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Theorem wlkl0 29311
Description: There is exactly one walk of length 0 on each vertex 𝑋. (Contributed by AV, 4-Jun-2022.)
Hypothesis
Ref Expression
clwlknon2num.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
wlkl0 (𝑋𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})
Distinct variable groups:   𝑤,𝐺   𝑤,𝑉   𝑤,𝑋

Proof of Theorem wlkl0
StepHypRef Expression
1 clwlkwlk 28723 . . . . . . . 8 (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺))
2 wlkop 28576 . . . . . . . 8 (𝑤 ∈ (Walks‘𝐺) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
31, 2syl 17 . . . . . . 7 (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
4 fvex 6855 . . . . . . . . . . . . . . 15 (1st𝑤) ∈ V
5 hasheq0 14263 . . . . . . . . . . . . . . 15 ((1st𝑤) ∈ V → ((♯‘(1st𝑤)) = 0 ↔ (1st𝑤) = ∅))
64, 5ax-mp 5 . . . . . . . . . . . . . 14 ((♯‘(1st𝑤)) = 0 ↔ (1st𝑤) = ∅)
76biimpi 215 . . . . . . . . . . . . 13 ((♯‘(1st𝑤)) = 0 → (1st𝑤) = ∅)
87adantr 481 . . . . . . . . . . . 12 (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → (1st𝑤) = ∅)
983ad2ant3 1135 . . . . . . . . . . 11 ((𝑋𝑉 ∧ (1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (1st𝑤) = ∅)
108adantl 482 . . . . . . . . . . . . . . . . 17 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (1st𝑤) = ∅)
1110breq1d 5115 . . . . . . . . . . . . . . . 16 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ↔ ∅(ClWalks‘𝐺)(2nd𝑤)))
12 clwlknon2num.v . . . . . . . . . . . . . . . . . . 19 𝑉 = (Vtx‘𝐺)
13121vgrex 27953 . . . . . . . . . . . . . . . . . 18 (𝑋𝑉𝐺 ∈ V)
14120clwlk 29074 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ V → (∅(ClWalks‘𝐺)(2nd𝑤) ↔ (2nd𝑤):(0...0)⟶𝑉))
1513, 14syl 17 . . . . . . . . . . . . . . . . 17 (𝑋𝑉 → (∅(ClWalks‘𝐺)(2nd𝑤) ↔ (2nd𝑤):(0...0)⟶𝑉))
1615adantr 481 . . . . . . . . . . . . . . . 16 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (∅(ClWalks‘𝐺)(2nd𝑤) ↔ (2nd𝑤):(0...0)⟶𝑉))
1711, 16bitrd 278 . . . . . . . . . . . . . . 15 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ↔ (2nd𝑤):(0...0)⟶𝑉))
18 fz0sn 13541 . . . . . . . . . . . . . . . . 17 (0...0) = {0}
1918feq2i 6660 . . . . . . . . . . . . . . . 16 ((2nd𝑤):(0...0)⟶𝑉 ↔ (2nd𝑤):{0}⟶𝑉)
20 c0ex 11149 . . . . . . . . . . . . . . . . . 18 0 ∈ V
2120fsn2 7082 . . . . . . . . . . . . . . . . 17 ((2nd𝑤):{0}⟶𝑉 ↔ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩}))
22 simprr 771 . . . . . . . . . . . . . . . . . . 19 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})
23 simprr 771 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((2nd𝑤)‘0) = 𝑋)
2423adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → ((2nd𝑤)‘0) = 𝑋)
2524opeq2d 4837 . . . . . . . . . . . . . . . . . . . 20 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → ⟨0, ((2nd𝑤)‘0)⟩ = ⟨0, 𝑋⟩)
2625sneqd 4598 . . . . . . . . . . . . . . . . . . 19 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → {⟨0, ((2nd𝑤)‘0)⟩} = {⟨0, 𝑋⟩})
2722, 26eqtrd 2776 . . . . . . . . . . . . . . . . . 18 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → (2nd𝑤) = {⟨0, 𝑋⟩})
2827ex 413 . . . . . . . . . . . . . . . . 17 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩}) → (2nd𝑤) = {⟨0, 𝑋⟩}))
2921, 28biimtrid 241 . . . . . . . . . . . . . . . 16 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((2nd𝑤):{0}⟶𝑉 → (2nd𝑤) = {⟨0, 𝑋⟩}))
3019, 29biimtrid 241 . . . . . . . . . . . . . . 15 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((2nd𝑤):(0...0)⟶𝑉 → (2nd𝑤) = {⟨0, 𝑋⟩}))
3117, 30sylbid 239 . . . . . . . . . . . . . 14 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) → (2nd𝑤) = {⟨0, 𝑋⟩}))
3231ex 413 . . . . . . . . . . . . 13 (𝑋𝑉 → (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) → (2nd𝑤) = {⟨0, 𝑋⟩})))
3332com23 86 . . . . . . . . . . . 12 (𝑋𝑉 → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) → (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → (2nd𝑤) = {⟨0, 𝑋⟩})))
34333imp 1111 . . . . . . . . . . 11 ((𝑋𝑉 ∧ (1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (2nd𝑤) = {⟨0, 𝑋⟩})
359, 34opeq12d 4838 . . . . . . . . . 10 ((𝑋𝑉 ∧ (1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ⟨(1st𝑤), (2nd𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)
36353exp 1119 . . . . . . . . 9 (𝑋𝑉 → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) → (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → ⟨(1st𝑤), (2nd𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)))
37 eleq1 2825 . . . . . . . . . . 11 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (ClWalks‘𝐺)))
