| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simpl2 1192 | . . . 4
⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
∧ (𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))) → 𝑋 ∈ 𝑉) | 
| 2 |  | extwwlkfab.v | . . . . . 6
⊢ 𝑉 = (Vtx‘𝐺) | 
| 3 | 2 | nbgrisvtx 29359 | . . . . 5
⊢ (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → 𝑌 ∈ 𝑉) | 
| 4 | 3 | ad2antll 729 | . . . 4
⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
∧ (𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))) → 𝑌 ∈ 𝑉) | 
| 5 |  | simpl3 1193 | . . . 4
⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
∧ (𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))) → 𝑁 ∈
(ℤ≥‘3)) | 
| 6 |  | nbgrsym 29381 | . . . . . . . 8
⊢ (𝑌 ∈ (𝐺 NeighbVtx 𝑋) ↔ 𝑋 ∈ (𝐺 NeighbVtx 𝑌)) | 
| 7 |  | eqid 2736 | . . . . . . . . . 10
⊢
(Edg‘𝐺) =
(Edg‘𝐺) | 
| 8 | 7 | nbusgreledg 29371 | . . . . . . . . 9
⊢ (𝐺 ∈ USGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) ↔ {𝑋, 𝑌} ∈ (Edg‘𝐺))) | 
| 9 | 8 | biimpd 229 | . . . . . . . 8
⊢ (𝐺 ∈ USGraph → (𝑋 ∈ (𝐺 NeighbVtx 𝑌) → {𝑋, 𝑌} ∈ (Edg‘𝐺))) | 
| 10 | 6, 9 | biimtrid 242 | . . . . . . 7
⊢ (𝐺 ∈ USGraph → (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → {𝑋, 𝑌} ∈ (Edg‘𝐺))) | 
| 11 | 10 | adantld 490 | . . . . . 6
⊢ (𝐺 ∈ USGraph → ((𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → {𝑋, 𝑌} ∈ (Edg‘𝐺))) | 
| 12 | 11 | 3ad2ant1 1133 | . . . . 5
⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
→ ((𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → {𝑋, 𝑌} ∈ (Edg‘𝐺))) | 
| 13 | 12 | imp 406 | . . . 4
⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
∧ (𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))) → {𝑋, 𝑌} ∈ (Edg‘𝐺)) | 
| 14 |  | simprl 770 | . . . . 5
⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
∧ (𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))) → 𝑊 ∈ 𝐹) | 
| 15 |  | extwwlkfab.f | . . . . 5
⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) | 
| 16 | 14, 15 | eleqtrdi 2850 | . . . 4
⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
∧ (𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))) → 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) | 
| 17 | 2, 7 | clwwlknonex2 30129 | . . . 4
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
∧ {𝑋, 𝑌} ∈ (Edg‘𝐺) ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) → ((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) ∈ (𝑁 ClWWalksN 𝐺)) | 
| 18 | 1, 4, 5, 13, 16, 17 | syl311anc 1385 | . . 3
⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
∧ (𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))) → ((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) ∈ (𝑁 ClWWalksN 𝐺)) | 
| 19 | 15 | eleq2i 2832 | . . . . . . . 8
⊢ (𝑊 ∈ 𝐹 ↔ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) | 
| 20 |  | uz3m2nn 12934 | . . . . . . . . . . 11
⊢ (𝑁 ∈
(ℤ≥‘3) → (𝑁 − 2) ∈ ℕ) | 
| 21 | 20 | nnne0d 12317 | . . . . . . . . . 10
⊢ (𝑁 ∈
(ℤ≥‘3) → (𝑁 − 2) ≠ 0) | 
| 22 | 2, 7 | clwwlknonel 30115 | . . . . . . . . . 10
⊢ ((𝑁 − 2) ≠ 0 → (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑊) = (𝑁 − 2) ∧ (𝑊‘0) = 𝑋))) | 
| 23 | 21, 22 | syl 17 | . . . . . . . . 9
⊢ (𝑁 ∈
(ℤ≥‘3) → (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑊) = (𝑁 − 2) ∧ (𝑊‘0) = 𝑋))) | 
| 24 | 23 | 3ad2ant3 1135 | . . . . . . . 8
⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
→ (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑊) = (𝑁 − 2) ∧ (𝑊‘0) = 𝑋))) | 
| 25 | 19, 24 | bitrid 283 | . . . . . . 7
⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
→ (𝑊 ∈ 𝐹 ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑊) = (𝑁 − 2) ∧ (𝑊‘0) = 𝑋))) | 
| 26 |  | 3simpa 1148 | . . . . . . . . . . . . . 14
⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2))) | 
| 27 | 26 | adantr 480 | . . . . . . . . . . . . 13
⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2))) | 
| 28 |  | simp32 1210 | . . . . . . . . . . . . . 14
⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
→ 𝑋 ∈ 𝑉) | 
| 29 | 28, 3 | anim12i 613 | . . . . . . . . . . . . 13
⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) | 
| 30 |  | simpl33 1256 | . . . . . . . . . . . . 13
⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → 𝑁 ∈
(ℤ≥‘3)) | 
| 31 | 27, 29, 30 | 3jca 1128 | . . . . . . . . . . . 12
⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2) ∧ (𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)))
∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑁 ∈
(ℤ≥‘3))) | 
| 32 | 31 | 3exp1 1352 | . . . . . . . . . . 11
⊢ (𝑊 ∈ Word 𝑉 → ((♯‘𝑊) = (𝑁 − 2) → ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
→ (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑁 ∈
(ℤ≥‘3)))))) | 
| 33 | 32 | 3ad2ant1 1133 | . . . . . . . . . 10
⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)) → ((♯‘𝑊) = (𝑁 − 2) → ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
→ (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑁 ∈
(ℤ≥‘3)))))) | 
| 34 | 33 | imp 406 | . . . . . . . . 9
⊢ (((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑊) = (𝑁 − 2)) → ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
