Theorem hashwwlksnext

Description: Number of walks (as words) extended by an edge as a sum over the prefixes. (Contributed by Alexander van der Vekens, 21-Aug-2018) (Revised by AV, 20-Apr-2021) (Revised by AV, 26-Oct-2022)

	Ref	Expression
Hypotheses	wwlksnextprop.x	⊢ 𝑋 = ( ( 𝑁 + 1 ) WWalksN 𝐺 )
	wwlksnextprop.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
	wwlksnextprop.y	⊢ 𝑌 = { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑃 }
Assertion	hashwwlksnext	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( ♯ ‘ { 𝑥 ∈ 𝑋 ∣ ∃ 𝑦 ∈ 𝑌 ( ( 𝑥 prefix 𝑀 ) = 𝑦 ∧ ( 𝑦 ‘ 0 ) = 𝑃 ∧ { ( lastS ‘ 𝑦 ) , ( lastS ‘ 𝑥 ) } ∈ 𝐸 ) } ) = Σ 𝑦 ∈ 𝑌 ( ♯ ‘ { 𝑥 ∈ 𝑋 ∣ ( ( 𝑥 prefix 𝑀 ) = 𝑦 ∧ ( 𝑦 ‘ 0 ) = 𝑃 ∧ { ( lastS ‘ 𝑦 ) , ( lastS ‘ 𝑥 ) } ∈ 𝐸 ) } ) )

Proof

Step	Hyp	Ref	Expression
1		wwlksnextprop.x	⊢ 𝑋 = ( ( 𝑁 + 1 ) WWalksN 𝐺 )
2		wwlksnextprop.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
3		wwlksnextprop.y	⊢ 𝑌 = { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑃 }
4		wwlksnfi	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝑁 WWalksN 𝐺 ) ∈ Fin )
5		ssrab2	⊢ { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑃 } ⊆ ( 𝑁 WWalksN 𝐺 )
6		ssfi	⊢ ( ( ( 𝑁 WWalksN 𝐺 ) ∈ Fin ∧ { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑃 } ⊆ ( 𝑁 WWalksN 𝐺 ) ) → { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑃 } ∈ Fin )
7	4 5 6	sylancl	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑃 } ∈ Fin )
8	3 7	eqeltrid	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → 𝑌 ∈ Fin )
9		wwlksnfi	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( ( 𝑁 + 1 ) WWalksN 𝐺 ) ∈ Fin )
10	1 9	eqeltrid	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → 𝑋 ∈ Fin )
11		rabfi	⊢ ( 𝑋 ∈ Fin → { 𝑥 ∈ 𝑋 ∣ ( ( 𝑥 prefix 𝑀 ) = 𝑦 ∧ ( 𝑦 ‘ 0 ) = 𝑃 ∧ { ( lastS ‘ 𝑦 ) , ( lastS ‘ 𝑥 ) } ∈ 𝐸 ) } ∈ Fin )
12	10 11	syl	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → { 𝑥 ∈ 𝑋 ∣ ( ( 𝑥 prefix 𝑀 ) = 𝑦 ∧ ( 𝑦 ‘ 0 ) = 𝑃 ∧ { ( lastS ‘ 𝑦 ) , ( lastS ‘ 𝑥 ) } ∈ 𝐸 ) } ∈ Fin )
13	12	adantr	⊢ ( ( ( Vtx ‘ 𝐺 ) ∈ Fin ∧ 𝑦 ∈ 𝑌 ) → { 𝑥 ∈ 𝑋 ∣ ( ( 𝑥 prefix 𝑀 ) = 𝑦 ∧ ( 𝑦 ‘ 0 ) = 𝑃 ∧ { ( lastS ‘ 𝑦 ) , ( lastS ‘ 𝑥 ) } ∈ 𝐸 ) } ∈ Fin )
14	1 2 3	disjxwwlkn	⊢ Disj 𝑦 ∈ 𝑌 { 𝑥 ∈ 𝑋 ∣ ( ( 𝑥 prefix 𝑀 ) = 𝑦 ∧ ( 𝑦 ‘ 0 ) = 𝑃 ∧ { ( lastS ‘ 𝑦 ) , ( lastS ‘ 𝑥 ) } ∈ 𝐸 ) }
15	14	a1i	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → Disj 𝑦 ∈ 𝑌 { 𝑥 ∈ 𝑋 ∣ ( ( 𝑥 prefix 𝑀 ) = 𝑦 ∧ ( 𝑦 ‘ 0 ) = 𝑃 ∧ { ( lastS ‘ 𝑦 ) , ( lastS ‘ 𝑥 ) } ∈ 𝐸 ) } )
16	8 13 15	hashrabrex	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( ♯ ‘ { 𝑥 ∈ 𝑋 ∣ ∃ 𝑦 ∈ 𝑌 ( ( 𝑥 prefix 𝑀 ) = 𝑦 ∧ ( 𝑦 ‘ 0 ) = 𝑃 ∧ { ( lastS ‘ 𝑦 ) , ( lastS ‘ 𝑥 ) } ∈ 𝐸 ) } ) = Σ 𝑦 ∈ 𝑌 ( ♯ ‘ { 𝑥 ∈ 𝑋 ∣ ( ( 𝑥 prefix 𝑀 ) = 𝑦 ∧ ( 𝑦 ‘ 0 ) = 𝑃 ∧ { ( lastS ‘ 𝑦 ) , ( lastS ‘ 𝑥 ) } ∈ 𝐸 ) } ) )