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Theorem numclwwlk2lem1 28165
 Description: In a friendship graph, for each walk of length 𝑛 starting at a fixed vertex 𝑣 and ending not at this vertex, there is a unique vertex so that the walk extended by an edge to this vertex and an edge from this vertex to the first vertex of the walk is a value of operation 𝐻. If the walk is represented as a word, it is sufficient to add one vertex to the word to obtain the closed walk contained in the value of operation 𝐻, since in a word representing a closed walk the starting vertex is not repeated at the end. This theorem generally holds only for friendship graphs, because these guarantee that for the first and last vertex there is a (unique) third vertex "in between". (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 30-May-2021.) (Revised by AV, 1-May-2022.)
Hypotheses
Ref Expression
numclwwlk.v 𝑉 = (Vtx‘𝐺)
numclwwlk.q 𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})
numclwwlk.h 𝐻 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})
Assertion
Ref Expression
numclwwlk2lem1 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) → ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑁,𝑣,𝑤   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑤   𝑤,𝑉   𝑣,𝑊,𝑤
Allowed substitution hints:   𝑄(𝑤,𝑣,𝑛)   𝐻(𝑤,𝑣,𝑛)   𝑊(𝑛)

Proof of Theorem numclwwlk2lem1
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 numclwwlk.v . . . . . 6 𝑉 = (Vtx‘𝐺)
2 numclwwlk.q . . . . . 6 𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})
31, 2numclwwlkovq 28163 . . . . 5 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
433adant1 1127 . . . 4 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
54eleq2d 2878 . . 3 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) ↔ 𝑊 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}))
6 fveq1 6648 . . . . . 6 (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0))
76eqeq1d 2803 . . . . 5 (𝑤 = 𝑊 → ((𝑤‘0) = 𝑋 ↔ (𝑊‘0) = 𝑋))
8 fveq2 6649 . . . . . 6 (𝑤 = 𝑊 → (lastS‘𝑤) = (lastS‘𝑊))
98neeq1d 3049 . . . . 5 (𝑤 = 𝑊 → ((lastS‘𝑤) ≠ 𝑋 ↔ (lastS‘𝑊) ≠ 𝑋))
107, 9anbi12d 633 . . . 4 (𝑤 = 𝑊 → (((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋) ↔ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)))
1110elrab 3631 . . 3 (𝑊 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)} ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)))
125, 11syl6bb 290 . 2 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))))
13 simpl1 1188 . . . . 5 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → 𝐺 ∈ FriendGraph )
14 eqid 2801 . . . . . . . . . . . . 13 (Edg‘𝐺) = (Edg‘𝐺)
151, 14wwlknp 27633 . . . . . . . . . . . 12 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
16 peano2nn 11641 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
1716adantl 485 . . . . . . . . . . . . . . 15 (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ)
18 simpl 486 . . . . . . . . . . . . . . 15 (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)))
1917, 18jca 515 . . . . . . . . . . . . . 14 (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1))))
2019ex 416 . . . . . . . . . . . . 13 ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)))))
21203adant3 1129 . . . . . . . . . . . 12 ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)))))
2215, 21syl 17 . . . . . . . . . . 11 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)))))
23 lswlgt0cl 13916 . . . . . . . . . . 11 (((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1))) → (lastS‘𝑊) ∈ 𝑉)
2422, 23syl6 35 . . . . . . . . . 10 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ → (lastS‘𝑊) ∈ 𝑉))
2524adantr 484 . . . . . . . . 9 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (𝑁 ∈ ℕ → (lastS‘𝑊) ∈ 𝑉))
2625com12 32 . . . . . . . 8 (𝑁 ∈ ℕ → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (lastS‘𝑊) ∈ 𝑉))
27263ad2ant3 1132 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (lastS‘𝑊) ∈ 𝑉))
2827imp 410 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → (lastS‘𝑊) ∈ 𝑉)
29 eleq1 2880 . . . . . . . . . . 11 ((𝑊‘0) = 𝑋 → ((𝑊‘0) ∈ 𝑉𝑋𝑉))
3029biimprd 251 . . . . . . . . . 10 ((𝑊‘0) = 𝑋 → (𝑋𝑉 → (𝑊‘0) ∈ 𝑉))
3130ad2antrl 727 . . . . . . . . 9 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (𝑋𝑉 → (𝑊‘0) ∈ 𝑉))
3231com12 32 . . . . . . . 8 (𝑋𝑉 → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (𝑊‘0) ∈ 𝑉))
33323ad2ant2 1131 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (𝑊‘0) ∈ 𝑉))
3433imp 410 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → (𝑊‘0) ∈ 𝑉)
35 neeq2 3053 . . . . . . . . . 10 (𝑋 = (𝑊‘0) → ((lastS‘𝑊) ≠ 𝑋 ↔ (lastS‘𝑊) ≠ (𝑊‘0)))
3635eqcoms 2809 . . . . . . . . 9 ((𝑊‘0) = 𝑋 → ((lastS‘𝑊) ≠ 𝑋 ↔ (lastS‘𝑊) ≠ (𝑊‘0)))
3736biimpa 480 . . . . . . . 8 (((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋) → (lastS‘𝑊) ≠ (𝑊‘0))
3837adantl 485 . . . . . . 7 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (lastS‘𝑊) ≠ (𝑊‘0))
3938adantl 485 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → (lastS‘𝑊) ≠ (𝑊‘0))