→ (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑁 ∈
(ℤ≥‘3))))) | 
| 35 | 34 | 3adant3 1132 | . . . . . . . 8
⊢ (((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑊) = (𝑁 − 2) ∧ (𝑊‘0) = 𝑋) → ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
→ (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑁 ∈
(ℤ≥‘3))))) | 
| 36 | 35 | com12 32 | . . . . . . 7
⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
→ (((𝑊 ∈ Word
𝑉 ∧ ∀𝑖 ∈
(0..^((♯‘𝑊)
− 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ (Edg‘𝐺)) ∧ (♯‘𝑊) = (𝑁 − 2) ∧ (𝑊‘0) = 𝑋) → (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑁 ∈
(ℤ≥‘3))))) | 
| 37 | 25, 36 | sylbid 240 | . . . . . 6
⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
→ (𝑊 ∈ 𝐹 → (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑁 ∈
(ℤ≥‘3))))) | 
| 38 | 37 | imp32 418 | . . . . 5
⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
∧ (𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑁 ∈
(ℤ≥‘3))) | 
| 39 |  | numclwwlk1lem2foalem 30371 | . . . . 5
⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑁 ∈ (ℤ≥‘3))
→ ((((𝑊 ++
〈“𝑋”〉) ++ 〈“𝑌”〉) prefix (𝑁 − 2)) = 𝑊 ∧ (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 1)) = 𝑌 ∧ (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 2)) = 𝑋)) | 
| 40 | 38, 39 | syl 17 | . . . 4
⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
∧ (𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))) → ((((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) prefix (𝑁 − 2)) = 𝑊 ∧ (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 1)) = 𝑌 ∧ (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 2)) = 𝑋)) | 
| 41 |  | eleq1a 2835 | . . . . . 6
⊢ (𝑊 ∈ 𝐹 → ((((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) prefix (𝑁 − 2)) = 𝑊 → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) prefix (𝑁 − 2)) ∈ 𝐹)) | 
| 42 | 14, 41 | syl 17 | . . . . 5
⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
∧ (𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))) → ((((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) prefix (𝑁 − 2)) = 𝑊 → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) prefix (𝑁 − 2)) ∈ 𝐹)) | 
| 43 |  | eleq1a 2835 | . . . . . 6
⊢ (𝑌 ∈ (𝐺 NeighbVtx 𝑋) → ((((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 1)) = 𝑌 → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx 𝑋))) | 
| 44 | 43 | ad2antll 729 | . . . . 5
⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
∧ (𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))) → ((((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 1)) = 𝑌 → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx 𝑋))) | 
| 45 |  | idd 24 | . . . . 5
⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
∧ (𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))) → ((((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 2)) = 𝑋 → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 2)) = 𝑋)) | 
| 46 | 42, 44, 45 | 3anim123d 1444 | . . . 4
⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
∧ (𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))) → (((((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) prefix (𝑁 − 2)) = 𝑊 ∧ (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 1)) = 𝑌 ∧ (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 2)) = 𝑋) → ((((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) prefix (𝑁 − 2)) ∈ 𝐹 ∧ (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 2)) = 𝑋))) | 
| 47 | 40, 46 | mpd 15 | . . 3
⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
∧ (𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))) → ((((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) prefix (𝑁 − 2)) ∈ 𝐹 ∧ (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 2)) = 𝑋)) | 
| 48 |  | extwwlkfab.c | . . . . 5
⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) | 
| 49 | 2, 48, 15 | extwwlkfabel 30373 | . . . 4
⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
→ (((𝑊 ++
〈“𝑋”〉) ++ 〈“𝑌”〉) ∈ (𝑋𝐶𝑁) ↔ (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) ∈ (𝑁 ClWWalksN 𝐺) ∧ ((((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) prefix (𝑁 − 2)) ∈ 𝐹 ∧ (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 2)) = 𝑋)))) | 
| 50 | 49 | adantr 480 | . . 3
⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
∧ (𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) ∈ (𝑋𝐶𝑁) ↔ (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) ∈ (𝑁 ClWWalksN 𝐺) ∧ ((((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) prefix (𝑁 − 2)) ∈ 𝐹 ∧ (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘(𝑁 − 2)) = 𝑋)))) | 
| 51 | 18, 47, 50 | mpbir2and 713 | . 2
⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
∧ (𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋))) → ((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) ∈ (𝑋𝐶𝑁)) | 
| 52 | 51 | ex 412 | 1
⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))
→ ((𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → ((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) ∈ (𝑋𝐶𝑁))) |