Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ismntoplly Structured version   Visualization version   GIF version

Theorem ismntoplly 30409
Description: Property of being a manifold. (Contributed by Thierry Arnoux, 28-Dec-2019.)
Assertion
Ref Expression
ismntoplly ((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))

Proof of Theorem ismntoplly
Dummy variables 𝑗 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 468 . . . . 5 ((𝑛 = 𝑁𝑗 = 𝐽) → 𝑛 = 𝑁)
21eleq1d 2835 . . . 4 ((𝑛 = 𝑁𝑗 = 𝐽) → (𝑛 ∈ ℕ0𝑁 ∈ ℕ0))
3 simpr 471 . . . . . 6 ((𝑛 = 𝑁𝑗 = 𝐽) → 𝑗 = 𝐽)
43eleq1d 2835 . . . . 5 ((𝑛 = 𝑁𝑗 = 𝐽) → (𝑗 ∈ 2nd𝜔 ↔ 𝐽 ∈ 2nd𝜔))
53eleq1d 2835 . . . . 5 ((𝑛 = 𝑁𝑗 = 𝐽) → (𝑗 ∈ Haus ↔ 𝐽 ∈ Haus))
6 fveq2 6332 . . . . . . . . . 10 (𝑛 = 𝑁 → (𝔼hil𝑛) = (𝔼hil𝑁))
76fveq2d 6336 . . . . . . . . 9 (𝑛 = 𝑁 → (TopOpen‘(𝔼hil𝑛)) = (TopOpen‘(𝔼hil𝑁)))
87eceq1d 7935 . . . . . . . 8 (𝑛 = 𝑁 → [(TopOpen‘(𝔼hil𝑛))] ≃ = [(TopOpen‘(𝔼hil𝑁))] ≃ )
9 llyeq 21494 . . . . . . . 8 ([(TopOpen‘(𝔼hil𝑛))] ≃ = [(TopOpen‘(𝔼hil𝑁))] ≃ → Locally [(TopOpen‘(𝔼hil𝑛))] ≃ = Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )
108, 9syl 17 . . . . . . 7 (𝑛 = 𝑁 → Locally [(TopOpen‘(𝔼hil𝑛))] ≃ = Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )
1110adantr 466 . . . . . 6 ((𝑛 = 𝑁𝑗 = 𝐽) → Locally [(TopOpen‘(𝔼hil𝑛))] ≃ = Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )
123, 11eleq12d 2844 . . . . 5 ((𝑛 = 𝑁𝑗 = 𝐽) → (𝑗 ∈ Locally [(TopOpen‘(𝔼hil𝑛))] ≃ ↔ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ))
134, 5, 123anbi123d 1547 . . . 4 ((𝑛 = 𝑁𝑗 = 𝐽) → ((𝑗 ∈ 2nd𝜔 ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼hil𝑛))] ≃ ) ↔ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))
142, 13anbi12d 616 . . 3 ((𝑛 = 𝑁𝑗 = 𝐽) → ((𝑛 ∈ ℕ0 ∧ (𝑗 ∈ 2nd𝜔 ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼hil𝑛))] ≃ )) ↔ (𝑁 ∈ ℕ0 ∧ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ))))
15 df-mntop 30407 . . 3 ManTop = {⟨𝑛, 𝑗⟩ ∣ (𝑛 ∈ ℕ0 ∧ (𝑗 ∈ 2nd𝜔 ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼hil𝑛))] ≃ ))}
1614, 15brabga 5122 . 2 ((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝑁 ∈ ℕ0 ∧ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ))))
17 simpl 468 . . 3 ((𝑁 ∈ ℕ0𝐽𝑉) → 𝑁 ∈ ℕ0)
1817biantrurd 522 . 2 ((𝑁 ∈ ℕ0𝐽𝑉) → ((𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ) ↔ (𝑁 ∈ ℕ0 ∧ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ))))
1916, 18bitr4d 271 1 ((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145   class class class wbr 4786  cfv 6031  [cec 7894  0cn0 11494  TopOpenctopn 16290  Hauscha 21333  2nd𝜔c2ndc 21462  Locally clly 21488  chmph 21778  𝔼hilcehl 23391  ManTopcmntop 30406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fv 6039  df-ec 7898  df-lly 21490  df-mntop 30407
This theorem is referenced by:  ismntop  30410
  Copyright terms: Public domain W3C validator