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Theorem wlkl0 28060
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 27470 . . . . . . . 8 (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺))
2 wlkop 27323 . . . . . . . 8 (𝑤 ∈ (Walks‘𝐺) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
31, 2syl 17 . . . . . . 7 (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
4 fvex 6680 . . . . . . . . . . . . . . 15 (1st𝑤) ∈ V
5 hasheq0 13714 . . . . . . . . . . . . . . 15 ((1st𝑤) ∈ V → ((♯‘(1st𝑤)) = 0 ↔ (1st𝑤) = ∅))
64, 5ax-mp 5 . . . . . . . . . . . . . 14 ((♯‘(1st𝑤)) = 0 ↔ (1st𝑤) = ∅)
76biimpi 217 . . . . . . . . . . . . 13 ((♯‘(1st𝑤)) = 0 → (1st𝑤) = ∅)
87adantr 481 . . . . . . . . . . . 12 (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → (1st𝑤) = ∅)
983ad2ant3 1129 . . . . . . . . . . 11 ((𝑋𝑉 ∧ (1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (1st𝑤) = ∅)
108adantl 482 . . . . . . . . . . . . . . . . 17 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (1st𝑤) = ∅)
1110breq1d 5073 . . . . . . . . . . . . . . . 16 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ↔ ∅(ClWalks‘𝐺)(2nd𝑤)))
12 clwlknon2num.v . . . . . . . . . . . . . . . . . . 19 𝑉 = (Vtx‘𝐺)
13121vgrex 26701 . . . . . . . . . . . . . . . . . 18 (𝑋𝑉𝐺 ∈ V)
14120clwlk 27823 . . . . . . . . . . . . . . . . . 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 280 . . . . . . . . . . . . . . 15 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ↔ (2nd𝑤):(0...0)⟶𝑉))
18 fz0sn 12997 . . . . . . . . . . . . . . . . 17 (0...0) = {0}
1918feq2i 6503 . . . . . . . . . . . . . . . 16 ((2nd𝑤):(0...0)⟶𝑉 ↔ (2nd𝑤):{0}⟶𝑉)
20 c0ex 10624 . . . . . . . . . . . . . . . . . 18 0 ∈ V
2120fsn2 6894 . . . . . . . . . . . . . . . . 17 ((2nd𝑤):{0}⟶𝑉 ↔ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩}))
22 simprr 769 . . . . . . . . . . . . . . . . . . 19 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})
23 simprr 769 . . . . . . . . . . . . . . . . . . . . . 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 4809 . . . . . . . . . . . . . . . . . . . 20 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → ⟨0, ((2nd𝑤)‘0)⟩ = ⟨0, 𝑋⟩)
2625sneqd 4576 . . . . . . . . . . . . . . . . . . 19 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → {⟨0, ((2nd𝑤)‘0)⟩} = {⟨0, 𝑋⟩})
2722, 26eqtrd 2861 . . . . . . . . . . . . . . . . . 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, 28syl5bi 243 . . . . . . . . . . . . . . . 16 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((2nd𝑤):{0}⟶𝑉 → (2nd𝑤) = {⟨0, 𝑋⟩}))
3019, 29syl5bi 243 . . . . . . . . . . . . . . 15 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((2nd𝑤):(0...0)⟶𝑉 → (2nd𝑤) = {⟨0, 𝑋⟩}))
3117, 30sylbid 241 . . . . . . . . . . . . . 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 1105 . . . . . . . . . . 11 ((𝑋𝑉 ∧ (1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (2nd𝑤) = {⟨0, 𝑋⟩})
359, 34opeq12d 4810 . . . . . . . . . 10 ((𝑋𝑉 ∧ (1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ⟨(1st𝑤), (2nd𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)
36353exp 1113 . . . . . . . . 9 (𝑋𝑉 → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) → (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → ⟨(1st𝑤), (2nd𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)))
37 eleq1 2905 . . . . . . . . . . 11 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (ClWalks‘𝐺)))
38 df-br 5064 . . . . . . . . . . 11 ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (ClWalks‘𝐺))
3937, 38syl6bbr 290 . . . . . . . . . 10 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ (1st𝑤)(ClWalks‘𝐺)(2nd𝑤)))
