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Theorem numclwwlk2lem1 30395
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 30393 . . . . 5 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
433adant1 1131 . . . 4 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
54eleq2d 2827 . . 3 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) ↔ 𝑊 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}))
6 fveq1 6905 . . . . . 6 (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0))
76eqeq1d 2739 . . . . 5 (𝑤 = 𝑊 → ((𝑤‘0) = 𝑋 ↔ (𝑊‘0) = 𝑋))
8 fveq2 6906 . . . . . 6 (𝑤 = 𝑊 → (lastS‘𝑤) = (lastS‘𝑊))
98neeq1d 3000 . . . . 5 (𝑤 = 𝑊 → ((lastS‘𝑤) ≠ 𝑋 ↔ (lastS‘𝑊) ≠ 𝑋))
107, 9anbi12d 632 . . . 4 (𝑤 = 𝑊 → (((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋) ↔ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)))
1110elrab 3692 . . 3 (𝑊 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)} ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)))
125, 11bitrdi 287 . 2 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))))
13 simpl1 1192 . . . . 5 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → 𝐺 ∈ FriendGraph )
14 eqid 2737 . . . . . . . . . . . . 13 (Edg‘𝐺) = (Edg‘𝐺)
151, 14wwlknp 29863 . . . . . . . . . . . 12 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
16 peano2nn 12278 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
1716adantl 481 . . . . . . . . . . . . . . 15 (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑁 + 1) ∈ ℕ)
18 simpl 482 . . . . . . . . . . . . . . 15 (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)))
1917, 18jca 511 . . . . . . . . . . . . . 14 (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1))))
2019ex 412 . . . . . . . . . . . . 13 ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑁 ∈ ℕ → ((𝑁 + 1) ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)))))
21203adant3 1133 . . . . . . . . . . . 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 14607 . . . . . . . . . . 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 480 . . . . . . . . 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 1136 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (lastS‘𝑊) ∈ 𝑉))
2827imp 406 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → (lastS‘𝑊) ∈ 𝑉)
29 eleq1 2829 . . . . . . . . . . 11 ((𝑊‘0) = 𝑋 → ((𝑊‘0) ∈ 𝑉𝑋𝑉))
3029biimprd 248 . . . . . . . . . 10 ((𝑊‘0) = 𝑋 → (𝑋𝑉 → (𝑊‘0) ∈ 𝑉))
3130ad2antrl 728 . . . . . . . . 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 1135 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (𝑊‘0) ∈ 𝑉))
3433imp 406 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → (𝑊‘0) ∈ 𝑉)
35 neeq2 3004 . . . . . . . . . 10 (𝑋 = (𝑊‘0) → ((lastS‘𝑊) ≠ 𝑋 ↔ (lastS‘𝑊) ≠ (𝑊‘0)))
3635eqcoms 2745 . . . . . . . . 9 ((𝑊‘0) = 𝑋 → ((lastS‘𝑊) ≠ 𝑋 ↔ (lastS‘𝑊) ≠ (𝑊‘0)))
3736biimpa 476 . . . . . . . 8 (((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋) → (lastS‘𝑊) ≠ (𝑊‘0))
3837adantl 481 . . . . . . 7 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (lastS‘𝑊) ≠ (𝑊‘0))
3938adantl 481 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → (lastS‘𝑊) ≠ (𝑊‘0))
4028, 34, 393jca 1129 . . . . 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 30286 . . . . 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 482 . . . . . . . . . 10 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → 𝑊 ∈ (𝑁 WWalksN 𝐺))
4443ad2antlr 727 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → 𝑊 ∈ (𝑁 WWalksN 𝐺))
45 simpr 484 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → 𝑣𝑉)
46 nnnn0 12533 . . . . . . . . . . 11 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
47463ad2ant3 1136 . . . . . . . . . 10 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)
4847ad2antrr 726 . . . . . . . . 9 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → 𝑁 ∈ ℕ0)
4944, 45, 483jca 1129 . . . . . . . 8 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑣𝑉𝑁 ∈ ℕ0))
501, 14wwlksext2clwwlk 30076 . . . . . . . . . 10 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑣𝑉) → (({(lastS‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))
51503adant3 1133 . . . . . . . . 9 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑣𝑉𝑁 ∈ ℕ0) → (({(lastS‘𝑊), 𝑣} ∈ (Edg‘𝐺) ∧ {𝑣, (𝑊‘0)} ∈ (Edg‘𝐺)) → (𝑊 ++ ⟨“𝑣”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))
5251imp 406 . . . . . . . 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 580 . . . . . . 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 29862 . . . . . . . . . . 11 (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉))
5554simp3d 1145 . . . . . . . . . 10 (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ Word 𝑉)
5655ad2antrl 728 . . . . . . . . 9 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → 𝑊 ∈ Word 𝑉)
5756ad2antrr 726 . . . . . . . 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 480 . . . . . . . 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 12649 . . . . . . . . . . 11 2 ∈ ℤ
60 nn0pzuz 12947 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ 2 ∈ ℤ) → (𝑁 + 2) ∈ (ℤ‘2))
6146, 59, 60sylancl 586 . . . . . . . . . 10 (𝑁 ∈ ℕ → (𝑁 + 2) ∈ (ℤ‘2))
62613ad2ant3 1136 . . . . . . . . 9 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑁 + 2) ∈ (ℤ‘2))
