MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clwlksfoclwwlk Structured version   Visualization version   GIF version

Theorem clwlksfoclwwlk 27050
Description: There is an onto function between the set of closed walks (defined as words) of a fixed prime length 𝑁 and the set of closed walks of length 𝑁. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 2-May-2021.) (Proof shortened by AV, 23-Mar-2022.)
Hypotheses
Ref Expression
clwlksfclwwlk.1 𝐴 = (1st𝑐)
clwlksfclwwlk.2 𝐵 = (2nd𝑐)
clwlksfclwwlk.c 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}
clwlksfclwwlk.f 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
Assertion
Ref Expression
clwlksfoclwwlk ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶onto→(𝑁 ClWWalksN 𝐺))
Distinct variable groups:   𝐺,𝑐   𝑁,𝑐   𝐶,𝑐   𝐹,𝑐
Allowed substitution hints:   𝐴(𝑐)   𝐵(𝑐)

Proof of Theorem clwlksfoclwwlk
Dummy variables 𝑓 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clwlksfclwwlk.1 . . 3 𝐴 = (1st𝑐)
2 clwlksfclwwlk.2 . . 3 𝐵 = (2nd𝑐)
3 clwlksfclwwlk.c . . 3 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}
4 clwlksfclwwlk.f . . 3 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
51, 2, 3, 4clwlksfclwwlk 27049 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶(𝑁 ClWWalksN 𝐺))
6 eqid 2651 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
76clwwlknbp 26997 . . . . 5 (𝑤 ∈ (𝑁 ClWWalksN 𝐺) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))
87adantl 481 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))
9 isclwwlkn 26987 . . . . . . 7 (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑤 ∈ (ClWWalks‘𝐺) ∧ (#‘𝑤) = 𝑁))
10 fusgrusgr 26259 . . . . . . . . . . . . 13 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)
11 usgruspgr 26118 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
1210, 11syl 17 . . . . . . . . . . . 12 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USPGraph)
1312adantr 480 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐺 ∈ USPGraph)
1413adantr 480 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 𝐺 ∈ USPGraph)
15 simprl 809 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 𝑤 ∈ Word (Vtx‘𝐺))
16 eleq1 2718 . . . . . . . . . . . . . . 15 ((#‘𝑤) = 𝑁 → ((#‘𝑤) ∈ ℙ ↔ 𝑁 ∈ ℙ))
17 prmnn 15435 . . . . . . . . . . . . . . . 16 ((#‘𝑤) ∈ ℙ → (#‘𝑤) ∈ ℕ)
1817nnge1d 11101 . . . . . . . . . . . . . . 15 ((#‘𝑤) ∈ ℙ → 1 ≤ (#‘𝑤))
1916, 18syl6bir 244 . . . . . . . . . . . . . 14 ((#‘𝑤) = 𝑁 → (𝑁 ∈ ℙ → 1 ≤ (#‘𝑤)))
2019adantl 481 . . . . . . . . . . . . 13 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁) → (𝑁 ∈ ℙ → 1 ≤ (#‘𝑤)))
2120com12 32 . . . . . . . . . . . 12 (𝑁 ∈ ℙ → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁) → 1 ≤ (#‘𝑤)))
2221adantl 481 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁) → 1 ≤ (#‘𝑤)))
2322imp 444 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 1 ≤ (#‘𝑤))
24 eqid 2651 . . . . . . . . . . 11 (iEdg‘𝐺) = (iEdg‘𝐺)
256, 24clwlkclwwlk2 26969 . . . . . . . . . 10 ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (ClWWalks‘𝐺)))
2614, 15, 23, 25syl3anc 1366 . . . . . . . . 9 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (ClWWalks‘𝐺)))
2726bicomd 213 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (𝑤 ∈ (ClWWalks‘𝐺) ↔ ∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩)))
2827anbi1d 741 . . . . . . 7 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → ((𝑤 ∈ (ClWWalks‘𝐺) ∧ (#‘𝑤) = 𝑁) ↔ (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ∧ (#‘𝑤) = 𝑁)))
299, 28syl5bb 272 . . . . . 6 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ↔ (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ∧ (#‘𝑤) = 𝑁)))
30 df-br 4686 . . . . . . . . 9 (𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
31 simpl 472 . . . . . . . . . . . . 13 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
32 prmnn 15435 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
3332nnge1d 11101 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℙ → 1 ≤ 𝑁)
3433ad2antlr 763 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 1 ≤ 𝑁)
35 breq2 4689 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑤) = 𝑁 → (1 ≤ (#‘𝑤) ↔ 1 ≤ 𝑁))
3635ad2antll 765 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (1 ≤ (#‘𝑤) ↔ 1 ≤ 𝑁))
3734, 36mpbird 247 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 1 ≤ (#‘𝑤))
3815, 37jca 553 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)))
39 clwlkwlk 26727 . . . . . . . . . . . . . . . 16 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (Walks‘𝐺))
40 wlklenvclwlk 26607 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (Walks‘𝐺) → (#‘𝑓) = (#‘𝑤)))
4138, 39, 40syl2im 40 . . . . . . . . . . . . . . 15 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (#‘𝑓) = (#‘𝑤)))
4241impcom 445 . . . . . . . . . . . . . 14 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → (#‘𝑓) = (#‘𝑤))
43 vex 3234 . . . . . . . . . . . . . . . . . 18 𝑓 ∈ V
44 ovex 6718 . . . . . . . . . . . . . . . . . 18 (𝑤 ++ ⟨“(𝑤‘0)”⟩) ∈ V
4543, 44op1st 7218 . . . . . . . . . . . . . . . . 17 (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = 𝑓