40 eqeq1 2830 . . . . . . . . . . 11 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ ↔ ⟨(1st𝑤), (2nd𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩))
4140imbi2d 342 . . . . . . . . . 10 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → ((((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩) ↔ (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → ⟨(1st𝑤), (2nd𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)))
4239, 41imbi12d 346 . . . . . . . . 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 247 . . . . . . . 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 2827 . . . . . . 7 (𝑋𝑉 → ∅ = ∅)
4920a1i 11 . . . . . . . 8 (𝑋𝑉 → 0 ∈ V)
50 snidg 4596 . . . . . . . 8 (𝑋𝑉𝑋 ∈ {𝑋})
5149, 50fsnd 6654 . . . . . . 7 (𝑋𝑉 → {⟨0, 𝑋⟩}:{0}⟶{𝑋})
52120clwlkv 27824 . . . . . . 7 ((𝑋𝑉 ∧ ∅ = ∅ ∧ {⟨0, 𝑋⟩}:{0}⟶{𝑋}) → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
5348, 51, 52mpd3an23 1456 . . . . . 6 (𝑋𝑉 → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
54 hash0 13718 . . . . . . 7 (♯‘∅) = 0
5554a1i 11 . . . . . 6 (𝑋𝑉 → (♯‘∅) = 0)
56 fvsng 6938 . . . . . . 7 ((0 ∈ V ∧ 𝑋𝑉) → ({⟨0, 𝑋⟩}‘0) = 𝑋)
5720, 56mpan 686 . . . . . 6 (𝑋𝑉 → ({⟨0, 𝑋⟩}‘0) = 𝑋)
5853, 55, 57jca32 516 . . . . 5 (𝑋𝑉 → (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ∧ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋)))
59 eleq1 2905 . . . . . . 7 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺)))
60 df-br 5064 . . . . . . 7 (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺))
6159, 60syl6bbr 290 . . . . . 6 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ∅(ClWalks‘𝐺){⟨0, 𝑋⟩}))
62 0ex 5208 . . . . . . . . 9 ∅ ∈ V
63 snex 5328 . . . . . . . . 9 {⟨0, 𝑋⟩} ∈ V
6462, 63op1std 7690 . . . . . . . 8 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (1st𝑤) = ∅)
6564fveqeq2d 6675 . . . . . . 7 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((♯‘(1st𝑤)) = 0 ↔ (♯‘∅) = 0))
6662, 63op2ndd 7691 . . . . . . . . 9 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (2nd𝑤) = {⟨0, 𝑋⟩})
6766fveq1d 6669 . . . . . . . 8 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((2nd𝑤)‘0) = ({⟨0, 𝑋⟩}‘0))
6867eqeq1d 2828 . . . . . . 7 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (((2nd𝑤)‘0) = 𝑋 ↔ ({⟨0, 𝑋⟩}‘0) = 𝑋))
6965, 68anbi12d 630 . . . . . 6 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋)))
7061, 69anbi12d 630 . . . . 5 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ∧ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋))))
7158, 70syl5ibrcom 248 . . . 4 (𝑋𝑉 → (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋))))
7247, 71impbid 213 . . 3 (𝑋𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
7372alrimiv 1921 . 2 (𝑋𝑉 → ∀𝑤((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
74 rabeqsn 4603 . 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 235 1 (𝑋𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081  wal 1528   = wceq 1530  wcel 2107  {crab 3147  Vcvv 3500  c0 4295  {csn 4564  cop 4570   class class class wbr 5063  wf 6348  cfv 6352  (class class class)co 7148  1st c1st 7678  2nd c2nd 7679  0cc0 10526  ...cfz 12882  chash 13680  Vtxcvtx 26695  Walkscwlks 27292  ClWalkscclwlks 27465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ifp 1057  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-er 8279  df-map 8398  df-pm 8399  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-n0 11887  df-z 11971  df-uz 12233  df-fz 12883  df-fzo 13024  df-hash 13681  df-word 13852  df-wlks 27295  df-clwlks 27466
This theorem is referenced by:  numclwlk1lem1  28062
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