Theorem ismntoplly 34283

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 486	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → 𝑁 ∈ ℕ0)
2		simpl 486	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → 𝑛 = 𝑁)
3	2	eleq1d 2846	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑛 ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
4		simpr 488	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
5	4	eleq1d 2846	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑗 ∈ 2ndω ↔ 𝐽 ∈ 2ndω))
6	4	eleq1d 2846	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑗 ∈ Haus ↔ 𝐽 ∈ Haus))
7		2fveq3 6867	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (TopOpen‘(𝔼hil‘𝑛)) = (TopOpen‘(𝔼hil‘𝑁)))
8	7	eceq1d 8713	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → [(TopOpen‘(𝔼hil‘𝑛))] ≃ = [(TopOpen‘(𝔼hil‘𝑁))] ≃ )
9		llyeq 23518	. . . . . . . 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 484	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ = Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )
12	4, 11	eleq12d 2855	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑗 = 𝐽) → (𝑗 ∈ Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ ↔ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ))
13	5, 6, 12	3anbi123d 1456	. . . 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 641	. . 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 34281	. . 3 ⊢ ManTop = {⟨𝑛, 𝑗⟩ ∣ (𝑛 ∈ ℕ0 ∧ (𝑗 ∈ 2ndω ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ ))}
16	14, 15	brabga 5501	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝑁 ∈ ℕ0 ∧ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ ))))
17	1, 16	mpbirand 717	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   class class class wbr 5097  ‘cfv 6516  [cec 8670  ℕ₀cn0 12475  TopOpenctopn 17441  Hauscha 23356  2^ndωc2ndc 23486  Locally clly 23512   ≃ chmph 23802  𝔼_hilcehl 25434  ManTopcmntop 34280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-xp 5649  df-cnv 5651  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fv 6524  df-ec 8674  df-lly 23514  df-mntop 34281

This theorem is referenced by:  ismntop  34284