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Theorem ismntoplly 31340
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 486 . 2 ((𝑁 ∈ ℕ0𝐽𝑉) → 𝑁 ∈ ℕ0)
2 simpl 486 . . . . 5 ((𝑛 = 𝑁𝑗 = 𝐽) → 𝑛 = 𝑁)
32eleq1d 2898 . . . 4 ((𝑛 = 𝑁𝑗 = 𝐽) → (𝑛 ∈ ℕ0𝑁 ∈ ℕ0))
4 simpr 488 . . . . . 6 ((𝑛 = 𝑁𝑗 = 𝐽) → 𝑗 = 𝐽)
54eleq1d 2898 . . . . 5 ((𝑛 = 𝑁𝑗 = 𝐽) → (𝑗 ∈ 2ndω ↔ 𝐽 ∈ 2ndω))
64eleq1d 2898 . . . . 5 ((𝑛 = 𝑁𝑗 = 𝐽) → (𝑗 ∈ Haus ↔ 𝐽 ∈ Haus))
7 2fveq3 6657 . . . . . . . . 9 (𝑛 = 𝑁 → (TopOpen‘(𝔼hil𝑛)) = (TopOpen‘(𝔼hil𝑁)))
87eceq1d 8315 . . . . . . . 8 (𝑛 = 𝑁 → [(TopOpen‘(𝔼hil𝑛))] ≃ = [(TopOpen‘(𝔼hil𝑁))] ≃ )
9 llyeq 22073 . . . . . . . 8 ([(TopOpen‘(𝔼hil𝑛))] ≃ = [(TopOpen‘(𝔼hil𝑁))] ≃ → Locally [(TopOpen‘(𝔼hil𝑛))] ≃ = Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )
108, 9syl 17 . . . . . . 7 (𝑛 = 𝑁 → Locally [(TopOpen‘(𝔼hil𝑛))] ≃ = Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )
1110adantr 484 . . . . . 6 ((𝑛 = 𝑁𝑗 = 𝐽) → Locally [(TopOpen‘(𝔼hil𝑛))] ≃ = Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )
124, 11eleq12d 2908 . . . . 5 ((𝑛 = 𝑁𝑗 = 𝐽) → (𝑗 ∈ Locally [(TopOpen‘(𝔼hil𝑛))] ≃ ↔ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ))
135, 6, 123anbi123d 1433 . . . 4 ((𝑛 = 𝑁𝑗 = 𝐽) → ((𝑗 ∈ 2ndω ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼hil𝑛))] ≃ ) ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))
143, 13anbi12d 633 . . 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 31338 . . 3 ManTop = {⟨𝑛, 𝑗⟩ ∣ (𝑛 ∈ ℕ0 ∧ (𝑗 ∈ 2ndω ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼hil𝑛))] ≃ ))}
1614, 15brabga 5398 . 2 ((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝑁 ∈ ℕ0 ∧ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ ))))
171, 16mpbirand 706 1 ((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114   class class class wbr 5042  cfv 6334  [cec 8274  0cn0 11885  TopOpenctopn 16686  Hauscha 21911  2ndωc2ndc 22041  Locally clly 22067  chmph 22357  𝔼hilcehl 23986  ManTopcmntop 31337
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-xp 5538  df-cnv 5540  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fv 6342  df-ec 8278  df-lly 22069  df-mntop 31338
This theorem is referenced by:  ismntop  31341
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