38 df-br 5106 . . . . . . . . . . 11 ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (ClWalks‘𝐺))
3937, 38bitr4di 288 . . . . . . . . . 10 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ (1st𝑤)(ClWalks‘𝐺)(2nd𝑤)))
40 eqeq1 2740 . . . . . . . . . . 11 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ ↔ ⟨(1st𝑤), (2nd𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩))
4140imbi2d 340 . . . . . . . . . 10 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → ((((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩) ↔ (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → ⟨(1st𝑤), (2nd𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)))
4239, 41imbi12d 344 . . . . . . . . 9 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → ((𝑤 ∈ (ClWalks‘𝐺) → (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩)) ↔ ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) → (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → ⟨(1st𝑤), (2nd𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩))))
4336, 42syl5ibr 245 . . . . . . . 8 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → (𝑋𝑉 → (𝑤 ∈ (ClWalks‘𝐺) → (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))))
4443com23 86 . . . . . . 7 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) → (𝑋𝑉 → (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))))
453, 44mpcom 38 . . . . . 6 (𝑤 ∈ (ClWalks‘𝐺) → (𝑋𝑉 → (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩)))
4645com12 32 . . . . 5 (𝑋𝑉 → (𝑤 ∈ (ClWalks‘𝐺) → (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩)))
4746impd 411 . . . 4 (𝑋𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
48 eqidd 2737 . . . . . . 7 (𝑋𝑉 → ∅ = ∅)
4920a1i 11 . . . . . . . 8 (𝑋𝑉 → 0 ∈ V)
50 snidg 4620 . . . . . . . 8 (𝑋𝑉𝑋 ∈ {𝑋})
5149, 50fsnd 6827 . . . . . . 7 (𝑋𝑉 → {⟨0, 𝑋⟩}:{0}⟶{𝑋})
52120clwlkv 29075 . . . . . . 7 ((𝑋𝑉 ∧ ∅ = ∅ ∧ {⟨0, 𝑋⟩}:{0}⟶{𝑋}) → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
5348, 51, 52mpd3an23 1463 . . . . . 6 (𝑋𝑉 → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
54 hash0 14267 . . . . . . 7 (♯‘∅) = 0
5554a1i 11 . . . . . 6 (𝑋𝑉 → (♯‘∅) = 0)
56 fvsng 7126 . . . . . . 7 ((0 ∈ V ∧ 𝑋𝑉) → ({⟨0, 𝑋⟩}‘0) = 𝑋)
5720, 56mpan 688 . . . . . 6 (𝑋𝑉 → ({⟨0, 𝑋⟩}‘0) = 𝑋)
5853, 55, 57jca32 516 . . . . 5 (𝑋𝑉 → (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ∧ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋)))
59 eleq1 2825 . . . . . . 7 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺)))
60 df-br 5106 . . . . . . 7 (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺))
6159, 60bitr4di 288 . . . . . 6 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ∅(ClWalks‘𝐺){⟨0, 𝑋⟩}))
62 0ex 5264 . . . . . . . . 9 ∅ ∈ V
63 snex 5388 . . . . . . . . 9 {⟨0, 𝑋⟩} ∈ V
6462, 63op1std 7931 . . . . . . . 8 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (1st𝑤) = ∅)
6564fveqeq2d 6850 . . . . . . 7 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((♯‘(1st𝑤)) = 0 ↔ (♯‘∅) = 0))
6662, 63op2ndd 7932 . . . . . . . . 9 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (2nd𝑤) = {⟨0, 𝑋⟩})
6766fveq1d 6844 . . . . . . . 8 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((2nd𝑤)‘0) = ({⟨0, 𝑋⟩}‘0))
6867eqeq1d 2738 . . . . . . 7 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (((2nd𝑤)‘0) = 𝑋 ↔ ({⟨0, 𝑋⟩}‘0) = 𝑋))
6965, 68anbi12d 631 . . . . . 6 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋)))
7061, 69anbi12d 631 . . . . 5 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ∧ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋))))
7158, 70syl5ibrcom 246 . . . 4 (𝑋𝑉 → (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋))))
7247, 71impbid 211 . . 3 (𝑋𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
7372alrimiv 1930 . 2 (𝑋𝑉 → ∀𝑤((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
74 rabeqsn 4627 . 2 ({𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩} ↔ ∀𝑤((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
7573, 74sylibr 233 1 (𝑋𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087  wal 1539   = wceq 1541  wcel 2106  {crab 3407  Vcvv 3445  c0 4282  {csn 4586  cop 4592   class class class wbr 5105  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  0cc0 11051  ...cfz 13424  chash 14230  Vtxcvtx 27947  Walkscwlks 28544  ClWalkscclwlks 28718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-wlks 28547  df-clwlks 28719
This theorem is referenced by:  numclwlk1lem1  29313
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