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

Theorem wlkl0 30387
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 29796 . . . . . . . 8 (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺))
2 wlkop 29647 . . . . . . . 8 (𝑤 ∈ (Walks‘𝐺) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
31, 2syl 17 . . . . . . 7 (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
4 fvex 6918 . . . . . . . . . . . . . . 15 (1st𝑤) ∈ V
5 hasheq0 14403 . . . . . . . . . . . . . . 15 ((1st𝑤) ∈ V → ((♯‘(1st𝑤)) = 0 ↔ (1st𝑤) = ∅))
64, 5ax-mp 5 . . . . . . . . . . . . . 14 ((♯‘(1st𝑤)) = 0 ↔ (1st𝑤) = ∅)
76biimpi 216 . . . . . . . . . . . . 13 ((♯‘(1st𝑤)) = 0 → (1st𝑤) = ∅)
87adantr 480 . . . . . . . . . . . 12 (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) → (1st𝑤) = ∅)
983ad2ant3 1135 . . . . . . . . . . 11 ((𝑋𝑉 ∧ (1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (1st𝑤) = ∅)
108adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (1st𝑤) = ∅)
1110breq1d 5152 . . . . . . . . . . . . . . . 16 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ↔ ∅(ClWalks‘𝐺)(2nd𝑤)))
12 clwlknon2num.v . . . . . . . . . . . . . . . . . . 19 𝑉 = (Vtx‘𝐺)
13121vgrex 29020 . . . . . . . . . . . . . . . . . 18 (𝑋𝑉𝐺 ∈ V)
14120clwlk 30150 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ V → (∅(ClWalks‘𝐺)(2nd𝑤) ↔ (2nd𝑤):(0...0)⟶𝑉))
1513, 14syl 17 . . . . . . . . . . . . . . . . 17 (𝑋𝑉 → (∅(ClWalks‘𝐺)(2nd𝑤) ↔ (2nd𝑤):(0...0)⟶𝑉))
1615adantr 480 . . . . . . . . . . . . . . . 16 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (∅(ClWalks‘𝐺)(2nd𝑤) ↔ (2nd𝑤):(0...0)⟶𝑉))
1711, 16bitrd 279 . . . . . . . . . . . . . . 15 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ↔ (2nd𝑤):(0...0)⟶𝑉))
18 fz0sn 13668 . . . . . . . . . . . . . . . . 17 (0...0) = {0}
1918feq2i 6727 . . . . . . . . . . . . . . . 16 ((2nd𝑤):(0...0)⟶𝑉 ↔ (2nd𝑤):{0}⟶𝑉)
20 c0ex 11256 . . . . . . . . . . . . . . . . . 18 0 ∈ V
2120fsn2 7155 . . . . . . . . . . . . . . . . 17 ((2nd𝑤):{0}⟶𝑉 ↔ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩}))
22 simprr 772 . . . . . . . . . . . . . . . . . . 19 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})
23 simprr 772 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((2nd𝑤)‘0) = 𝑋)
2423adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → ((2nd𝑤)‘0) = 𝑋)
2524opeq2d 4879 . . . . . . . . . . . . . . . . . . . 20 (((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ∧ (((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩})) → ⟨0, ((2nd𝑤)‘0)⟩ = ⟨0, 𝑋⟩)
2625sneqd 4637 . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . 17 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((((2nd𝑤)‘0) ∈ 𝑉 ∧ (2nd𝑤) = {⟨0, ((2nd𝑤)‘0)⟩}) → (2nd𝑤) = {⟨0, 𝑋⟩}))
2921, 28biimtrid 242 . . . . . . . . . . . . . . . 16 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((2nd𝑤):{0}⟶𝑉 → (2nd𝑤) = {⟨0, 𝑋⟩}))
3019, 29biimtrid 242 . . . . . . . . . . . . . . 15 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((2nd𝑤):(0...0)⟶𝑉 → (2nd𝑤) = {⟨0, 𝑋⟩}))
3117, 30sylbid 240 . . . . . . . . . . . . . 14 ((𝑋𝑉 ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) → (2nd𝑤) = {⟨0, 𝑋⟩}))
3231ex 412 . . . . . . . . . . . . 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 1110 . . . . . . . . . . 11 ((𝑋𝑉 ∧ (1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) → (2nd𝑤) = {⟨0, 𝑋⟩})
