ismntop - Mathbox for Thierry Arnoux

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Theorem ismntop 34480

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 34479	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2^ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼_hil‘𝑁))] ≃ )))
2		haustop 23538	. . . . . . . . 9 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
3	2	adantl 487	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Top)
4	3	biantrurd 542	. . . . . . 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 23992	. . . . . . . . . . . . 13 ⊢ ≃ Er Top
6		errel 8710	. . . . . . . . . . . . 13 ⊢ ( ≃ Er Top → Rel ≃ )
7		relelec 8748	. . . . . . . . . . . . 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 23994	. . . . . . . . . . . 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 642	. . . . . . . . 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 3191	. . . . . . . 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 3231	. . . . . . 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 23676	. . . . . . . 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 590	. . . . 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 642	. . . 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 486	. 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 ∈ wcel 2146 ∀wral 3081 ∃wrex 3091 ∩ cin 3905 𝒫 cpw 4564 class class class wbr 5111 Rel wrel 5668 ‘cfv 6540 (class class class)co 7419 Er wer 8697 [cec 8698 ℕ₀cn0 12519 ↾_t crest 17495 TopOpenctopn 17496 Topctop 23100 Hauscha 23515 2^ndωc2ndc 23645 Locally clly 23672 ≃ chmph 23962 𝔼_hilcehl 25594 ManTopcmntop 34476

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7992 df-2nd 7993 df-1o 8459 df-er 8700 df-ec 8702 df-map 8832 df-top 23101 df-topon 23118 df-cn 23434 df-haus 23522 df-lly 23674 df-hmeo 23963 df-hmph 23964 df-mntop 34477

This theorem is used by: (None)

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