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Metamath Proof Explorer

Theorem uhgrwkspth

Description: Any walk of length 1 between two different vertices is a simple path. (Contributed by AV, 25-Jan-2021) (Proof shortened by AV, 31-Oct-2021) (Revised by AV, 7-Jul-2022)

Ref Expression
Assertion uhgrwkspth ( ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) → ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 ↔ 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 ) )

Proof

Step Hyp Ref Expression
1 simpl31 ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) ) → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
2 uhgrwkspthlem1 ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( ♯ ‘ 𝐹 ) = 1 ) → Fun ◡ 𝐹 )
3 2 expcom ⊢ ( ( ♯ ‘ 𝐹 ) = 1 → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → Fun ◡ 𝐹 ) )
4 3 3ad2ant2 ⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → Fun ◡ 𝐹 ) )
5 4 com12 ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) → Fun ◡ 𝐹 ) )
6 5 3ad2ant1 ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) → ( ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) → Fun ◡ 𝐹 ) )
7 6 3ad2ant3 ⊢ ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) → ( ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) → Fun ◡ 𝐹 ) )
8 7 imp ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) ) → Fun ◡ 𝐹 )
9 istrl ⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝐹 ) )
10 1 8 9 sylanbrc ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) ) → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 )
11 3simpc ⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) → ( ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) )
12 11 adantl ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) ) → ( ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) )
13 3simpc ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) → ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) )
14 13 3ad2ant3 ⊢ ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) → ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) )
15 14 adantr ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) ) → ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) )
16 uhgrwkspthlem2 ⊢ ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) ∧ ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) → Fun ◡ 𝑃 )
17 1 12 15 16 syl3anc ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) ) → Fun ◡ 𝑃 )
18 isspth ⊢ ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝑃 ) )
19 10 17 18 sylanbrc ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) ) → 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 )
20 3anass ⊢ ( ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ↔ ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ∧ ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) )
21 19 15 20 sylanbrc ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) ) → ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) )
22 3simpa ⊢ ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) → ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ) )
23 22 adantr ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) ) → ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ) )
24 eqid ⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
25 24 isspthonpth ⊢ ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ) → ( 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 ↔ ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) )
26 23 25 syl ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) ) → ( 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 ↔ ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) )
27 21 26 mpbird ⊢ ( ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) ∧ ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) ) → 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 )
28 27 ex ⊢ ( ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) → ( ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) → 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 ) )
29 24 wlkonprop ⊢ ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 → ( ( 𝐺 ∈ V ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) )
30 3simpc ⊢ ( ( 𝐺 ∈ V ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) → ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) )
31 30 3anim1i ⊢ ( ( ( 𝐺 ∈ V ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) → ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) )
32 29 31 syl ⊢ ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 → ( ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) )
33 28 32 syl11 ⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) → ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 → 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 ) )
34 spthonpthon ⊢ ( 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 → 𝐹 ( 𝐴 ( PathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 )
35 pthontrlon ⊢ ( 𝐹 ( 𝐴 ( PathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 → 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐵 ) 𝑃 )
36 trlsonwlkon ⊢ ( 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐵 ) 𝑃 → 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 )
37 34 35 36 3syl ⊢ ( 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 → 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 )
38 33 37 impbid1 ⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( ♯ ‘ 𝐹 ) = 1 ∧ 𝐴 ≠ 𝐵 ) → ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 ↔ 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 ) )