Metamath Proof Explorer

Theorem lfgrwlkprop

Description: Two adjacent vertices in a walk are different in a loop-free graph. (Contributed by AV, 28-Jan-2021)

Ref Expression
Hypothesis lfgrwlkprop.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
Assertion lfgrwlkprop ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) )

Proof

Step Hyp Ref Expression
1 lfgrwlkprop.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
2 wlkv ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) )
3 eqid ⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
4 3 1 iswlk ⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) )
5 2 4 syl ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) )
6 ifptru ⊢ ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) → ( if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ↔ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } ) )
7 6 adantr ⊢ ( ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ↔ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } ) )
8 simplr ⊢ ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } )
9 wrdsymbcl ⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ‘ 𝑘 ) ∈ dom 𝐼 )
10 9 ad4ant14 ⊢ ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ‘ 𝑘 ) ∈ dom 𝐼 )
11 8 10 ffvelrnd ⊢ ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } )
12 fveq2 ⊢ ( 𝑥 = ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) → ( ♯ ‘ 𝑥 ) = ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) )
13 12 breq2d ⊢ ( 𝑥 = ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) → ( 2 ≤ ( ♯ ‘ 𝑥 ) ↔ 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) )
14 13 elrab ⊢ ( ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ↔ ( ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ 𝒫 𝑉 ∧ 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) )
15 fveq2 ⊢ ( ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } → ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) = ( ♯ ‘ { ( 𝑃 ‘ 𝑘 ) } ) )
16 15 breq2d ⊢ ( ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } → ( 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ↔ 2 ≤ ( ♯ ‘ { ( 𝑃 ‘ 𝑘 ) } ) ) )
17 fvex ⊢ ( 𝑃 ‘ 𝑘 ) ∈ V
18 hashsng ⊢ ( ( 𝑃 ‘ 𝑘 ) ∈ V → ( ♯ ‘ { ( 𝑃 ‘ 𝑘 ) } ) = 1 )
19 17 18 ax-mp ⊢ ( ♯ ‘ { ( 𝑃 ‘ 𝑘 ) } ) = 1
20 19 breq2i ⊢ ( 2 ≤ ( ♯ ‘ { ( 𝑃 ‘ 𝑘 ) } ) ↔ 2 ≤ 1 )
21 1lt2 ⊢ 1 < 2
22 1re ⊢ 1 ∈ ℝ
23 2re ⊢ 2 ∈ ℝ
24 22 23 ltnlei ⊢ ( 1 < 2 ↔ ¬ 2 ≤ 1 )
25 pm2.21 ⊢ ( ¬ 2 ≤ 1 → ( 2 ≤ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) )
26 24 25 sylbi ⊢ ( 1 < 2 → ( 2 ≤ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) )
27 21 26 ax-mp ⊢ ( 2 ≤ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) )
28 20 27 sylbi ⊢ ( 2 ≤ ( ♯ ‘ { ( 𝑃 ‘ 𝑘 ) } ) → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) )
29 16 28 syl6bi ⊢ ( ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } → ( 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) )
30 29 com12 ⊢ ( 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) )
31 30 adantl ⊢ ( ( ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ 𝒫 𝑉 ∧ 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) )
32 31 a1i ⊢ ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ 𝒫 𝑉 ∧ 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) )
33 14 32 syl5bi ⊢ ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } → ( ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) )
34 11 33 mpd ⊢ ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) )
35 34 adantl ⊢ ( ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) )
36 7 35 sylbid ⊢ ( ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) )
37 36 ex ⊢ ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) → ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) )
38 neqne ⊢ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) )
39 38 2a1d ⊢ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) → ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) )
40 37 39 pm2.61i ⊢ ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) )
41 40 ralimdva ⊢ ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) )
42 41 ex ⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) )
43 42 com23 ⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) )
44 43 3impia ⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) )
45 5 44 syl6bi ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) )
46 45 pm2.43i ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) )
47 46 imp ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) )