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Theorem numclwwlk2lem1 27929
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 27927 . . . . 5 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
433adant1 1110 . . . 4 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
54eleq2d 2852 . . 3 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) ↔ 𝑊 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}))
6 fveq1 6498 . . . . . 6 (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0))
76eqeq1d 2781 . . . . 5 (𝑤 = 𝑊 → ((𝑤‘0) = 𝑋 ↔ (𝑊‘0) = 𝑋))
8 fveq2 6499 . . . . . 6 (𝑤 = 𝑊 → (lastS‘𝑤) = (lastS‘𝑊))
98neeq1d 3027 . . . . 5 (𝑤 = 𝑊 → ((lastS‘𝑤) ≠ 𝑋 ↔ (lastS‘𝑊) ≠ 𝑋))
107, 9anbi12d 621 . . . 4 (𝑤 = 𝑊 → (((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋) ↔ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)))
1110elrab 3596 . . 3 (𝑊 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)} ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)))
125, 11syl6bb 279 . 2 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))))
13 simpl1 1171 . . . . 5 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → 𝐺 ∈ FriendGraph )
14 eqid 2779 . . . . . . . . . . . . 13 (Edg‘𝐺) = (Edg‘𝐺)
151, 14wwlknp 27329 . . . . . . . . . . . 12 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
16 peano2nn 11453 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
1716adantl 474 . . . . . . . . . . . . . . 15 (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ)
18 simpl 475 . . . . . . . . . . . . . . 15 (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)))
1917, 18jca 504 . . . . . . . . . . . . . 14 (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1))))
2019ex 405 . . . . . . . . . . . . 13 ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)))))
21203adant3 1112 . . . . . . . . . . . 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 13732 . . . . . . . . . . 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 473 . . . . . . . . 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 1115 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (lastS‘𝑊) ∈ 𝑉))
2827imp 398 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → (lastS‘𝑊) ∈ 𝑉)
29 eleq1 2854 . . . . . . . . . . 11 ((𝑊‘0) = 𝑋 → ((𝑊‘0) ∈ 𝑉𝑋𝑉))
3029biimprd 240 . . . . . . . . . 10 ((𝑊‘0) = 𝑋 → (𝑋𝑉 → (𝑊‘0) ∈ 𝑉))
3130ad2antrl 715 . . . . . . . . 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 1114 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (𝑊‘0) ∈ 𝑉))
3433imp 398 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → (𝑊‘0) ∈ 𝑉)
35 neeq2 3031 . . . . . . . . . 10 (𝑋 = (𝑊‘0) → ((lastS‘𝑊) ≠ 𝑋 ↔ (lastS‘𝑊) ≠ (𝑊‘0)))
3635eqcoms 2787 . . . . . . . . 9 ((𝑊‘0) = 𝑋 → ((lastS‘𝑊) ≠ 𝑋 ↔ (lastS‘𝑊) ≠ (𝑊‘0)))
3736biimpa 469 . . . . . . . 8 (((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋) → (lastS‘𝑊) ≠ (𝑊‘0))
3837adantl 474 . . . . . . 7 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (lastS‘𝑊) ≠ (𝑊‘0))
3938adantl 474 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → (lastS‘𝑊) ≠ (𝑊‘0))
4028, 34, 393jca 1108 . . . . 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 27801 . . . . 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 475 . . . . . . . . . 10 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → 𝑊 ∈ (𝑁 WWalksN 𝐺))
4443ad2antlr 714 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → 𝑊 ∈ (𝑁 WWalksN 𝐺))
45 simpr 477 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → 𝑣𝑉)
46 nnnn0 11715 . . . . . . . . . . 11 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
47463ad2ant3 1115 . . . . . . . . . 10 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
4847ad2antrr 713 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → 𝑁 ∈ ℕ0)
4944, 45, 483jca 1108 . . . . . . . 8 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑣𝑉𝑁 ∈ ℕ0))
501, 14wwlksext2clwwlk 27580 . . . . . . . . . 10 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑣𝑉) → (({(lastS‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))
51503adant3 1112 . . . . . . . . 9 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑣𝑉𝑁 ∈ ℕ0) → (({(lastS‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))
5251imp 398 . . . . . . . 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 572 . . . . . . 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 27328 . . . . . . . . . . 11 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉))
5554simp3d 1124 . . . . . . . . . 10 (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ Word 𝑉)
5655ad2antrl 715 . . . . . . . . 9 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → 𝑊 ∈ Word 𝑉)
5756ad2antrr 713 . . . . . . . 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 473 . . . . . . . 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 11827 . . . . . . . . . . 11 2 ∈ ℤ
60 nn0pzuz 12119 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ 2 ∈ ℤ) → (𝑁 + 2) ∈ (ℤ‘2))
6146, 59, 60sylancl 577 . . . . . . . . . 10 (𝑁 ∈ ℕ → (𝑁 + 2) ∈ (ℤ‘2))
62613ad2ant3 1115 . . . . . . . . 9 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑁 + 2) ∈ (ℤ‘2))
