Theorem ismntoplly 34455

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 488	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → 𝑁 ∈ ℕ0)
2		simpl 488	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → 𝑛 = 𝑁)
3	2	eleq1d 2850	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑛 ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
4		simpr 490	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
5	4	eleq1d 2850	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑗 ∈ 2ndω ↔ 𝐽 ∈ 2ndω))
6	4	eleq1d 2850	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑗 ∈ Haus ↔ 𝐽 ∈ Haus))
7		2fveq3 6890	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (TopOpen‘(𝔼hil‘𝑛)) = (TopOpen‘(𝔼hil‘𝑁)))
8	7	eceq1d 8737	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → [(TopOpen‘(𝔼hil‘𝑛))] ≃ = [(TopOpen‘(𝔼hil‘𝑁))] ≃ )
9		llyeq 23658	. . . . . . . 8 ⊢ ([(TopOpen‘(𝔼hil‘𝑛))] ≃ = [(TopOpen‘(𝔼hil‘𝑁))] ≃ → Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ = Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )
10	8, 9	syl 18	. . . . . . 7 ⊢ (𝑛 = 𝑁 → Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ = Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )
11	10	adantr 486	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ = Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )
12	4, 11	eleq12d 2859	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑗 ∈ Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ ↔ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ))
13	5, 6, 12	3anbi123d 1464	. . . 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 644	. . 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 34453	. . 3 ⊢ ManTop = {⟨𝑛, 𝑗⟩ ∣ (𝑛 ∈ ℕ0 ∧ (𝑗 ∈ 2ndω ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ ))}
16	14, 15	brabga 5520	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝑁 ∈ ℕ0 ∧ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ))))
17	1, 16	mpbirand 720	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   class class class wbr 5111  ‘cfv 6540  [cec 8694  ℕ₀cn0 12515  TopOpenctopn 17492  Hauscha 23495  2^ndωc2ndc 23625  Locally clly 23652   ≃ chmph 23942  𝔼_hilcehl 25574  ManTopcmntop 34452

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fv 6548  df-ec 8698  df-lly 23654  df-mntop 34453

This theorem is used by:  ismntop  34456