359, 34opeq12d 4880 . . . . . . . . . 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 2828 . . . . . . . . . . 11 (𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (ClWalks‘𝐺)))
38 df-br 5143 . . . . . . . . . . 11 ((1st𝑤)(ClWalks‘𝐺)(2nd𝑤) ↔ ⟨(1st𝑤), (2nd𝑤)⟩ ∈ (ClWalks‘𝐺))
3937, 38bitr4di 289 . . . . . . . . . 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, 42imbitrrid 246 . . . . . . . 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 410 . . . 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 4659 . . . . . . . 8 (𝑋𝑉𝑋 ∈ {𝑋})
5149, 50fsnd 6890 . . . . . . 7 (𝑋𝑉 → {⟨0, 𝑋⟩}:{0}⟶{𝑋})
52120clwlkv 30151 . . . . . . 7 ((𝑋𝑉 ∧ ∅ = ∅ ∧ {⟨0, 𝑋⟩}:{0}⟶{𝑋}) → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
5348, 51, 52mpd3an23 1464 . . . . . 6 (𝑋𝑉 → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
54 hash0 14407 . . . . . . 7 (♯‘∅) = 0
5554a1i 11 . . . . . 6 (𝑋𝑉 → (♯‘∅) = 0)
56 fvsng 7201 . . . . . . 7 ((0 ∈ V ∧ 𝑋𝑉) → ({⟨0, 𝑋⟩}‘0) = 𝑋)
5720, 56mpan 690 . . . . . 6 (𝑋𝑉 → ({⟨0, 𝑋⟩}‘0) = 𝑋)
5853, 55, 57jca32 515 . . . . 5 (𝑋𝑉 → (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ∧ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋)))
59 eleq1 2828 . . . . . . 7 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺)))
60 df-br 5143 . . . . . . 7 (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺))
6159, 60bitr4di 289 . . . . . 6 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ∅(ClWalks‘𝐺){⟨0, 𝑋⟩}))
62 0ex 5306 . . . . . . . . 9 ∅ ∈ V
63 snex 5435 . . . . . . . . 9 {⟨0, 𝑋⟩} ∈ V
6462, 63op1std 8025 . . . . . . . 8 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (1st𝑤) = ∅)
6564fveqeq2d 6913 . . . . . . 7 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((♯‘(1st𝑤)) = 0 ↔ (♯‘∅) = 0))
6662, 63op2ndd 8026 . . . . . . . . 9 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (2nd𝑤) = {⟨0, 𝑋⟩})
6766fveq1d 6907 . . . . . . . 8 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((2nd𝑤)‘0) = ({⟨0, 𝑋⟩}‘0))
6867eqeq1d 2738 . . . . . . 7 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (((2nd𝑤)‘0) = 𝑋 ↔ ({⟨0, 𝑋⟩}‘0) = 𝑋))
6965, 68anbi12d 632 . . . . . 6 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋)))
7061, 69anbi12d 632 . . . . 5 (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ∧ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋))))
7158, 70syl5ibrcom 247 . . . 4 (𝑋𝑉 → (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋))))
7247, 71impbid 212 . . 3 (𝑋𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
7372alrimiv 1926 . 2 (𝑋𝑉 → ∀𝑤((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
74 rabeqsn 4666 . 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 234 1 (𝑋𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wal 1537   = wceq 1539  wcel 2107  {crab 3435  Vcvv 3479  c0 4332  {csn 4625  cop 4631   class class class wbr 5142  wf 6556  cfv 6560  (class class class)co 7432  1st c1st 8013  2nd c2nd 8014  0cc0 11156  ...cfz 13548  chash 14370  Vtxcvtx 29014  Walkscwlks 29615  ClWalkscclwlks 29791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-pm 8870  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-n0 12529  df-z 12616  df-uz 12880  df-fz 13549  df-fzo 13696  df-hash 14371  df-word 14554  df-wlks 29618  df-clwlks 29792
This theorem is referenced by:  numclwlk1lem1  30389
  Copyright terms: Public domain W3C validator