ismntop - Mathbox for Thierry Arnoux

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Theorem ismntop 33001

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 33000	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2^ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼_hil‘𝑁))] ≃ )))
2		haustop 22834	. . . . . . . . 9 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
3	2	adantl 482	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Top)
4	3	biantrurd 533	. . . . . . 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 23287	. . . . . . . . . . . . 13 ⊢ ≃ Er Top
6		errel 8711	. . . . . . . . . . . . 13 ⊢ ( ≃ Er Top → Rel ≃ )
7		relelec 8747	. . . . . . . . . . . . 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 23289	. . . . . . . . . . . 12 ⊢ ((TopOpen‘(𝔼_hil‘𝑁)) ≃ (𝐽 ↾_t 𝑢) ↔ (𝐽 ↾_t 𝑢) ≃ (TopOpen‘(𝔼_hil‘𝑁)))
10	8, 9	bitr2i 275	. . . . . . . . . . 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 629	. . . . . . . . 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 3178	. . . . . . . 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 3218	. . . . . . 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 22971	. . . . . . . 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 311	. . . . . 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 579	. . . . 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 629	. . . 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 313	. . 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 481	. 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 278	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 396 ∧ w3a 1087 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 ∩ cin 3947 𝒫 cpw 4602 class class class wbr 5148 Rel wrel 5681 ‘cfv 6543 (class class class)co 7408 Er wer 8699 [cec 8700 ℕ₀cn0 12471 ↾_t crest 17365 TopOpenctopn 17366 Topctop 22394 Hauscha 22811 2^ndωc2ndc 22941 Locally clly 22967 ≃ chmph 23257 𝔼_hilcehl 24900 ManTopcmntop 32997

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-1o 8465 df-er 8702 df-ec 8704 df-map 8821 df-top 22395 df-topon 22412 df-cn 22730 df-haus 22818 df-lly 22969 df-hmeo 23258 df-hmph 23259 df-mntop 32998

This theorem is referenced by: (None)

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