Theorem frlmfzolen

Description: The dimension of a vector of a module with indices from 0 to N - 1 . (Contributed by SN, 1-Sep-2023)

	Ref	Expression
Hypotheses	frlmfzowrd.w	⊢ 𝑊 = ( 𝐾 freeLMod ( 0 ..^ 𝑁 ) )
	frlmfzowrd.b	⊢ 𝐵 = ( Base ‘ 𝑊 )
	frlmfzowrd.s	⊢ 𝑆 = ( Base ‘ 𝐾 )
Assertion	frlmfzolen	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) → ( ♯ ‘ 𝑋 ) = 𝑁 )

Proof

Step	Hyp	Ref	Expression
1		frlmfzowrd.w	⊢ 𝑊 = ( 𝐾 freeLMod ( 0 ..^ 𝑁 ) )
2		frlmfzowrd.b	⊢ 𝐵 = ( Base ‘ 𝑊 )
3		frlmfzowrd.s	⊢ 𝑆 = ( Base ‘ 𝐾 )
4		ovexd	⊢ ( 𝑁 ∈ ℕ0 → ( 0 ..^ 𝑁 ) ∈ V )
5	1 3 2	frlmbasf	⊢ ( ( ( 0 ..^ 𝑁 ) ∈ V ∧ 𝑋 ∈ 𝐵 ) → 𝑋 : ( 0 ..^ 𝑁 ) ⟶ 𝑆 )
6	4 5	sylan	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 : ( 0 ..^ 𝑁 ) ⟶ 𝑆 )
7		fnfzo0hash	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑋 : ( 0 ..^ 𝑁 ) ⟶ 𝑆 ) → ( ♯ ‘ 𝑋 ) = 𝑁 )
8	6 7	syldan	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) → ( ♯ ‘ 𝑋 ) = 𝑁 )