Theorem ismntoplly 34209

Description: Property of being a manifold. (Contributed by Thierry Arnoux, 28-Dec-2019.)

Assertion

Ref	Expression
ismntoplly	⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))

Proof of Theorem ismntoplly

Dummy variables 𝑗 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 483	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → 𝑁 ∈ ℕ0)
2		simpl 483	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → 𝑛 = 𝑁)
3	2	eleq1d 2824	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑛 ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
4		simpr 485	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
5	4	eleq1d 2824	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑗 ∈ 2ndω ↔ 𝐽 ∈ 2ndω))
6	4	eleq1d 2824	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑗 ∈ Haus ↔ 𝐽 ∈ Haus))
7		2fveq3 6832	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (TopOpen‘(𝔼hil‘𝑛)) = (TopOpen‘(𝔼hil‘𝑁)))
8	7	eceq1d 8674	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → [(TopOpen‘(𝔼hil‘𝑛))] ≃ = [(TopOpen‘(𝔼hil‘𝑁))] ≃ )
9		llyeq 23453	. . . . . . . 8 ⊢ ([(TopOpen‘(𝔼hil‘𝑛))] ≃ = [(TopOpen‘(𝔼hil‘𝑁))] ≃ → Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ = Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )
10	8, 9	syl 17	. . . . . . 7 ⊢ (𝑛 = 𝑁 → Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ = Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )
11	10	adantr 481	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ = Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )
12	4, 11	eleq12d 2833	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑗 ∈ Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ ↔ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ))
13	5, 6, 12	3anbi123d 1444	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → ((𝑗 ∈ 2ndω ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ ) ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))
14	3, 13	anbi12d 638	. . 3 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → ((𝑛 ∈ ℕ0 ∧ (𝑗 ∈ 2ndω ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ )) ↔ (𝑁 ∈ ℕ0 ∧ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ))))
15		df-mntop 34207	. . 3 ⊢ ManTop = {⟨𝑛, 𝑗⟩ ∣ (𝑛 ∈ ℕ0 ∧ (𝑗 ∈ 2ndω ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ ))}
16	14, 15	brabga 5476	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝑁 ∈ ℕ0 ∧ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ))))
17	1, 16	mpbirand 713	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   class class class wbr 5072  ‘cfv 6485  [cec 8631  ℕ₀cn0 12428  TopOpenctopn 17375  Hauscha 23291  2^ndωc2ndc 23421  Locally clly 23447   ≃ chmph 23737  𝔼_hilcehl 25369  ManTopcmntop 34206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-xp 5624  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fv 6493  df-ec 8635  df-lly 23449  df-mntop 34207

This theorem is referenced by:  ismntop  34210