This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem crctcshtrl

Description: Cyclically shifting the indices of a circuit <. F , P >. results in a trail <. H , Q >. . (Contributed by AV, 14-Mar-2021) (Proof shortened by AV, 30-Oct-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 crctcshtrl ( 𝜑 → 𝐻 ( Trails ‘ 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 crctcshwlk ⊢ ( 𝜑 → 𝐻 ( Walks ‘ 𝐺 ) 𝑄 )
9 crctistrl ⊢ ( 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 )
10 2 trlf1 ⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 )
11 df-f1 ⊢ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 ↔ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐼 ∧ Fun ◡ 𝐹 ) )
12 iswrdi ⊢ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐼 → 𝐹 ∈ Word dom 𝐼 )
13 12 anim1i ⊢ ( ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ⟶ dom 𝐼 ∧ Fun ◡ 𝐹 ) → ( 𝐹 ∈ Word dom 𝐼 ∧ Fun ◡ 𝐹 ) )
14 11 13 sylbi ⊢ ( 𝐹 : ( 0 ..^ ( ♯ ‘ 𝐹 ) ) –1-1→ dom 𝐼 → ( 𝐹 ∈ Word dom 𝐼 ∧ Fun ◡ 𝐹 ) )
15 3 9 10 14 4syl ⊢ ( 𝜑 → ( 𝐹 ∈ Word dom 𝐼 ∧ Fun ◡ 𝐹 ) )
16 elfzoelz ⊢ ( 𝑆 ∈ ( 0 ..^ 𝑁 ) → 𝑆 ∈ ℤ )
17 5 16 syl ⊢ ( 𝜑 → 𝑆 ∈ ℤ )
18 df-3an ⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ Fun ◡ 𝐹 ∧ 𝑆 ∈ ℤ ) ↔ ( ( 𝐹 ∈ Word dom 𝐼 ∧ Fun ◡ 𝐹 ) ∧ 𝑆 ∈ ℤ ) )
19 15 17 18 sylanbrc ⊢ ( 𝜑 → ( 𝐹 ∈ Word dom 𝐼 ∧ Fun ◡ 𝐹 ∧ 𝑆 ∈ ℤ ) )
20 cshinj ⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ Fun ◡ 𝐹 ∧ 𝑆 ∈ ℤ ) → ( 𝐻 = ( 𝐹 cyclShift 𝑆 ) → Fun ◡ 𝐻 ) )
21 19 6 20 mpisyl ⊢ ( 𝜑 → Fun ◡ 𝐻 )
22 istrl ⊢ ( 𝐻 ( Trails ‘ 𝐺 ) 𝑄 ↔ ( 𝐻 ( Walks ‘ 𝐺 ) 𝑄 ∧ Fun ◡ 𝐻 ) )
23 8 21 22 sylanbrc ⊢ ( 𝜑 → 𝐻 ( Trails ‘ 𝐺 ) 𝑄 )