Theorem crctcshwlk

Description: Cyclically shifting the indices of a circuit <. F , P >. results in a walk <. H , Q >. . (Contributed by AV, 10-Mar-2021)

	Ref	Expression
Hypotheses	crctcsh.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	crctcsh.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
	crctcsh.d	⊢ ( 𝜑 → 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 )
	crctcsh.n	⊢ 𝑁 = ( ♯ ‘ 𝐹 )
	crctcsh.s	⊢ ( 𝜑 → 𝑆 ∈ ( 0 ..^ 𝑁 ) )
	crctcsh.h	⊢ 𝐻 = ( 𝐹 cyclShift 𝑆 )
	crctcsh.q	⊢ 𝑄 = ( 𝑥 ∈ ( 0 ... 𝑁 ) ↦ if ( 𝑥 ≤ ( 𝑁 − 𝑆 ) , ( 𝑃 ‘ ( 𝑥 + 𝑆 ) ) , ( 𝑃 ‘ ( ( 𝑥 + 𝑆 ) − 𝑁 ) ) ) )
Assertion	crctcshwlk	⊢ ( 𝜑 → 𝐻 ( Walks ‘ 𝐺 ) 𝑄 )

Proof

Step	Hyp	Ref	Expression
1		crctcsh.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		crctcsh.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
3		crctcsh.d	⊢ ( 𝜑 → 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 )
4		crctcsh.n	⊢ 𝑁 = ( ♯ ‘ 𝐹 )
5		crctcsh.s	⊢ ( 𝜑 → 𝑆 ∈ ( 0 ..^ 𝑁 ) )
6		crctcsh.h	⊢ 𝐻 = ( 𝐹 cyclShift 𝑆 )
7		crctcsh.q	⊢ 𝑄 = ( 𝑥 ∈ ( 0 ... 𝑁 ) ↦ if ( 𝑥 ≤ ( 𝑁 − 𝑆 ) , ( 𝑃 ‘ ( 𝑥 + 𝑆 ) ) , ( 𝑃 ‘ ( ( 𝑥 + 𝑆 ) − 𝑁 ) ) ) )
8	1 2 3 4 5 6 7	crctcshlem4	⊢ ( ( 𝜑 ∧ 𝑆 = 0 ) → ( 𝐻 = 𝐹 ∧ 𝑄 = 𝑃 ) )
9		crctistrl	⊢ ( 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 )
10		trliswlk	⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
11	3 9 10	3syl	⊢ ( 𝜑 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
12		breq12	⊢ ( ( 𝐻 = 𝐹 ∧ 𝑄 = 𝑃 ) → ( 𝐻 ( Walks ‘ 𝐺 ) 𝑄 ↔ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) )
13	11 12	syl5ibrcom	⊢ ( 𝜑 → ( ( 𝐻 = 𝐹 ∧ 𝑄 = 𝑃 ) → 𝐻 ( Walks ‘ 𝐺 ) 𝑄 ) )
14	13	adantr	⊢ ( ( 𝜑 ∧ 𝑆 = 0 ) → ( ( 𝐻 = 𝐹 ∧ 𝑄 = 𝑃 ) → 𝐻 ( Walks ‘ 𝐺 ) 𝑄 ) )
15	8 14	mpd	⊢ ( ( 𝜑 ∧ 𝑆 = 0 ) → 𝐻 ( Walks ‘ 𝐺 ) 𝑄 )
16	1 2 3 4 5 6 7	crctcshwlkn0	⊢ ( ( 𝜑 ∧ 𝑆 ≠ 0 ) → 𝐻 ( Walks ‘ 𝐺 ) 𝑄 )
17	15 16	pm2.61dane	⊢ ( 𝜑 → 𝐻 ( Walks ‘ 𝐺 ) 𝑄 )