4645a1i 11 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = 𝑓)
4746fveq2d 6233 . . . . . . . . . . . . . . 15 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓))
4847adantl 481 . . . . . . . . . . . . . 14 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓))
49 eqcom 2658 . . . . . . . . . . . . . . . . 17 ((#‘𝑤) = 𝑁𝑁 = (#‘𝑤))
5049biimpi 206 . . . . . . . . . . . . . . . 16 ((#‘𝑤) = 𝑁𝑁 = (#‘𝑤))
5150ad2antll 765 . . . . . . . . . . . . . . 15 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 𝑁 = (#‘𝑤))
5251adantl 481 . . . . . . . . . . . . . 14 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → 𝑁 = (#‘𝑤))
5342, 48, 523eqtr4d 2695 . . . . . . . . . . . . 13 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁)
541fveq2i 6232 . . . . . . . . . . . . . . . 16 (#‘𝐴) = (#‘(1st𝑐))
5554eqeq1i 2656 . . . . . . . . . . . . . . 15 ((#‘𝐴) = 𝑁 ↔ (#‘(1st𝑐)) = 𝑁)
56 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (1st𝑐) = (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
5756fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (#‘(1st𝑐)) = (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
5857eqeq1d 2653 . . . . . . . . . . . . . . 15 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘(1st𝑐)) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
5955, 58syl5bb 272 . . . . . . . . . . . . . 14 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘𝐴) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
6059, 3elrab2 3399 . . . . . . . . . . . . 13 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
6131, 53, 60sylanbrc 699 . . . . . . . . . . . 12 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
6242adantr 480 . . . . . . . . . . . . . . . . . 18 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (#‘𝑓) = (#‘𝑤))
6362opeq2d 4440 . . . . . . . . . . . . . . . . 17 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨0, (#‘𝑓)⟩ = ⟨0, (#‘𝑤)⟩)
6463oveq2d 6706 . . . . . . . . . . . . . . . 16 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩))
65 simpr 476 . . . . . . . . . . . . . . . . . 18 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
6641adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (#‘𝑓) = (#‘𝑤)))
67 eqeq2 2662 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 = (#‘𝑤) → ((#‘𝑓) = 𝑁 ↔ (#‘𝑓) = (#‘𝑤)))
6867eqcoms 2659 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘𝑤) = 𝑁 → ((#‘𝑓) = 𝑁 ↔ (#‘𝑓) = (#‘𝑤)))
6968imbi2d 329 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘𝑤) = 𝑁 → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (#‘𝑓) = 𝑁) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (#‘𝑓) = (#‘𝑤))))
7069ad2antll 765 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (#‘𝑓) = 𝑁) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (#‘𝑓) = (#‘𝑤))))
7170adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (#‘𝑓) = 𝑁) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (#‘𝑓) = (#‘𝑤))))
7266, 71mpbird 247 . . . . . . . . . . . . . . . . . . 19 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (#‘𝑓) = 𝑁))
7372imp 444 . . . . . . . . . . . . . . . . . 18 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (#‘𝑓) = 𝑁)
7445a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = 𝑓)
7574fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓))
7657, 75eqtrd 2685 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (#‘(1st𝑐)) = (#‘𝑓))
7776eqeq1d 2653 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘(1st𝑐)) = 𝑁 ↔ (#‘𝑓) = 𝑁))
7855, 77syl5bb 272 . . . . . . . . . . . . . . . . . . 19 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘𝐴) = 𝑁 ↔ (#‘𝑓) = 𝑁))
7978, 3elrab2 3399 . . . . . . . . . . . . . . . . . 18 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (#‘𝑓) = 𝑁))
8065, 73, 79sylanbrc 699 . . . . . . . . . . . . . . . . 17 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
81 ovex 6718 . . . . . . . . . . . . . . . . 17 ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩) ∈ V
8254opeq2i 4437 . . . . . . . . . . . . . . . . . . . 20 ⟨0, (#‘𝐴)⟩ = ⟨0, (#‘(1st𝑐))⟩
832, 82oveq12i 6702 . . . . . . . . . . . . . . . . . . 19 (𝐵 substr ⟨0, (#‘𝐴)⟩) = ((2nd𝑐) substr ⟨0, (#‘(1st𝑐))⟩)
84 fveq2 6229 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (2nd𝑐) = (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
8557opeq2d 4440 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ⟨0, (#‘(1st𝑐))⟩ = ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩)
8684, 85oveq12d 6708 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd𝑐) substr ⟨0, (#‘(1st𝑐))⟩) = ((2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) substr ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩))
8743, 44op2nd 7219 . . . . . . . . . . . . . . . . . . . . 21 (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = (𝑤 ++ ⟨“(𝑤‘0)”⟩)
8845fveq2i 6232 . . . . . . . . . . . . . . . . . . . . . 22 (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓)
8988opeq2i 4437 . . . . . . . . . . . . . . . . . . . . 21 ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩ = ⟨0, (#‘𝑓)⟩
9087, 89oveq12i 6702 . . . . . . . . . . . . . . . . . . . 20 ((2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) substr ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩)
