MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wlkl0 Structured version   Visualization version   GIF version

Theorem wlkl0 27671
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 26963 . . . . . . . 8 (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺))
2 wlkop 26814 . . . . . . . 8 (𝑤 ∈ (Walks‘𝐺) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
31, 2syl 17 . . . . . . 7 (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
4 fvex 6388 . . . . . . . . . . . . . . 15 (1st𝑤) ∈ V
5 hasheq0 13356 . . . . . . . . . . . . . . 15 ((1st𝑤) ∈ V → ((♯‘(1st𝑤)) = 0 ↔ (1st𝑤) = ∅))
64, 5ax-mp 5 . . . . . . . . . . . . . 14 ((♯‘(1st𝑤)) = 0 ↔ (1st𝑤) = ∅)
76biimpi 207 . . . . . . . . . . . . 13 ((♯‘(1st𝑤)) = 0 → (1st𝑤) = ∅)
87adantr 472 . . . . . . . . . . . 12 (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → (1st𝑤) = ∅)
983ad2ant3 1165 . . . . . . . . . . 11 ((𝑋𝑉 ∧ (1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (1st𝑤) = ∅)
108adantl 473 . . . . . . . . . . . . . . . . 17 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (1st𝑤) = ∅)
1110breq1d 4819 . . . . . . . . . . . . . . . 16 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ↔ ∅(ClWalks‘𝐺)(2nd𝑤)))
12 clwlknon2num.v . . . . . . . . . . . . . . . . . . 19 𝑉 = (Vtx‘𝐺)
13121vgrex 26171 . . . . . . . . . . . . . . . . . 18 (𝑋𝑉𝐺 ∈ V)
14120clwlk 27408 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ V → (∅(ClWalks‘𝐺)(2nd𝑤) ↔ (2nd𝑤):(0...0)⟶𝑉))
1513, 14syl 17 . . . . . . . . . . . . . . . . 17 (𝑋𝑉 → (∅(ClWalks‘𝐺)(2nd𝑤) ↔ (2nd𝑤):(0...0)⟶𝑉))
1615adantr 472 . . . . . . . . . . . . . . . 16 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (∅(ClWalks‘𝐺)(2nd𝑤) ↔ (2nd𝑤):(0...0)⟶𝑉))
1711, 16bitrd 270 . . . . . . . . . . . . . . 15 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ↔ (2nd𝑤):(0...0)⟶𝑉))
18 fz0sn 12647 . . . . . . . . . . . . . . . . 17 (0...0) = {0}
1918feq2i 6215 . . . . . . . . . . . . . . . 16 ((2nd𝑤):(0...0)⟶𝑉 ↔ (2nd𝑤):{0}⟶𝑉)
20 c0ex 10287 . . . . . . . . . . . . . . . . . 18 0 ∈ V
2120fsn2 6594 . . . . . . . . . . . . . . . . 17 ((2nd𝑤):{0}⟶𝑉 ↔ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩}))
22 simprr 789 . . . . . . . . . . . . . . . . . . 19 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})
23 simprr 789 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((2nd𝑤)‘0) = 𝑋)
2423adantr 472 . . . . . . . . . . . . . . . . . . . . 21 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → ((2nd𝑤)‘0) = 𝑋)
2524opeq2d 4566 . . . . . . . . . . . . . . . . . . . 20 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → ⟨0, ((2nd𝑤)‘0)⟩ = ⟨0, 𝑋⟩)
2625sneqd 4346 . . . . . . . . . . . . . . . . . . 19 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → {⟨0, ((2nd𝑤)‘0)⟩} = {⟨0, 𝑋⟩})
2722, 26eqtrd 2799 . . . . . . . . . . . . . . . . . 18 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → (2nd𝑤) = {⟨0, 𝑋⟩})
2827ex 401 . . . . . . . . . . . . . . . . 17 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩}) → (2nd𝑤) = {⟨0, 𝑋⟩}))
2921, 28syl5bi 233 . . . . . . . . . . . . . . . 16 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((2nd𝑤):{0}⟶𝑉 → (2nd𝑤) = {⟨0, 𝑋⟩}))
3019, 29syl5bi 233 . . . . . . . . . . . . . . 15 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((2nd𝑤):(0...0)⟶𝑉 → (2nd𝑤) = {⟨0, 𝑋⟩}))
3117, 30sylbid 231 . . . . . . . . . . . . . 14 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) → (2nd𝑤) = {⟨0, 𝑋⟩}))
3231ex 401 . . . . . . . . . . . . 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 1137 . . . . . . . . . . 11 ((𝑋𝑉 ∧ (1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (2nd𝑤) = {⟨0, 𝑋⟩})
359, 34opeq12d 4567 . . . . . . . . . 10 ((𝑋𝑉 ∧ (1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ⟨(1st𝑤), (2nd𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)
36353exp 1148 . . . . . . . . 9 (𝑋𝑉 → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) → (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → ⟨(1st𝑤), (2nd𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)))
