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Theorem clwwlknonclwlknonf1oOLD 27790
 Description: Obsolete version of clwwlknonclwlknonf1o 27789 as of 12-Oct-2022. (Contributed by AV, 26-May-2022.) (Proof shortened by AV, 7-Aug-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
clwwlknonclwlknonf1oOLD.v 𝑉 = (Vtx‘𝐺)
clwwlknonclwlknonf1oOLD.w 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)}
clwwlknonclwlknonf1oOLD.f 𝐹 = (𝑐𝑊 ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩))
Assertion
Ref Expression
clwwlknonclwlknonf1oOLD ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝐹:𝑊1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
Distinct variable groups:   𝐺,𝑐,𝑤   𝑁,𝑐,𝑤   𝑉,𝑐   𝑊,𝑐   𝑋,𝑐,𝑤
Allowed substitution hints:   𝐹(𝑤,𝑐)   𝑉(𝑤)   𝑊(𝑤)

Proof of Theorem clwwlknonclwlknonf1oOLD
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 clwwlknonclwlknonf1oOLD.w . . 3 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)}
2 eqid 2778 . . 3 {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}
3 clwwlknonclwlknonf1oOLD.f . . 3 𝐹 = (𝑐𝑊 ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩))
4 eqid 2778 . . 3 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩))
5 eqid 2778 . . . . 5 (1st𝑐) = (1st𝑐)
6 eqid 2778 . . . . 5 (2nd𝑐) = (2nd𝑐)
75, 6, 2, 4clwlknf1oclwwlknOLD 27489 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)):{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}–1-1-onto→(𝑁 ClWWalksN 𝐺))
873adant2 1122 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)):{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}–1-1-onto→(𝑁 ClWWalksN 𝐺))
9 fveq1 6447 . . . . . . 7 (𝑠 = ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩) → (𝑠‘0) = (((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)‘0))
1093ad2ant3 1126 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∧ 𝑠 = ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) → (𝑠‘0) = (((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)‘0))
11 2fveq3 6453 . . . . . . . . . . . 12 (𝑤 = 𝑐 → (♯‘(1st𝑤)) = (♯‘(1st𝑐)))
1211eqeq1d 2780 . . . . . . . . . . 11 (𝑤 = 𝑐 → ((♯‘(1st𝑤)) = 𝑁 ↔ (♯‘(1st𝑐)) = 𝑁))
1312elrab 3572 . . . . . . . . . 10 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (♯‘(1st𝑐)) = 𝑁))
14 clwlkwlk 27131 . . . . . . . . . . . 12 (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
15 wlkcpr 26980 . . . . . . . . . . . . 13 (𝑐 ∈ (Walks‘𝐺) ↔ (1st𝑐)(Walks‘𝐺)(2nd𝑐))
16 eqid 2778 . . . . . . . . . . . . . . . . 17 (Vtx‘𝐺) = (Vtx‘𝐺)
1716wlkpwrd 26969 . . . . . . . . . . . . . . . 16 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (2nd𝑐) ∈ Word (Vtx‘𝐺))
18173ad2ant1 1124 . . . . . . . . . . . . . . 15 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → (2nd𝑐) ∈ Word (Vtx‘𝐺))
19 elnnuz 12034 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ‘1))
20 eluzfz2 12670 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘1) → 𝑁 ∈ (1...𝑁))
2119, 20sylbi 209 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ → 𝑁 ∈ (1...𝑁))
22 fzelp1 12714 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (1...𝑁) → 𝑁 ∈ (1...(𝑁 + 1)))
2321, 22syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ → 𝑁 ∈ (1...(𝑁 + 1)))
24233ad2ant3 1126 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝑁 ∈ (1...(𝑁 + 1)))
25243ad2ant3 1126 . . . . . . . . . . . . . . . . 17 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → 𝑁 ∈ (1...(𝑁 + 1)))
26 id 22 . . . . . . . . . . . . . . . . . . 19 ((♯‘(1st𝑐)) = 𝑁 → (♯‘(1st𝑐)) = 𝑁)
27 oveq1 6931 . . . . . . . . . . . . . . . . . . . 20 ((♯‘(1st𝑐)) = 𝑁 → ((♯‘(1st𝑐)) + 1) = (𝑁 + 1))
2827oveq2d 6940 . . . . . . . . . . . . . . . . . . 19 ((♯‘(1st𝑐)) = 𝑁 → (1...((♯‘(1st𝑐)) + 1)) = (1...(𝑁 + 1)))
2926, 28eleq12d 2853 . . . . . . . . . . . . . . . . . 18 ((♯‘(1st𝑐)) = 𝑁 → ((♯‘(1st𝑐)) ∈ (1...((♯‘(1st𝑐)) + 1)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
30293ad2ant2 1125 . . . . . . . . . . . . . . . . 17 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → ((♯‘(1st𝑐)) ∈ (1...((♯‘(1st𝑐)) + 1)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3125, 30mpbird 249 . . . . . . . . . . . . . . . 16 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → (♯‘(1st𝑐)) ∈ (1...((♯‘(1st𝑐)) + 1)))
32 wlklenvp1 26970 . . . . . . . . . . . . . . . . . . 19 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (♯‘(2nd𝑐)) = ((♯‘(1st𝑐)) + 1))
3332oveq2d 6940 . . . . . . . . . . . . . . . . . 18 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (1...(♯‘(2nd𝑐))) = (1...((♯‘(1st𝑐)) + 1)))
3433eleq2d 2845 . . . . . . . . . . . . . . . . 17 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → ((♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐))) ↔ (♯‘(1st𝑐)) ∈ (1...((♯‘(1st𝑐)) + 1))))
