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Theorem ismntop 30044
Description: Property of being a manifold. (Contributed by Thierry Arnoux, 5-Jan-2020.)
Assertion
Ref Expression
ismntop ((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))))))
Distinct variable groups:   𝑢,𝐽,𝑥,𝑦   𝑢,𝑁,𝑥,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑢)

Proof of Theorem ismntop
StepHypRef Expression
1 ismntoplly 30043 . 2 ((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))
2 haustop 21116 . . . . . . . . 9 (𝐽 ∈ Haus → 𝐽 ∈ Top)
32adantl 482 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → 𝐽 ∈ Top)
43biantrurd 529 . . . . . . 7 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ ) ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ ))))
5 hmpher 21568 . . . . . . . . . . . . 13 ≃ Er Top
6 errel 7736 . . . . . . . . . . . . 13 ( ≃ Er Top → Rel ≃ )
7 relelec 7772 . . . . . . . . . . . . 13 (Rel ≃ → ((𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ ↔ (TopOpen‘(𝔼hil𝑁)) ≃ (𝐽t 𝑢)))
85, 6, 7mp2b 10 . . . . . . . . . . . 12 ((𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ ↔ (TopOpen‘(𝔼hil𝑁)) ≃ (𝐽t 𝑢))
9 hmphsymb 21570 . . . . . . . . . . . 12 ((TopOpen‘(𝔼hil𝑁)) ≃ (𝐽t 𝑢) ↔ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)))
108, 9bitr2i 265 . . . . . . . . . . 11 ((𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)) ↔ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )
1110a1i 11 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → ((𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)) ↔ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ ))
1211anbi2d 739 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → ((𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))) ↔ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )))
1312rexbidv 3048 . . . . . . . 8 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))) ↔ ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )))
14132ralbidv 2986 . . . . . . 7 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))) ↔ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )))
15 islly 21252 . . . . . . . 8 (𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ )))
1615a1i 11 . . . . . . 7 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ [(TopOpen‘(𝔼hil𝑁))] ≃ ))))
174, 14, 163bitr4rd 301 . . . . . 6 ((𝑁 ∈ ℕ0𝐽 ∈ Haus) → (𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ↔ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)))))
1817pm5.32da 672 . . . . 5 (𝑁 ∈ ℕ0 → ((𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ) ↔ (𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))))))
1918anbi2d 739 . . . 4 (𝑁 ∈ ℕ0 → ((𝐽 ∈ 2nd𝜔 ∧ (𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )) ↔ (𝐽 ∈ 2nd𝜔 ∧ (𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)))))))
20 3anass 1040 . . . 4 ((𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ) ↔ (𝐽 ∈ 2nd𝜔 ∧ (𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))
21 3anass 1040 . . . 4 ((𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁)))) ↔ (𝐽 ∈ 2nd𝜔 ∧ (𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))))))
2219, 20, 213bitr4g 303 . . 3 (𝑁 ∈ ℕ0 → ((𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ) ↔ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))))))
2322adantr 481 . 2 ((𝑁 ∈ ℕ0𝐽𝑉) → ((𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ) ↔ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))))))
241, 23bitrd 268 1 ((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wcel 1988  wral 2909  wrex 2910  cin 3566  𝒫 cpw 4149   class class class wbr 4644  Rel wrel 5109  cfv 5876  (class class class)co 6635   Er wer 7724  [cec 7725  0cn0 11277  t crest 16062  TopOpenctopn 16063  Topctop 20679  Hauscha 21093  2nd𝜔c2ndc 21222  Locally clly 21248  chmph 21538  𝔼hilcehl 23153  ManTopcmntop 30040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-1o 7545  df-er 7727  df-ec 7729  df-map 7844  df-top 20680  df-topon 20697  df-cn 21012  df-haus 21100  df-lly 21250  df-hmeo 21539  df-hmph 21540  df-mntop 30041
This theorem is referenced by: (None)
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