ismntop - Mathbox for Thierry Arnoux

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Theorem ismntop 34186

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 34185	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2^ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼_hil‘𝑁))] ≃ )))
2		haustop 23306	. . . . . . . . 9 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
3	2	adantl 481	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Top)
4	3	biantrurd 532	. . . . . . 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 23759	. . . . . . . . . . . . 13 ⊢ ≃ Er Top
6		errel 8646	. . . . . . . . . . . . 13 ⊢ ( ≃ Er Top → Rel ≃ )
7		relelec 8684	. . . . . . . . . . . . 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 23761	. . . . . . . . . . . 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 631	. . . . . . . . 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 3162	. . . . . . . 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 3202	. . . . . . 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 23443	. . . . . . . 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 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 631	. . . 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 480	. 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 206 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 ∩ cin 3889 𝒫 cpw 4542 class class class wbr 5086 Rel wrel 5629 ‘cfv 6492 (class class class)co 7360 Er wer 8633 [cec 8634 ℕ₀cn0 12428 ↾_t crest 17374 TopOpenctopn 17375 Topctop 22868 Hauscha 23283 2^ndωc2ndc 23413 Locally clly 23439 ≃ chmph 23729 𝔼_hilcehl 25361 ManTopcmntop 34182

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7363 df-oprab 7364 df-mpo 7365 df-1st 7935 df-2nd 7936 df-1o 8398 df-er 8636 df-ec 8638 df-map 8768 df-top 22869 df-topon 22886 df-cn 23202 df-haus 23290 df-lly 23441 df-hmeo 23730 df-hmph 23731 df-mntop 34183

This theorem is referenced by: (None)

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