35343ad2ant1 1124 . . . . . . . . . . . . . . . 16 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → ((♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐))) ↔ (♯‘(1st𝑐)) ∈ (1...((♯‘(1st𝑐)) + 1))))
3631, 35mpbird 249 . . . . . . . . . . . . . . 15 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐))))
3718, 36jca 507 . . . . . . . . . . . . . 14 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐)))))
38373exp 1109 . . . . . . . . . . . . 13 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → ((♯‘(1st𝑐)) = 𝑁 → ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐)))))))
3915, 38sylbi 209 . . . . . . . . . . . 12 (𝑐 ∈ (Walks‘𝐺) → ((♯‘(1st𝑐)) = 𝑁 → ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐)))))))
4014, 39syl 17 . . . . . . . . . . 11 (𝑐 ∈ (ClWalks‘𝐺) → ((♯‘(1st𝑐)) = 𝑁 → ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐)))))))
4140imp 397 . . . . . . . . . 10 ((𝑐 ∈ (ClWalks‘𝐺) ∧ (♯‘(1st𝑐)) = 𝑁) → ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐))))))
4213, 41sylbi 209 . . . . . . . . 9 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} → ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐))))))
4342impcom 398 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐)))))
44 swrd0fv0OLD 13763 . . . . . . . 8 (((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐)))) → (((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)‘0) = ((2nd𝑐)‘0))
4543, 44syl 17 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}) → (((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)‘0) = ((2nd𝑐)‘0))
46453adant3 1123 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∧ 𝑠 = ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) → (((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)‘0) = ((2nd𝑐)‘0))
4710, 46eqtrd 2814 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∧ 𝑠 = ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) → (𝑠‘0) = ((2nd𝑐)‘0))
4847eqeq1d 2780 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∧ 𝑠 = ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) → ((𝑠‘0) = 𝑋 ↔ ((2nd𝑐)‘0) = 𝑋))
49 nfv 1957 . . . . 5 𝑤((2nd𝑐)‘0) = 𝑋
50 fveq2 6448 . . . . . . 7 (𝑤 = 𝑐 → (2nd𝑤) = (2nd𝑐))
5150fveq1d 6450 . . . . . 6 (𝑤 = 𝑐 → ((2nd𝑤)‘0) = ((2nd𝑐)‘0))
5251eqeq1d 2780 . . . . 5 (𝑤 = 𝑐 → (((2nd𝑤)‘0) = 𝑋 ↔ ((2nd𝑐)‘0) = 𝑋))
5349, 52sbie 2484 . . . 4 ([𝑐 / 𝑤]((2nd𝑤)‘0) = 𝑋 ↔ ((2nd𝑐)‘0) = 𝑋)
5448, 53syl6bbr 281 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∧ 𝑠 = ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) → ((𝑠‘0) = 𝑋 ↔ [𝑐 / 𝑤]((2nd𝑤)‘0) = 𝑋))
551, 2, 3, 4, 8, 54f1ossf1o 6662 . 2 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝐹:𝑊1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋})
56 clwwlknon 27496 . . 3 (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋}
57 f1oeq3 6384 . . 3 ((𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋} → (𝐹:𝑊1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ 𝐹:𝑊1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋}))
5856, 57ax-mp 5 . 2 (𝐹:𝑊1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ 𝐹:𝑊1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋})
5955, 58sylibr 226 1 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝐹:𝑊1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601  [wsb 2011   ∈ wcel 2107  {crab 3094  ⟨cop 4404   class class class wbr 4888   ↦ cmpt 4967  –1-1-onto→wf1o 6136  ‘cfv 6137  (class class class)co 6924  1st c1st 7445  2nd c2nd 7446  0cc0 10274  1c1 10275   + caddc 10277  ℕcn 11378  ℤ≥cuz 11996  ...cfz 12647  ♯chash 13439  Word cword 13603   substr csubstr 13734  Vtxcvtx 26348  USPGraphcuspgr 26501  Walkscwlks 26948  ClWalkscclwlks 27126   ClWWalksN cclwwlkn 27417  ClWWalksNOncclwwlknon 27493 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ifp 1047  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-2o 7846  df-oadd 7849  df-er 8028  df-map 8144  df-pm 8145  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-card 9100  df-cda 9327  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-nn 11379  df-2 11442  df-n0 11647  df-xnn0 11719  df-z 11733  df-uz 11997  df-rp 12142  df-fz 12648  df-fzo 12789  df-hash 13440  df-word 13604  df-lsw 13657  df-concat 13665  df-s1 13690  df-substr 13735  df-pfx 13784  df-edg 26400  df-uhgr 26410  df-upgr 26434  df-uspgr 26503  df-wlks 26951  df-clwlks 27127  df-clwwlk 27366  df-clwwlkn 27418  df-clwwlknon 27494 This theorem is referenced by:  clwwlknonclwlknonenOLD  27792  dlwwlknondlwlknonf1oOLD  27796
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