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Theorem ismntoplly 30585
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 475 . . . . 5 ((𝑛 = 𝑁𝑗 = 𝐽) → 𝑛 = 𝑁)
21eleq1d 2863 . . . 4 ((𝑛 = 𝑁𝑗 = 𝐽) → (𝑛 ∈ ℕ0𝑁 ∈ ℕ0))
3 simpr 478 . . . . . 6 ((𝑛 = 𝑁𝑗 = 𝐽) → 𝑗 = 𝐽)
43eleq1d 2863 . . . . 5 ((𝑛 = 𝑁𝑗 = 𝐽) → (𝑗 ∈ 2nd𝜔 ↔ 𝐽 ∈ 2nd𝜔))
53eleq1d 2863 . . . . 5 ((𝑛 = 𝑁𝑗 = 𝐽) → (𝑗 ∈ Haus ↔ 𝐽 ∈ Haus))
6 2fveq3 6416 . . . . . . . . 9 (𝑛 = 𝑁 → (TopOpen‘(𝔼hil𝑛)) = (TopOpen‘(𝔼hil𝑁)))
76eceq1d 8021 . . . . . . . 8 (𝑛 = 𝑁 → [(TopOpen‘(𝔼hil𝑛))] ≃ = [(TopOpen‘(𝔼hil𝑁))] ≃ )
8 llyeq 21602 . . . . . . . 8 ([(TopOpen‘(𝔼hil𝑛))] ≃ = [(TopOpen‘(𝔼hil𝑁))] ≃ → Locally [(TopOpen‘(𝔼hil𝑛))] ≃ = Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )
97, 8syl 17 . . . . . . 7 (𝑛 = 𝑁 → Locally [(TopOpen‘(𝔼hil𝑛))] ≃ = Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )
109adantr 473 . . . . . 6 ((𝑛 = 𝑁𝑗 = 𝐽) → Locally [(TopOpen‘(𝔼hil𝑛))] ≃ = Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )
113, 10eleq12d 2872 . . . . 5 ((𝑛 = 𝑁𝑗 = 𝐽) → (𝑗 ∈ Locally [(TopOpen‘(𝔼hil𝑛))] ≃ ↔ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ))
124, 5, 113anbi123d 1561 . . . 4 ((𝑛 = 𝑁𝑗 = 𝐽) → ((𝑗 ∈ 2nd𝜔 ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼hil𝑛))] ≃ ) ↔ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))
132, 12anbi12d 625 . . 3 ((𝑛 = 𝑁𝑗 = 𝐽) → ((𝑛 ∈ ℕ0 ∧ (𝑗 ∈ 2nd𝜔 ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼hil𝑛))] ≃ )) ↔ (𝑁 ∈ ℕ0 ∧ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ))))
14 df-mntop 30583 . . 3 ManTop = {⟨𝑛, 𝑗⟩ ∣ (𝑛 ∈ ℕ0 ∧ (𝑗 ∈ 2nd𝜔 ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼hil𝑛))] ≃ ))}
1513, 14brabga 5185 . 2 ((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝑁 ∈ ℕ0 ∧ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ))))
16 simpl 475 . . 3 ((𝑁 ∈ ℕ0𝐽𝑉) → 𝑁 ∈ ℕ0)
1716biantrurd 529 . 2 ((𝑁 ∈ ℕ0𝐽𝑉) → ((𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ) ↔ (𝑁 ∈ ℕ0 ∧ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ))))
1815, 17bitr4d 274 1 ((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157   class class class wbr 4843  cfv 6101  [cec 7980  0cn0 11580  TopOpenctopn 16397  Hauscha 21441  2nd𝜔c2ndc 21570  Locally clly 21596  chmph 21886  𝔼hilcehl 23510  ManTopcmntop 30582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-xp 5318  df-cnv 5320  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fv 6109  df-ec 7984  df-lly 21598  df-mntop 30583
This theorem is referenced by:  ismntop  30586
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