4028, 34, 393jca 1125 . . . . 5 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → ((lastS‘𝑊) ∈ 𝑉 ∧ (𝑊‘0) ∈ 𝑉 ∧ (lastS‘𝑊) ≠ (𝑊‘0)))
411, 14frcond2 28056 . . . . 5 (𝐺 ∈ FriendGraph → (((lastS‘𝑊) ∈ 𝑉 ∧ (𝑊‘0) ∈ 𝑉 ∧ (lastS‘𝑊) ≠ (𝑊‘0)) → ∃!𝑣𝑉 ({(lastS‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺))))
4213, 40, 41sylc 65 . . . 4 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → ∃!𝑣𝑉 ({(lastS‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)))
43 simpl 486 . . . . . . . . . 10 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → 𝑊 ∈ (𝑁 WWalksN 𝐺))
4443ad2antlr 726 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → 𝑊 ∈ (𝑁 WWalksN 𝐺))
45 simpr 488 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → 𝑣𝑉)
46 nnnn0 11896 . . . . . . . . . . 11 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
47463ad2ant3 1132 . . . . . . . . . 10 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
4847ad2antrr 725 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → 𝑁 ∈ ℕ0)
4944, 45, 483jca 1125 . . . . . . . 8 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑣𝑉𝑁 ∈ ℕ0))
501, 14wwlksext2clwwlk 27846 . . . . . . . . . 10 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑣𝑉) → (({(lastS‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))
51503adant3 1129 . . . . . . . . 9 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑣𝑉𝑁 ∈ ℕ0) → (({(lastS‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))
5251imp 410 . . . . . . . 8 (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑣𝑉𝑁 ∈ ℕ0) ∧ ({(lastS‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺))) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺))
5349, 52sylan 583 . . . . . . 7 (((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) ∧ ({(lastS‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺))) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺))
541wwlknbp 27632 . . . . . . . . . . 11 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉))
5554simp3d 1141 . . . . . . . . . 10 (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ Word 𝑉)
5655ad2antrl 727 . . . . . . . . 9 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → 𝑊 ∈ Word 𝑉)
5756ad2antrr 725 . . . . . . . 8 (((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)) → 𝑊 ∈ Word 𝑉)
5845adantr 484 . . . . . . . 8 (((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)) → 𝑣𝑉)
59 2z 12006 . . . . . . . . . . 11 2 ∈ ℤ
60 nn0pzuz 12297 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ 2 ∈ ℤ) → (𝑁 + 2) ∈ (ℤ‘2))
6146, 59, 60sylancl 589 . . . . . . . . . 10 (𝑁 ∈ ℕ → (𝑁 + 2) ∈ (ℤ‘2))
62613ad2ant3 1132 . . . . . . . . 9 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑁 + 2) ∈ (ℤ‘2))
6362ad3antrrr 729 . . . . . . . 8 (((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)) → (𝑁 + 2) ∈ (ℤ‘2))
64 simpr 488 . . . . . . . 8 (((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺))
651, 14clwwlkext2edg 27845 . . . . . . . 8 (((𝑊 ∈ Word 𝑉𝑣𝑉 ∧ (𝑁 + 2) ∈ (ℤ‘2)) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)) → ({(lastS‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)))
6657, 58, 63, 64, 65syl31anc 1370 . . . . . . 7 (((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)) → ({(lastS‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)))
6753, 66impbida 800 . . . . . 6 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (({(lastS‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))
6845, 1eleqtrdi 2903 . . . . . . . . . 10 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → 𝑣 ∈ (Vtx‘𝐺))
6937anim2i 619 . . . . . . . . . . . 12 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑊) ≠ (𝑊‘0)))
7069ad2antlr 726 . . . . . . . . . . 11 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑊) ≠ (𝑊‘0)))
7170simprd 499 . . . . . . . . . 10 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (lastS‘𝑊) ≠ (𝑊‘0))
72 numclwwlk2lem1lem 28131 . . . . . . . . . 10 ((𝑣 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑊) ≠ (𝑊‘0)) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0) ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ (𝑊‘0)))
7368, 44, 71, 72syl3anc 1368 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0) ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ (𝑊‘0)))
74 eqeq2 2813 . . . . . . . . . . . . 13 (𝑋 = (𝑊‘0) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
7574eqcoms 2809 . . . . . . . . . . . 12 ((𝑊‘0) = 𝑋 → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
7675ad2antrl 727 . . . . . . . . . . 11 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
7776ad2antlr 726 . . . . . . . . . 10 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
7873simpld 498 . . . . . . . . . . 11 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0))
7978neeq2d 3050 . . . . . . . . . 10 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0) ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ (𝑊‘0)))
8077, 79anbi12d 633 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → ((((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)) ↔ (((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0) ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ (𝑊‘0))))
