Theorem gpgprismgr4cycllem4

Description: Lemma 4 for gpgprismgr4cycl0 : the cycle <. P , F >. consists of 5 vertices (the first and the last vertex are identical, see gpgprismgr4cycllem6 . (Contributed by AV, 1-Nov-2025)

	Ref	Expression
Hypothesis	gpgprismgr4cycl.p	⊢ 𝑃 = ⟨“ ⟨ 0 , 0 ⟩ ⟨ 0 , 1 ⟩ ⟨ 1 , 1 ⟩ ⟨ 1 , 0 ⟩ ⟨ 0 , 0 ⟩ ”⟩
Assertion	gpgprismgr4cycllem4	⊢ ( ♯ ‘ 𝑃 ) = 5

Proof

Step	Hyp	Ref	Expression
1		gpgprismgr4cycl.p	⊢ 𝑃 = ⟨“ ⟨ 0 , 0 ⟩ ⟨ 0 , 1 ⟩ ⟨ 1 , 1 ⟩ ⟨ 1 , 0 ⟩ ⟨ 0 , 0 ⟩ ”⟩
2	1	fveq2i	⊢ ( ♯ ‘ 𝑃 ) = ( ♯ ‘ ⟨“ ⟨ 0 , 0 ⟩ ⟨ 0 , 1 ⟩ ⟨ 1 , 1 ⟩ ⟨ 1 , 0 ⟩ ⟨ 0 , 0 ⟩ ”⟩ )
3		s5len	⊢ ( ♯ ‘ ⟨“ ⟨ 0 , 0 ⟩ ⟨ 0 , 1 ⟩ ⟨ 1 , 1 ⟩ ⟨ 1 , 0 ⟩ ⟨ 0 , 0 ⟩ ”⟩ ) = 5
4	2 3	eqtri	⊢ ( ♯ ‘ 𝑃 ) = 5