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Metamath Proof Explorer

Theorem crctcshlem3

Description: Lemma for crctcsh . (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 crctcshlem3 ( 𝜑 → ( 𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V ) )

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 crctistrl ⊢ ( 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 )
9 trliswlk ⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
10 wlkv ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) )
11 simp1 ⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) → 𝐺 ∈ V )
12 9 10 11 3syl ⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → 𝐺 ∈ V )
13 3 8 12 3syl ⊢ ( 𝜑 → 𝐺 ∈ V )
14 6 ovexi ⊢ 𝐻 ∈ V
15 14 a1i ⊢ ( 𝜑 → 𝐻 ∈ V )
16 ovex ⊢ ( 0 ... 𝑁 ) ∈ V
17 16 mptex ⊢ ( 𝑥 ∈ ( 0 ... 𝑁 ) ↦ if ( 𝑥 ≤ ( 𝑁 − 𝑆 ) , ( 𝑃 ‘ ( 𝑥 + 𝑆 ) ) , ( 𝑃 ‘ ( ( 𝑥 + 𝑆 ) − 𝑁 ) ) ) ) ∈ V
18 7 17 eqeltri ⊢ 𝑄 ∈ V
19 18 a1i ⊢ ( 𝜑 → 𝑄 ∈ V )
20 13 15 19 3jca ⊢ ( 𝜑 → ( 𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V ) )