8173, 80mpbird 260 . . . . . . . 8 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
82 nncn 11637 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
83 2cnd 11707 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ → 2 ∈ ℂ)
8482, 83pncand 10991 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → ((𝑁 + 2) − 2) = 𝑁)
85843ad2ant3 1132 . . . . . . . . . . . 12 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((𝑁 + 2) − 2) = 𝑁)
8685ad2antrr 725 . . . . . . . . . . 11 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → ((𝑁 + 2) − 2) = 𝑁)
8786fveq2d 6653 . . . . . . . . . 10 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) = ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁))
8887neeq1d 3049 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0) ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
8988anbi2d 631 . . . . . . . 8 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → ((((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)) ↔ (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘𝑁) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0))))
9081, 89mpbird 260 . . . . . . 7 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
9190biantrud 535 . . . . . 6 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → ((𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺) ↔ ((𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))))
9261anim2i 619 . . . . . . . . . . 11 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑉 ∧ (𝑁 + 2) ∈ (ℤ‘2)))
93923adant1 1127 . . . . . . . . . 10 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑉 ∧ (𝑁 + 2) ∈ (ℤ‘2)))
9493ad2antrr 725 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (𝑋𝑉 ∧ (𝑁 + 2) ∈ (ℤ‘2)))
95 numclwwlk.h . . . . . . . . . 10 𝐻 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})
9695numclwwlkovh 28162 . . . . . . . . 9 ((𝑋𝑉 ∧ (𝑁 + 2) ∈ (ℤ‘2)) → (𝑋𝐻(𝑁 + 2)) = {𝑤 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))})
9794, 96syl 17 . . . . . . . 8 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (𝑋𝐻(𝑁 + 2)) = {𝑤 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))})
9897eleq2d 2878 . . . . . . 7 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → ((𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ {𝑤 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))}))
99 fveq1 6648 . . . . . . . . . 10 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑤‘0) = ((𝑊 ++ ⟨“𝑣”⟩)‘0))
10099eqeq1d 2803 . . . . . . . . 9 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ((𝑤‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋))
101 fveq1 6648 . . . . . . . . . 10 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑤‘((𝑁 + 2) − 2)) = ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)))
102101, 99neeq12d 3051 . . . . . . . . 9 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ((𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0) ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
103100, 102anbi12d 633 . . . . . . . 8 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0)) ↔ (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0))))
104103elrab 3631 . . . . . . 7 ((𝑊 ++ ⟨“𝑣”⟩) ∈ {𝑤 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))} ↔ ((𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0))))
10598, 104syl6rbb 291 . . . . . 6 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (((𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0))) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
10667, 91, 1053bitrd 308 . . . . 5 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (({(lastS‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)) ↔ (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
107106reubidva 3344 . . . 4 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → (∃!𝑣𝑉 ({(lastS‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)) ↔ ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
10842, 107mpbid 235 . . 3 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)))
109108ex 416 . 2 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
11012, 109sylbid 243 1 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) → ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  ∀wral 3109  ∃!wreu 3111  {crab 3113  Vcvv 3444  {cpr 4530  ‘cfv 6328  (class class class)co 7139   ∈ cmpo 7141  0cc0 10530  1c1 10531   + caddc 10533   − cmin 10863  ℕcn 11629  2c2 11684  ℕ0cn0 11889  ℤcz 11973  ℤ≥cuz 12235  ..^cfzo 13032  ♯chash 13690  Word cword 13861  lastSclsw 13909   ++ cconcat 13917  ⟨“cs1 13944  Vtxcvtx 26793  Edgcedg 26844   WWalksN cwwlksn 27616   ClWWalksN cclwwlkn 27813  ClWWalksNOncclwwlknon 27876   FriendGraph cfrgr 28047 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-rp 12382  df-fz 12890  df-fzo 13033  df-hash 13691  df-word 13862  df-lsw 13910  df-concat 13918  df-s1 13945  df-wwlks 27620  df-wwlksn 27621  df-clwwlk 27771  df-clwwlkn 27814  df-clwwlknon 27877  df-frgr 28048 This theorem is referenced by:  numclwlk2lem2f1o  28168
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