Theorem ismntoplly 33973

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 482	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → 𝑁 ∈ ℕ0)
2		simpl 482	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → 𝑛 = 𝑁)
3	2	eleq1d 2829	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑛 ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
4		simpr 484	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
5	4	eleq1d 2829	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑗 ∈ 2ndω ↔ 𝐽 ∈ 2ndω))
6	4	eleq1d 2829	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑗 ∈ Haus ↔ 𝐽 ∈ Haus))
7		2fveq3 6927	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (TopOpen‘(𝔼hil‘𝑛)) = (TopOpen‘(𝔼hil‘𝑁)))
8	7	eceq1d 8805	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → [(TopOpen‘(𝔼hil‘𝑛))] ≃ = [(TopOpen‘(𝔼hil‘𝑁))] ≃ )
9		llyeq 23501	. . . . . . . 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 480	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ = Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )
12	4, 11	eleq12d 2838	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑗 ∈ Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ ↔ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ))
13	5, 6, 12	3anbi123d 1436	. . . 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 631	. . 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 33971	. . 3 ⊢ ManTop = {⟨𝑛, 𝑗⟩ ∣ (𝑛 ∈ ℕ0 ∧ (𝑗 ∈ 2ndω ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ ))}
16	14, 15	brabga 5553	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝑁 ∈ ℕ0 ∧ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ))))
17	1, 16	mpbirand 706	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   class class class wbr 5166  ‘cfv 6575  [cec 8763  ℕ₀cn0 12555  TopOpenctopn 17483  Hauscha 23339  2^ndωc2ndc 23469  Locally clly 23495   ≃ chmph 23785  𝔼_hilcehl 25439  ManTopcmntop 33970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fv 6583  df-ec 8767  df-lly 23497  df-mntop 33971

This theorem is referenced by:  ismntop  33974