9186, 90syl6eq 2701 . . . . . . . . . . . . . . . . . . 19 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd𝑐) substr ⟨0, (#‘(1st𝑐))⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩))
9283, 91syl5eq 2697 . . . . . . . . . . . . . . . . . 18 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐵 substr ⟨0, (#‘𝐴)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩))
9392, 4fvmptg 6319 . . . . . . . . . . . . . . . . 17 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ∧ ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩) ∈ V) → (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩))
9480, 81, 93sylancl 695 . . . . . . . . . . . . . . . 16 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩))
9538ad2antlr 763 . . . . . . . . . . . . . . . . 17 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)))
96 simpl 472 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → 𝑤 ∈ Word (Vtx‘𝐺))
97 wrdsymb1 13375 . . . . . . . . . . . . . . . . . . . 20 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → (𝑤‘0) ∈ (Vtx‘𝐺))
9897s1cld 13419 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺))
99 eqidd 2652 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → (#‘𝑤) = (#‘𝑤))
100 swrdccatid 13543 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = (#‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩) = 𝑤)
10196, 98, 99, 100syl3anc 1366 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩) = 𝑤)
102101eqcomd 2657 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → 𝑤 = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩))
10395, 102syl 17 . . . . . . . . . . . . . . . 16 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 𝑤 = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩))
10464, 94, 1033eqtr4rd 2696 . . . . . . . . . . . . . . 15 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
105104ex 449 . . . . . . . . . . . . . 14 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
106105adantr 480 . . . . . . . . . . . . 13 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
107 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐹𝑐) = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
108107eqeq2d 2661 . . . . . . . . . . . . . . 15 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝑤 = (𝐹𝑐) ↔ 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
109108imbi2d 329 . . . . . . . . . . . . . 14 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹𝑐)) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))))
110109adantl 481 . . . . . . . . . . . . 13 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹𝑐)) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))))
111106, 110mpbird 247 . . . . . . . . . . . 12 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹𝑐)))
11261, 111rspcimedv 3342 . . . . . . . . . . 11 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
113112ex 449 . . . . . . . . . 10 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ∃𝑐𝐶 𝑤 = (𝐹𝑐))))
114113pm2.43b 55 . . . . . . . . 9 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
11530, 114syl5bi 232 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
116115exlimdv 1901 . . . . . . 7 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
117116adantrd 483 . . . . . 6 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → ((∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ∧ (#‘𝑤) = 𝑁) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
11829, 117sylbid 230 . . . . 5 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
119118impancom 455 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ (𝑁 ClWWalksN 𝐺)) → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
1208, 119mpd 15 . . 3 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑐𝐶 𝑤 = (𝐹𝑐))
121120ralrimiva 2995 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → ∀𝑤 ∈ (𝑁 ClWWalksN 𝐺)∃𝑐𝐶 𝑤 = (𝐹𝑐))
122 dffo3 6414 . 2 (𝐹:𝐶onto→(𝑁 ClWWalksN 𝐺) ↔ (𝐹:𝐶⟶(𝑁 ClWWalksN 𝐺) ∧ ∀𝑤 ∈ (𝑁 ClWWalksN 𝐺)∃𝑐𝐶 𝑤 = (𝐹𝑐)))
1235, 121, 122sylanbrc 699 1 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶onto→(𝑁 ClWWalksN 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cop 4216   class class class wbr 4685  cmpt 4762  wf 5922  ontowfo 5924  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  0cc0 9974  1c1 9975  cle 10113  #chash 13157  Word cword 13323   ++ cconcat 13325  ⟨“cs1 13326   substr csubstr 13327  cprime 15432  Vtxcvtx 25919  iEdgciedg 25920  USPGraphcuspgr 26088  USGraphcusgr 26089  FinUSGraphcfusgr 26253  Walkscwlks 26548  ClWalkscclwlks 26722  ClWWalkscclwwlk 26949   ClWWalksN cclwwlkn 26981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-word 13331  df-lsw 13332  df-concat 13333  df-s1 13334  df-substr 13335  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-dvds 15028  df-prm 15433  df-edg 25985  df-uhgr 25998  df-upgr 26022  df-uspgr 26090  df-usgr 26091  df-fusgr 26254  df-wlks 26551  df-clwlks 26723  df-clwwlk 26950  df-clwwlkn 26983
This theorem is referenced by:  clwlksf1oclwwlk  27057
  Copyright terms: Public domain W3C validator