6362ad3antrrr 717 . . . . . . . 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 477 . . . . . . . 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 27579 . . . . . . . 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 1353 . . . . . . 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 788 . . . . . 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, 1syl6eleq 2877 . . . . . . . . . 10 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → 𝑣 ∈ (Vtx‘𝐺))
6937anim2i 607 . . . . . . . . . . . 12 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑊) ≠ (𝑊‘0)))
7069ad2antlr 714 . . . . . . . . . . 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 488 . . . . . . . . . 10 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (lastS‘𝑊) ≠ (𝑊‘0))
72 numclwwlk2lem1lem 27876 . . . . . . . . . 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 1351 . . . . . . . . 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 2790 . . . . . . . . . . . . 13 (𝑋 = (𝑊‘0) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
7574eqcoms 2787 . . . . . . . . . . . 12 ((𝑊‘0) = 𝑋 → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
7675ad2antrl 715 . . . . . . . . . . 11 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
7776ad2antlr 714 . . . . . . . . . 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 487 . . . . . . . . . . 11 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0))
7978neeq2d 3028 . . . . . . . . . 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 621 . . . . . . . . 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 249 . . . . . . . 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 11448 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
83 2cnd 11518 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ → 2 ∈ ℂ)
8482, 83pncand 10799 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → ((𝑁 + 2) − 2) = 𝑁)
85843ad2ant3 1115 . . . . . . . . . . . 12 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((𝑁 + 2) − 2) = 𝑁)
8685ad2antrr 713 . . . . . . . . . . 11 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → ((𝑁 + 2) − 2) = 𝑁)
8786fveq2d 6503 . . . . . . . . . 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 3027 . . . . . . . . 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 619 . . . . . . . 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 249 . . . . . . 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 524 . . . . . 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 607 . . . . . . . . . . 11 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑉 ∧ (𝑁 + 2) ∈ (ℤ‘2)))
93923adant1 1110 . . . . . . . . . 10 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑉 ∧ (𝑁 + 2) ∈ (ℤ‘2)))
9493ad2antrr 713 . . . . . . . . 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 27926 . . . . . . . . 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 2852 . . . . . . 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 6498 . . . . . . . . . 10 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑤‘0) = ((𝑊 ++ ⟨“𝑣”⟩)‘0))
10099eqeq1d 2781 . . . . . . . . 9 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ((𝑤‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋))
101 fveq1 6498 . . . . . . . . . 10 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑤‘((𝑁 + 2) − 2)) = ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)))
102101, 99neeq12d 3029 . . . . . . . . 9 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ((𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0) ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
103100, 102anbi12d 621 . . . . . . . 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 3596 . . . . . . 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 280 . . . . . 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 297 . . . . 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 3328 . . . 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 224 . . 3 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)))
109108ex 405 . 2 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
11012, 109sylbid 232 1 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) → ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wne 2968  wral 3089  ∃!wreu 3091  {crab 3093  Vcvv 3416  {cpr 4443  cfv 6188  (class class class)co 6976  cmpo 6978  0cc0 10335  1c1 10336   + caddc 10338  cmin 10670  cn 11439  2c2 11495  0cn0 11707  cz 11793  cuz 12058  ..^cfzo 12849  chash 13505  Word cword 13672  lastSclsw 13725   ++ cconcat 13733  ⟨“cs1 13758  Vtxcvtx 26484  Edgcedg 26535   WWalksN cwwlksn 27312   ClWWalksN cclwwlkn 27539  ClWWalksNOncclwwlknon 27615   FriendGraph cfrgr 27790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-pss 3846  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-oadd 7909  df-er 8089  df-map 8208  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-card 9162  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-nn 11440  df-2 11503  df-n0 11708  df-xnn0 11780  df-z 11794  df-uz 12059  df-rp 12205  df-fz 12709  df-fzo 12850  df-hash 13506  df-word 13673  df-lsw 13726  df-concat 13734  df-s1 13759  df-wwlks 27316  df-wwlksn 27317  df-clwwlk 27488  df-clwwlkn 27540  df-clwwlknon 27616  df-frgr 27791
This theorem is referenced by:  numclwlk2lem2f1o  27932  numclwlk2lem2f1oOLD  27935
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