Theorem tmachlem-exlargecover

Description: Product topology of tapes has an open cover. (Contributed by Ender Ting, 28-Jul-2026)

	Ref	Expression
Hypotheses	tmach.finalph	⊢ ( 𝜑 → 𝑈 ∈ Fin )
	tmach.exindex	⊢ ( 𝜑 → 𝐼 ∈ V )
	tmach.tapelist	⊢ ( 𝜑 → 𝑇 = ( 𝑈 ↑m 𝐼 ) )
	tmach.scanmap	⊢ ( 𝜑 → 𝑆 : 𝑇 ⟶ ( 𝒫 𝐼 ∩ Fin ) )
	tmach.agreemap	⊢ ( 𝜑 → 𝐴 = ( 𝑧 ∈ 𝑇 ↦ { 𝑦 ∈ 𝑇 ∣ ( 𝑦 ↾ ( 𝑆 ‘ 𝑧 ) ) = ( 𝑧 ↾ ( 𝑆 ‘ 𝑧 ) ) } ) )
	tmach.agreement	⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝑇 ∀ 𝑦 ∈ ( 𝐴 ‘ 𝑧 ) ( 𝑆 ‘ 𝑦 ) = ( 𝑆 ‘ 𝑧 ) )
Assertion	tmachlem-exlargecover	⊢ ( 𝜑 → ∪ ran 𝐴 = 𝑇 )

Proof

Step	Hyp	Ref	Expression
1		tmach.finalph	⊢ ( 𝜑 → 𝑈 ∈ Fin )
2		tmach.exindex	⊢ ( 𝜑 → 𝐼 ∈ V )
3		tmach.tapelist	⊢ ( 𝜑 → 𝑇 = ( 𝑈 ↑m 𝐼 ) )
4		tmach.scanmap	⊢ ( 𝜑 → 𝑆 : 𝑇 ⟶ ( 𝒫 𝐼 ∩ Fin ) )
5		tmach.agreemap	⊢ ( 𝜑 → 𝐴 = ( 𝑧 ∈ 𝑇 ↦ { 𝑦 ∈ 𝑇 ∣ ( 𝑦 ↾ ( 𝑆 ‘ 𝑧 ) ) = ( 𝑧 ↾ ( 𝑆 ‘ 𝑧 ) ) } ) )
6		tmach.agreement	⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝑇 ∀ 𝑦 ∈ ( 𝐴 ‘ 𝑧 ) ( 𝑆 ‘ 𝑦 ) = ( 𝑆 ‘ 𝑧 ) )
7	1 2 3 4 5 6	tmachlem-uassst	⊢ ( 𝜑 → ∪ ran 𝐴 ⊆ 𝑇 )
8	1 2 3 4 5 6	tmachlem-agreeself	⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑇 ) → 𝑎 ∈ ( 𝐴 ‘ 𝑎 ) )
9		elfvunirn	⊢ ( 𝑎 ∈ ( 𝐴 ‘ 𝑎 ) → 𝑎 ∈ ∪ ran 𝐴 )
10	8 9	syl	⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑇 ) → 𝑎 ∈ ∪ ran 𝐴 )
11	7 10	eqelssd	⊢ ( 𝜑 → ∪ ran 𝐴 = 𝑇 )