Theorem clwlknf1oclwwlknlem3

Description: Lemma 3 for clwlknf1oclwwlkn : The bijective function of clwlknf1oclwwlkn is the bijective function of clwlkclwwlkf1o restricted to the closed walks with a fixed positive length. (Contributed by AV, 26-May-2022) (Revised by AV, 1-Nov-2022)

	Ref	Expression
Hypotheses	clwlknf1oclwwlkn.a	⊢ 𝐴 = ( 1st ‘ 𝑐 )
	clwlknf1oclwwlkn.b	⊢ 𝐵 = ( 2nd ‘ 𝑐 )
	clwlknf1oclwwlkn.c	⊢ 𝐶 = { 𝑤 ∈ ( ClWalks ‘ 𝐺 ) ∣ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) = 𝑁 }
	clwlknf1oclwwlkn.f	⊢ 𝐹 = ( 𝑐 ∈ 𝐶 ↦ ( 𝐵 prefix ( ♯ ‘ 𝐴 ) ) )
Assertion	clwlknf1oclwwlknlem3	⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ ) → 𝐹 = ( ( 𝑐 ∈ { 𝑤 ∈ ( ClWalks ‘ 𝐺 ) ∣ 1 ≤ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) } ↦ ( 𝐵 prefix ( ♯ ‘ 𝐴 ) ) ) ↾ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		clwlknf1oclwwlkn.a	⊢ 𝐴 = ( 1st ‘ 𝑐 )
2		clwlknf1oclwwlkn.b	⊢ 𝐵 = ( 2nd ‘ 𝑐 )
3		clwlknf1oclwwlkn.c	⊢ 𝐶 = { 𝑤 ∈ ( ClWalks ‘ 𝐺 ) ∣ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) = 𝑁 }
4		clwlknf1oclwwlkn.f	⊢ 𝐹 = ( 𝑐 ∈ 𝐶 ↦ ( 𝐵 prefix ( ♯ ‘ 𝐴 ) ) )
5		nnge1	⊢ ( 𝑁 ∈ ℕ → 1 ≤ 𝑁 )
6		breq2	⊢ ( ( ♯ ‘ ( 1st ‘ 𝑤 ) ) = 𝑁 → ( 1 ≤ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) ↔ 1 ≤ 𝑁 ) )
7	5 6	syl5ibrcom	⊢ ( 𝑁 ∈ ℕ → ( ( ♯ ‘ ( 1st ‘ 𝑤 ) ) = 𝑁 → 1 ≤ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) ) )
8	7	ad2antlr	⊢ ( ( ( 𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ ) ∧ 𝑤 ∈ ( ClWalks ‘ 𝐺 ) ) → ( ( ♯ ‘ ( 1st ‘ 𝑤 ) ) = 𝑁 → 1 ≤ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) ) )
9	8	ss2rabdv	⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ ) → { 𝑤 ∈ ( ClWalks ‘ 𝐺 ) ∣ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) = 𝑁 } ⊆ { 𝑤 ∈ ( ClWalks ‘ 𝐺 ) ∣ 1 ≤ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) } )
10	3 9	eqsstrid	⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ ) → 𝐶 ⊆ { 𝑤 ∈ ( ClWalks ‘ 𝐺 ) ∣ 1 ≤ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) } )
11	10	resmptd	⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ ) → ( ( 𝑐 ∈ { 𝑤 ∈ ( ClWalks ‘ 𝐺 ) ∣ 1 ≤ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) } ↦ ( 𝐵 prefix ( ♯ ‘ 𝐴 ) ) ) ↾ 𝐶 ) = ( 𝑐 ∈ 𝐶 ↦ ( 𝐵 prefix ( ♯ ‘ 𝐴 ) ) ) )
12	4 11	eqtr4id	⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ ) → 𝐹 = ( ( 𝑐 ∈ { 𝑤 ∈ ( ClWalks ‘ 𝐺 ) ∣ 1 ≤ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) } ↦ ( 𝐵 prefix ( ♯ ‘ 𝐴 ) ) ) ↾ 𝐶 ) )