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 32025

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 32024	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2^ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼_hil‘𝑁))] ≃ )))
2		haustop 22531	. . . . . . . . 9 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
3	2	adantl 483	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Top)
4	3	biantrurd 534	. . . . . . 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 22984	. . . . . . . . . . . . 13 ⊢ ≃ Er Top
6		errel 8538	. . . . . . . . . . . . 13 ⊢ ( ≃ Er Top → Rel ≃ )
7		relelec 8574	. . . . . . . . . . . . 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 22986	. . . . . . . . . . . 12 ⊢ ((TopOpen‘(𝔼_hil‘𝑁)) ≃ (𝐽 ↾_t 𝑢) ↔ (𝐽 ↾_t 𝑢) ≃ (TopOpen‘(𝔼_hil‘𝑁)))
10	8, 9	bitr2i 276	. . . . . . . . . . 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 630	. . . . . . . . 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 3171	. . . . . . . 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 3208	. . . . . . 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 22668	. . . . . . . 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 312	. . . . . 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 580	. . . . 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 630	. . . 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 1095	. . . 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 1095	. . . 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 314	. . 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 482	. 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 279	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 205 ∧ wa 397 ∧ w3a 1087 ∈ wcel 2104 ∀wral 3061 ∃wrex 3070 ∩ cin 3891 𝒫 cpw 4539 class class class wbr 5081 Rel wrel 5605 ‘cfv 6458 (class class class)co 7307 Er wer 8526 [cec 8527 ℕ₀cn0 12283 ↾_t crest 17180 TopOpenctopn 17181 Topctop 22091 Hauscha 22508 2^ndωc2ndc 22638 Locally clly 22664 ≃ chmph 22954 𝔼_hilcehl 24597 ManTopcmntop 32021

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3333 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5500 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-ov 7310 df-oprab 7311 df-mpo 7312 df-1st 7863 df-2nd 7864 df-1o 8328 df-er 8529 df-ec 8531 df-map 8648 df-top 22092 df-topon 22109 df-cn 22427 df-haus 22515 df-lly 22666 df-hmeo 22955 df-hmph 22956 df-mntop 32022

This theorem is referenced by: (None)

W3C validator