝐽 ↾_t 𝑢) ∈ [(TopOpen‘(𝔼_hil‘𝑁))] ≃ ))
12	11	anbi2d 641	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ Haus) → ((𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ≃ (TopOpen‘(𝔼_hil‘𝑁))) ↔ (𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ∈ [(TopOpen‘(𝔼_hil‘𝑁))] ≃ )))
13	12	rexbidv 3189	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ Haus) → (∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ≃ (TopOpen‘(𝔼_hil‘𝑁))) ↔ ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ∈ [(TopOpen‘(𝔼_hil‘𝑁))] ≃ )))
14	13	2ralbidv 3229	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ Haus) → (∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ≃ (TopOpen‘(𝔼_hil‘𝑁))) ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ∈ [(TopOpen‘(𝔼_hil‘𝑁))] ≃ )))
15		islly 23625	. . . . . . . 8 ⊢ (𝐽 ∈ Locally [(TopOpen‘(𝔼_hil‘𝑁))] ≃ ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ∈ [(TopOpen‘(𝔼_hil‘𝑁))] ≃ )))
16	15	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ Haus) → (𝐽 ∈ Locally [(TopOpen‘(𝔼_hil‘𝑁))] ≃ ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ∈ [(TopOpen‘(𝔼_hil‘𝑁))] ≃ ))))
17	4, 14, 16	3bitr4rd 315	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ Haus) → (𝐽 ∈ Locally [(TopOpen‘(𝔼_hil‘𝑁))] ≃ ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ≃ (TopOpen‘(𝔼_hil‘𝑁)))))
18	17	pm5.32da 589	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → ((𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼_hil‘𝑁))] ≃ ) ↔ (𝐽 ∈ Haus ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ≃ (TopOpen‘(𝔼_hil‘𝑁))))))
19	18	anbi2d 641	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → ((𝐽 ∈ 2^ndω ∧ (𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼_hil‘𝑁))] ≃ )) ↔ (𝐽 ∈ 2^ndω ∧ (𝐽 ∈ Haus ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ≃ (TopOpen‘(𝔼_hil‘𝑁)))))))
20		3anass 1111	. . . 4 ⊢ ((𝐽 ∈ 2^ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼_hil‘𝑁))] ≃ ) ↔ (𝐽 ∈ 2^ndω ∧ (𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼_hil‘𝑁))] ≃ )))
21		3anass 1111	. . . 4 ⊢ ((𝐽 ∈ 2^ndω ∧ 𝐽 ∈ Haus ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ≃ (TopOpen‘(𝔼_hil‘𝑁)))) ↔ (𝐽 ∈ 2^ndω ∧ (𝐽 ∈ Haus ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ≃ (TopOpen‘(𝔼_hil‘𝑁))))))
22	19, 20, 21	3bitr4g 317	. . 3 ⊢ (𝑁 ∈ ℕ₀ → ((𝐽 ∈ 2^ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼_hil‘𝑁))] ≃ ) ↔ (𝐽 ∈ 2^ndω ∧ 𝐽 ∈ Haus ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ≃ (TopOpen‘(𝔼_hil‘𝑁))))))
23	22	adantr 485	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ 𝑉) → ((𝐽 ∈ 2^ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼_hil‘𝑁))] ≃ ) ↔ (𝐽 ∈ 2^ndω ∧ 𝐽 ∈ Haus ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ≃ (TopOpen‘(𝔼_hil‘𝑁))))))
24	1, 23	bitrd 282	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2^ndω ∧ 𝐽 ∈ Haus ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾_t 𝑢) ≃ (TopOpen‘(𝔼_hil‘𝑁))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 ∩ cin 3904 𝒫 cpw 4562 class class class wbr 5109 Rel wrel 5666 ‘cfv 6536 (class class class)co 7410 Er wer 8687 [cec 8688 ℕ₀cn0 12499 ↾_t crest 17468 TopOpenctopn 17469 Topctop 23050 Hauscha 23465 2^ndωc2ndc 23595 Locally clly 23621 ≃ chmph 23911 𝔼_hilcehl 25543 ManTopcmntop 34412

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-1st 7982 df-2nd 7983 df-1o 8449 df-er 8690 df-ec 8692 df-map 8822 df-top 23051 df-topon 23068 df-cn 23384 df-haus 23472 df-lly 23623 df-hmeo 23912 df-hmph 23913 df-mntop 34413

This theorem is referenced by: (None)

W3C validator