Theorem ismntoplly

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

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

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉 ) → 𝑁 ∈ ℕ0 )
2		simpl	⊢ ( ( 𝑛 = 𝑁 ∧ 𝑗 = 𝐽 ) → 𝑛 = 𝑁 )
3	2	eleq1d	⊢ ( ( 𝑛 = 𝑁 ∧ 𝑗 = 𝐽 ) → ( 𝑛 ∈ ℕ0 ↔ 𝑁 ∈ ℕ0 ) )
4		simpr	⊢ ( ( 𝑛 = 𝑁 ∧ 𝑗 = 𝐽 ) → 𝑗 = 𝐽 )
5	4	eleq1d	⊢ ( ( 𝑛 = 𝑁 ∧ 𝑗 = 𝐽 ) → ( 𝑗 ∈ 2ndω ↔ 𝐽 ∈ 2ndω ) )
6	4	eleq1d	⊢ ( ( 𝑛 = 𝑁 ∧ 𝑗 = 𝐽 ) → ( 𝑗 ∈ Haus ↔ 𝐽 ∈ Haus ) )
7		2fveq3	⊢ ( 𝑛 = 𝑁 → ( TopOpen ‘ ( 𝔼hil ‘ 𝑛 ) ) = ( TopOpen ‘ ( 𝔼hil ‘ 𝑁 ) ) )
8	7	eceq1d	⊢ ( 𝑛 = 𝑁 → [ ( TopOpen ‘ ( 𝔼hil ‘ 𝑛 ) ) ] ≃ = [ ( TopOpen ‘ ( 𝔼hil ‘ 𝑁 ) ) ] ≃ )
9		llyeq	⊢ ( [ ( TopOpen ‘ ( 𝔼hil ‘ 𝑛 ) ) ] ≃ = [ ( TopOpen ‘ ( 𝔼hil ‘ 𝑁 ) ) ] ≃ → Locally [ ( TopOpen ‘ ( 𝔼hil ‘ 𝑛 ) ) ] ≃ = Locally [ ( TopOpen ‘ ( 𝔼hil ‘ 𝑁 ) ) ] ≃ )
10	8 9	syl	⊢ ( 𝑛 = 𝑁 → Locally [ ( TopOpen ‘ ( 𝔼hil ‘ 𝑛 ) ) ] ≃ = Locally [ ( TopOpen ‘ ( 𝔼hil ‘ 𝑁 ) ) ] ≃ )
11	10	adantr	⊢ ( ( 𝑛 = 𝑁 ∧ 𝑗 = 𝐽 ) → Locally [ ( TopOpen ‘ ( 𝔼hil ‘ 𝑛 ) ) ] ≃ = Locally [ ( TopOpen ‘ ( 𝔼hil ‘ 𝑁 ) ) ] ≃ )
12	4 11	eleq12d	⊢ ( ( 𝑛 = 𝑁 ∧ 𝑗 = 𝐽 ) → ( 𝑗 ∈ Locally [ ( TopOpen ‘ ( 𝔼hil ‘ 𝑛 ) ) ] ≃ ↔ 𝐽 ∈ Locally [ ( TopOpen ‘ ( 𝔼hil ‘ 𝑁 ) ) ] ≃ ) )
13	5 6 12	3anbi123d	⊢ ( ( 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	⊢ ( ( 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	⊢ ManTop = { ⟨ 𝑛 , 𝑗 ⟩ ∣ ( 𝑛 ∈ ℕ0 ∧ ( 𝑗 ∈ 2ndω ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [ ( TopOpen ‘ ( 𝔼hil ‘ 𝑛 ) ) ] ≃ ) ) }
16	14 15	brabga	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉 ) → ( 𝑁 ManTop 𝐽 ↔ ( 𝑁 ∈ ℕ0 ∧ ( 𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [ ( TopOpen ‘ ( 𝔼hil ‘ 𝑁 ) ) ] ≃ ) ) ) )
17	1 16	mpbirand	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉 ) → ( 𝑁 ManTop 𝐽 ↔ ( 𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [ ( TopOpen ‘ ( 𝔼hil ‘ 𝑁 ) ) ] ≃ ) ) )