6362ad3antrrr 730 . . . . . . . 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 484 . . . . . . . 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 30075 . . . . . . . 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 1375 . . . . . . 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 801 . . . . . 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 2851 . . . . . . . . . 10 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → 𝑣 ∈ (Vtx‘𝐺))
6937anim2i 617 . . . . . . . . . . . 12 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑊) ≠ (𝑊‘0)))
7069ad2antlr 727 . . . . . . . . . . 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 495 . . . . . . . . . 10 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → (lastS‘𝑊) ≠ (𝑊‘0))
72 numclwwlk2lem1lem 30361 . . . . . . . . . 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 1373 . . . . . . . . 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 2749 . . . . . . . . . . . . 13 (𝑋 = (𝑊‘0) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
7574eqcoms 2745 . . . . . . . . . . . 12 ((𝑊‘0) = 𝑋 → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
7675ad2antrl 728 . . . . . . . . . . 11 ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → (((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0)))
7776ad2antlr 727 . . . . . . . . . 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 494 . . . . . . . . . . 11 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → ((𝑊 ++ ⟨“𝑣”⟩)‘0) = (𝑊‘0))
7978neeq2d 3001 . . . . . . . . . 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 632 . . . . . . . . 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 257 . . . . . . . 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 12274 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
83 2cnd 12344 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ → 2 ∈ ℂ)
8482, 83pncand 11621 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → ((𝑁 + 2) − 2) = 𝑁)
85843ad2ant3 1136 . . . . . . . . . . . 12 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((𝑁 + 2) − 2) = 𝑁)
8685ad2antrr 726 . . . . . . . . . . 11 ((((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) ∧ 𝑣𝑉) → ((𝑁 + 2) − 2) = 𝑁)
8786fveq2d 6910 . . . . . . . . . 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 3000 . . . . . . . . 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 630 . . . . . . . 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 257 . . . . . . 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 531 . . . . . 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 617 . . . . . . . . . . 11 ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑉 ∧ (𝑁 + 2) ∈ (ℤ‘2)))
93923adant1 1131 . . . . . . . . . 10 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑉 ∧ (𝑁 + 2) ∈ (ℤ‘2)))
9493ad2antrr 726 . . . . . . . . 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 30392 . . . . . . . . 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 2827 . . . . . . 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 6905 . . . . . . . . . 10 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑤‘0) = ((𝑊 ++ ⟨“𝑣”⟩)‘0))
10099eqeq1d 2739 . . . . . . . . 9 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ((𝑤‘0) = 𝑋 ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘0) = 𝑋))
101 fveq1 6905 . . . . . . . . . 10 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑤‘((𝑁 + 2) − 2)) = ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)))
102101, 99neeq12d 3002 . . . . . . . . 9 (𝑤 = (𝑊 ++ ⟨“𝑣”⟩) → ((𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0) ↔ ((𝑊 ++ ⟨“𝑣”⟩)‘((𝑁 + 2) − 2)) ≠ ((𝑊 ++ ⟨“𝑣”⟩)‘0)))
103100, 102anbi12d 632 . . . . . . . 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 3692 . . . . . . 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, 104bitr2di 288 . . . . . 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 305 . . . . 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 3396 . . . 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 232 . . 3 (((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋))) → ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)))
109108ex 412 . 2 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊‘0) = 𝑋 ∧ (lastS‘𝑊) ≠ 𝑋)) → ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
11012, 109sylbid 240 1 ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) → ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  ∃!wreu 3378  {crab 3436  Vcvv 3480  {cpr 4628  cfv 6561  (class class class)co 7431  cmpo 7433  0cc0 11155  1c1 11156   + caddc 11158  cmin 11492  cn 12266  2c2 12321  0cn0 12526  cz 12613  cuz 12878  ..^cfzo 13694  chash 14369  Word cword 14552  lastSclsw 14600   ++ cconcat 14608  ⟨“cs1 14633  Vtxcvtx 29013  Edgcedg 29064   WWalksN cwwlksn 29846   ClWWalksN cclwwlkn 30043  ClWWalksNOncclwwlknon 30106   FriendGraph cfrgr 30277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-lsw 14601  df-concat 14609  df-s1 14634  df-wwlks 29850  df-wwlksn 29851  df-clwwlk 30001  df-clwwlkn 30044  df-clwwlknon 30107  df-frgr 30278
This theorem is referenced by:  numclwlk2lem2f1o  30398
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