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Metamath Proof Explorer

Theorem lfgrwlknloop

Description: In a loop-free graph, each walk has no loops! (Contributed by AV, 2-Feb-2021)

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

Proof

Step Hyp Ref Expression
1 lfgrwlkprop.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
2 lfgriswlk.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
3 wlkv ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) )
4 1 2 lfgriswlk ⊢ ( ( 𝐺 ∈ V ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) )
5 simpl ⊢ ( ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) )
6 5 ralimi ⊢ ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) )
7 6 3ad2ant3 ⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) )
8 4 7 biimtrdi ⊢ ( ( 𝐺 ∈ V ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) )
9 8 ex ⊢ ( 𝐺 ∈ V → ( 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) )
10 9 com23 ⊢ ( 𝐺 ∈ V → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) )
11 10 3ad2ant1 ⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) )
12 3 11 mpcom ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) )
13 12 impcom ⊢ ( ( 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ∧ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) )