37 eleq1 2832 . . . . . . . . . . 11 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (ClWalks‘𝐺)))
38 df-br 4810 . . . . . . . . . . 11 ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (ClWalks‘𝐺))
3937, 38syl6bbr 280 . . . . . . . . . 10 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ (1st𝑤)(ClWalks‘𝐺)(2nd𝑤)))
40 eqeq1 2769 . . . . . . . . . . 11 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ ↔ ⟨(1st𝑤), (2nd𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩))
4140imbi2d 331 . . . . . . . . . 10 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → ((((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩) ↔ (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → ⟨(1st𝑤), (2nd𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)))
4239, 41imbi12d 335 . . . . . . . . 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 237 . . . . . . . 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 398 . . . 4 (𝑋𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
48 eqidd 2766 . . . . . . 7 (𝑋𝑉 → ∅ = ∅)
4920a1i 11 . . . . . . . 8 (𝑋𝑉 → 0 ∈ V)
50 snidg 4364 . . . . . . . 8 (𝑋𝑉𝑋 ∈ {𝑋})
5149, 50fsnd 6362 . . . . . . 7 (𝑋𝑉 → {⟨0, 𝑋⟩}:{0}⟶{𝑋})
52120clwlkv 27409 . . . . . . 7 ((𝑋𝑉 ∧ ∅ = ∅ ∧ {⟨0, 𝑋⟩}:{0}⟶{𝑋}) → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
5348, 51, 52mpd3an23 1587 . . . . . 6 (𝑋𝑉 → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
54 hash0 13360 . . . . . . 7 (♯‘∅) = 0
5554a1i 11 . . . . . 6 (𝑋𝑉 → (♯‘∅) = 0)
56 fvsng 6640 . . . . . . 7 ((0 ∈ V ∧ 𝑋𝑉) → ({⟨0, 𝑋⟩}‘0) = 𝑋)
5720, 56mpan 681 . . . . . 6 (𝑋𝑉 → ({⟨0, 𝑋⟩}‘0) = 𝑋)
5853, 55, 57jca32 511 . . . . 5 (𝑋𝑉 → (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ∧ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋)))
59 eleq1 2832 . . . . . . 7 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺)))
60 df-br 4810 . . . . . . 7 (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺))
6159, 60syl6bbr 280 . . . . . 6 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ∅(ClWalks‘𝐺){⟨0, 𝑋⟩}))
62 0ex 4950 . . . . . . . . 9 ∅ ∈ V
63 snex 5064 . . . . . . . . 9 {⟨0, 𝑋⟩} ∈ V
6462, 63op1std 7376 . . . . . . . 8 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (1st𝑤) = ∅)
6564fveqeq2d 6383 . . . . . . 7 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((♯‘(1st𝑤)) = 0 ↔ (♯‘∅) = 0))
6662, 63op2ndd 7377 . . . . . . . . 9 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (2nd𝑤) = {⟨0, 𝑋⟩})
6766fveq1d 6377 . . . . . . . 8 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((2nd𝑤)‘0) = ({⟨0, 𝑋⟩}‘0))
6867eqeq1d 2767 . . . . . . 7 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (((2nd𝑤)‘0) = 𝑋 ↔ ({⟨0, 𝑋⟩}‘0) = 𝑋))
6965, 68anbi12d 624 . . . . . 6 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋)))
7061, 69anbi12d 624 . . . . 5 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ∧ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋))))
7158, 70syl5ibrcom 238 . . . 4 (𝑋𝑉 → (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋))))
7247, 71impbid 203 . . 3 (𝑋𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
7372alrimiv 2022 . 2 (𝑋𝑉 → ∀𝑤((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
74 rabeqsn 4371 . 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 225 1 (𝑋𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107  wal 1650   = wceq 1652  wcel 2155  {crab 3059  Vcvv 3350  c0 4079  {csn 4334  cop 4340   class class class wbr 4809  wf 6064  cfv 6068  (class class class)co 6842  1st c1st 7364  2nd c2nd 7365  0cc0 10189  ...cfz 12533  chash 13321  Vtxcvtx 26165  Walkscwlks 26783  ClWalkscclwlks 26958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-ifp 1086  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13487  df-wlks 26786  df-clwlks 26959
This theorem is referenced by:  numclwlk1lem1  27673
  Copyright terms: Public domain W3C validator