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Theorem wlkl0 27558
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 26906 . . . . . . . 8 (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺))
2 wlkop 26758 . . . . . . . 8 (𝑤 ∈ (Walks‘𝐺) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
31, 2syl 17 . . . . . . 7 (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
4 fvex 6342 . . . . . . . . . . . . . . 15 (1st𝑤) ∈ V
5 hasheq0 13356 . . . . . . . . . . . . . . 15 ((1st𝑤) ∈ V → ((♯‘(1st𝑤)) = 0 ↔ (1st𝑤) = ∅))
64, 5ax-mp 5 . . . . . . . . . . . . . 14 ((♯‘(1st𝑤)) = 0 ↔ (1st𝑤) = ∅)
76biimpi 206 . . . . . . . . . . . . 13 ((♯‘(1st𝑤)) = 0 → (1st𝑤) = ∅)
87adantr 466 . . . . . . . . . . . 12 (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → (1st𝑤) = ∅)
983ad2ant3 1129 . . . . . . . . . . 11 ((𝑋𝑉 ∧ (1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (1st𝑤) = ∅)
108adantl 467 . . . . . . . . . . . . . . . . 17 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (1st𝑤) = ∅)
1110breq1d 4796 . . . . . . . . . . . . . . . 16 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ↔ ∅(ClWalks‘𝐺)(2nd𝑤)))
12 clwlknon2num.v . . . . . . . . . . . . . . . . . . 19 𝑉 = (Vtx‘𝐺)
13121vgrex 26103 . . . . . . . . . . . . . . . . . 18 (𝑋𝑉𝐺 ∈ V)
14120clwlk 27310 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ V → (∅(ClWalks‘𝐺)(2nd𝑤) ↔ (2nd𝑤):(0...0)⟶𝑉))
1513, 14syl 17 . . . . . . . . . . . . . . . . 17 (𝑋𝑉 → (∅(ClWalks‘𝐺)(2nd𝑤) ↔ (2nd𝑤):(0...0)⟶𝑉))
1615adantr 466 . . . . . . . . . . . . . . . 16 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (∅(ClWalks‘𝐺)(2nd𝑤) ↔ (2nd𝑤):(0...0)⟶𝑉))
1711, 16bitrd 268 . . . . . . . . . . . . . . 15 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ↔ (2nd𝑤):(0...0)⟶𝑉))
18 fz0sn 12647 . . . . . . . . . . . . . . . . 17 (0...0) = {0}
1918feq2i 6177 . . . . . . . . . . . . . . . 16 ((2nd𝑤):(0...0)⟶𝑉 ↔ (2nd𝑤):{0}⟶𝑉)
20 c0ex 10236 . . . . . . . . . . . . . . . . . 18 0 ∈ V
2120fsn2 6546 . . . . . . . . . . . . . . . . 17 ((2nd𝑤):{0}⟶𝑉 ↔ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩}))
22 simprr 756 . . . . . . . . . . . . . . . . . . 19 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})
23 simprr 756 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((2nd𝑤)‘0) = 𝑋)
2423adantr 466 . . . . . . . . . . . . . . . . . . . . 21 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → ((2nd𝑤)‘0) = 𝑋)
2524opeq2d 4546 . . . . . . . . . . . . . . . . . . . 20 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → ⟨0, ((2nd𝑤)‘0)⟩ = ⟨0, 𝑋⟩)
2625sneqd 4328 . . . . . . . . . . . . . . . . . . 19 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → {⟨0, ((2nd𝑤)‘0)⟩} = {⟨0, 𝑋⟩})
2722, 26eqtrd 2805 . . . . . . . . . . . . . . . . . 18 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → (2nd𝑤) = {⟨0, 𝑋⟩})
2827ex 397 . . . . . . . . . . . . . . . . 17 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩}) → (2nd𝑤) = {⟨0, 𝑋⟩}))
2921, 28syl5bi 232 . . . . . . . . . . . . . . . 16 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((2nd𝑤):{0}⟶𝑉 → (2nd𝑤) = {⟨0, 𝑋⟩}))
3019, 29syl5bi 232 . . . . . . . . . . . . . . 15 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((2nd𝑤):(0...0)⟶𝑉 → (2nd𝑤) = {⟨0, 𝑋⟩}))
3117, 30sylbid 230 . . . . . . . . . . . . . 14 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) → (2nd𝑤) = {⟨0, 𝑋⟩}))
3231ex 397 . . . . . . . . . . . . 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 1101 . . . . . . . . . . 11 ((𝑋𝑉 ∧ (1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (2nd𝑤) = {⟨0, 𝑋⟩})
359, 34opeq12d 4547 . . . . . . . . . 10 ((𝑋𝑉 ∧ (1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ⟨(1st𝑤), (2nd𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)
36353exp 1112 . . . . . . . . 9 (𝑋𝑉 → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) → (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → ⟨(1st𝑤), (2nd𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)))
37 eleq1 2838 . . . . . . . . . . 11 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (ClWalks‘𝐺)))
