Theorem 3wlkdlem2

Description: Lemma 2 for 3wlkd . (Contributed by AV, 7-Feb-2021)

	Ref	Expression
Hypotheses	3wlkd.p	⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩
	3wlkd.f	⊢ 𝐹 = ⟨“ 𝐽 𝐾 𝐿 ”⟩
Assertion	3wlkdlem2	⊢ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = { 0 , 1 , 2 }

Proof

Step	Hyp	Ref	Expression
1		3wlkd.p	⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩
2		3wlkd.f	⊢ 𝐹 = ⟨“ 𝐽 𝐾 𝐿 ”⟩
3	2	fveq2i	⊢ ( ♯ ‘ 𝐹 ) = ( ♯ ‘ ⟨“ 𝐽 𝐾 𝐿 ”⟩ )
4		s3len	⊢ ( ♯ ‘ ⟨“ 𝐽 𝐾 𝐿 ”⟩ ) = 3
5	3 4	eqtri	⊢ ( ♯ ‘ 𝐹 ) = 3
6	5	oveq2i	⊢ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 3 )
7		fzo0to3tp	⊢ ( 0 ..^ 3 ) = { 0 , 1 , 2 }
8	6 7	eqtri	⊢ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = { 0 , 1 , 2 }