Theorem ismntoplly 33985

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 2818	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑛 ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
4		simpr 484	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
5	4	eleq1d 2818	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑗 ∈ 2ndω ↔ 𝐽 ∈ 2ndω))
6	4	eleq1d 2818	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑗 ∈ Haus ↔ 𝐽 ∈ Haus))
7		2fveq3 6891	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (TopOpen‘(𝔼hil‘𝑛)) = (TopOpen‘(𝔼hil‘𝑁)))
8	7	eceq1d 8767	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → [(TopOpen‘(𝔼hil‘𝑛))] ≃ = [(TopOpen‘(𝔼hil‘𝑁))] ≃ )
9		llyeq 23424	. . . . . . . 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 2827	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑗 ∈ Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ ↔ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ))
13	5, 6, 12	3anbi123d 1437	. . . 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 632	. . 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 33983	. . 3 ⊢ ManTop = {⟨𝑛, 𝑗⟩ ∣ (𝑛 ∈ ℕ0 ∧ (𝑗 ∈ 2ndω ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ ))}
16	14, 15	brabga 5519	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝑁 ∈ ℕ0 ∧ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ))))
17	1, 16	mpbirand 707	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   class class class wbr 5123  ‘cfv 6541  [cec 8725  ℕ₀cn0 12509  TopOpenctopn 17437  Hauscha 23262  2^ndωc2ndc 23392  Locally clly 23418   ≃ chmph 23708  𝔼_hilcehl 25354  ManTopcmntop 33982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-xp 5671  df-cnv 5673  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fv 6549  df-ec 8729  df-lly 23420  df-mntop 33983

This theorem is referenced by:  ismntop  33986