38 df-br 4787 . . . . . . . . . . 11 ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (ClWalks‘𝐺))
3937, 38syl6bbr 278 . . . . . . . . . 10 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ (1st𝑤)(ClWalks‘𝐺)(2nd𝑤)))
40 eqeq1 2775 . . . . . . . . . . 11 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ ↔ ⟨(1st𝑤), (2nd𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩))
4140imbi2d 329 . . . . . . . . . 10 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → ((((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩) ↔ (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → ⟨(1st𝑤), (2nd𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)))
4239, 41imbi12d 333 . . . . . . . . 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 236 . . . . . . . 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 396 . . . 4 (𝑋𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
48 eqidd 2772 . . . . . . 7 (𝑋𝑉 → ∅ = ∅)
4920a1i 11 . . . . . . . 8 (𝑋𝑉 → 0 ∈ V)
50 snidg 4345 . . . . . . . 8 (𝑋𝑉𝑋 ∈ {𝑋})
5149, 50fsnd 6320 . . . . . . 7 (𝑋𝑉 → {⟨0, 𝑋⟩}:{0}⟶{𝑋})
52120clwlkv 27311 . . . . . . 7 ((𝑋𝑉 ∧ ∅ = ∅ ∧ {⟨0, 𝑋⟩}:{0}⟶{𝑋}) → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
5348, 51, 52mpd3an23 1574 . . . . . 6 (𝑋𝑉 → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
54 hash0 13360 . . . . . . 7 (♯‘∅) = 0
5554a1i 11 . . . . . 6 (𝑋𝑉 → (♯‘∅) = 0)
56 fvsng 6591 . . . . . . 7 ((0 ∈ V ∧ 𝑋𝑉) → ({⟨0, 𝑋⟩}‘0) = 𝑋)
5720, 56mpan 670 . . . . . 6 (𝑋𝑉 → ({⟨0, 𝑋⟩}‘0) = 𝑋)
5853, 55, 57jca32 505 . . . . 5 (𝑋𝑉 → (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ∧ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋)))
59 eleq1 2838 . . . . . . 7 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺)))
60 df-br 4787 . . . . . . 7 (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺))
6159, 60syl6bbr 278 . . . . . 6 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ∅(ClWalks‘𝐺){⟨0, 𝑋⟩}))
62 0ex 4924 . . . . . . . . . 10 ∅ ∈ V
63 snex 5036 . . . . . . . . . 10 {⟨0, 𝑋⟩} ∈ V
6462, 63op1std 7325 . . . . . . . . 9 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (1st𝑤) = ∅)
6564fveq2d 6336 . . . . . . . 8 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (♯‘(1st𝑤)) = (♯‘∅))
6665eqeq1d 2773 . . . . . . 7 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((♯‘(1st𝑤)) = 0 ↔ (♯‘∅) = 0))
6762, 63op2ndd 7326 . . . . . . . . 9 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (2nd𝑤) = {⟨0, 𝑋⟩})
6867fveq1d 6334 . . . . . . . 8 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((2nd𝑤)‘0) = ({⟨0, 𝑋⟩}‘0))
6968eqeq1d 2773 . . . . . . 7 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (((2nd𝑤)‘0) = 𝑋 ↔ ({⟨0, 𝑋⟩}‘0) = 𝑋))
7066, 69anbi12d 616 . . . . . 6 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋)))
7161, 70anbi12d 616 . . . . 5 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ∧ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋))))
7258, 71syl5ibrcom 237 . . . 4 (𝑋𝑉 → (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋))))
7347, 72impbid 202 . . 3 (𝑋𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
7473alrimiv 2007 . 2 (𝑋𝑉 → ∀𝑤((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
75 rabeqsn 4352 . 2 ({𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩} ↔ ∀𝑤((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
7674, 75sylibr 224 1 (𝑋𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071  wal 1629   = wceq 1631  wcel 2145  {crab 3065  Vcvv 3351  c0 4063  {csn 4316  cop 4322   class class class wbr 4786  wf 6027  cfv 6031  (class class class)co 6793  1st c1st 7313  2nd c2nd 7314  0cc0 10138  ...cfz 12533  chash 13321  Vtxcvtx 26095  Walkscwlks 26727  ClWalkscclwlks 26901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-ifp 1050  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-wlks 26730  df-clwlks 26902
This theorem is referenced by:  numclwlk1lem1  27560
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