Theorem gpgprismgr4cycllem1

Description: Lemma 1 for gpgprismgr4cycl0 : the cycle <. P , F >. consists of 4 edges (i.e., has length 4). (Contributed by AV, 1-Nov-2025)

	Ref	Expression
Hypothesis	gpgprismgr4cycllem1.f	⊢ 𝐹 = ⟨“ { ⟨ 0 , 0 ⟩ , ⟨ 0 , 1 ⟩ } { ⟨ 0 , 1 ⟩ , ⟨ 1 , 1 ⟩ } { ⟨ 1 , 1 ⟩ , ⟨ 1 , 0 ⟩ } { ⟨ 1 , 0 ⟩ , ⟨ 0 , 0 ⟩ } ”⟩
Assertion	gpgprismgr4cycllem1	⊢ ( ♯ ‘ 𝐹 ) = 4

Proof

Step	Hyp	Ref	Expression
1		gpgprismgr4cycllem1.f	⊢ 𝐹 = ⟨“ { ⟨ 0 , 0 ⟩ , ⟨ 0 , 1 ⟩ } { ⟨ 0 , 1 ⟩ , ⟨ 1 , 1 ⟩ } { ⟨ 1 , 1 ⟩ , ⟨ 1 , 0 ⟩ } { ⟨ 1 , 0 ⟩ , ⟨ 0 , 0 ⟩ } ”⟩
2	1	fveq2i	⊢ ( ♯ ‘ 𝐹 ) = ( ♯ ‘ ⟨“ { ⟨ 0 , 0 ⟩ , ⟨ 0 , 1 ⟩ } { ⟨ 0 , 1 ⟩ , ⟨ 1 , 1 ⟩ } { ⟨ 1 , 1 ⟩ , ⟨ 1 , 0 ⟩ } { ⟨ 1 , 0 ⟩ , ⟨ 0 , 0 ⟩ } ”⟩ )
3		s4len	⊢ ( ♯ ‘ ⟨“ { ⟨ 0 , 0 ⟩ , ⟨ 0 , 1 ⟩ } { ⟨ 0 , 1 ⟩ , ⟨ 1 , 1 ⟩ } { ⟨ 1 , 1 ⟩ , ⟨ 1 , 0 ⟩ } { ⟨ 1 , 0 ⟩ , ⟨ 0 , 0 ⟩ } ”⟩ ) = 4
4	2 3	eqtri	⊢ ( ♯ ‘ 𝐹 ) = 4