numclwlk2lem2f1o - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > numclwlk2lem2f1o		Structured version Visualization version GIF version

Theorem numclwlk2lem2f1o 30176

Description: 𝑅 is a 1-1 onto function. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 21-Jan-2022.) (Proof shortened by AV, 17-Mar-2022.) (Revised by AV, 1-Nov-2022.)

**Hypotheses**
Ref	Expression
numclwwlk.v	⊢ 𝑉 = (Vtx‘𝐺)
numclwwlk.q	⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})
numclwwlk.h	⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})
numclwwlk.r	⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 prefix (𝑁 + 1)))

**Assertion**
Ref	Expression
numclwlk2lem2f1o	⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))–1-1-onto→(𝑋𝑄𝑁))

Distinct variable groups: 𝑛,𝐺,𝑣,𝑤 𝑛,𝑁,𝑣,𝑤 𝑛,𝑉,𝑣 𝑛,𝑋,𝑣,𝑤 𝑤,𝑉 𝑥,𝐺,𝑤 𝑥,𝐻 𝑥,𝑁 𝑥,𝑄 𝑥,𝑉 𝑥,𝑋,𝑣 Allowed substitution hints: 𝑄(𝑤,𝑣,𝑛) 𝑅(𝑥,𝑤,𝑣,𝑛) 𝐻(𝑤,𝑣,𝑛)

Proof of Theorem numclwlk2lem2f1o Dummy variables 𝑦 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2811	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (𝑋𝐻(𝑁 + 2)) ↔ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))))
2		fveq2 6891	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑅‘𝑦) = (𝑅‘𝑥))
3		oveq1 7421	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)))
4	2, 3	eqeq12d 2743	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝑅‘𝑦) = (𝑦 prefix (𝑁 + 1)) ↔ (𝑅‘𝑥) = (𝑥 prefix (𝑁 + 1))))
5	1, 4	imbi12d 344	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝑦 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑦) = (𝑦 prefix (𝑁 + 1))) ↔ (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑥) = (𝑥 prefix (𝑁 + 1)))))
6	5	imbi2d 340	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑦 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑦) = (𝑦 prefix (𝑁 + 1)))) ↔ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑥) = (𝑥 prefix (𝑁 + 1))))))
7		numclwwlk.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
8		numclwwlk.q	. . . . . . . 8 ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})
9		numclwwlk.h	. . . . . . . 8 ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})
10		numclwwlk.r	. . . . . . . 8 ⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 prefix (𝑁 + 1)))
11	7, 8, 9, 10	numclwlk2lem2fv 30175	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑦 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑦) = (𝑦 prefix (𝑁 + 1))))
12	6, 11	chvarvv 1995	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑥) = (𝑥 prefix (𝑁 + 1))))
13	12	3adant1 1128	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑥) = (𝑥 prefix (𝑁 + 1))))
14	13	imp 406	. . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑅‘𝑥) = (𝑥 prefix (𝑁 + 1)))
15	7, 8, 9, 10	numclwlk2lem2f 30174	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))⟶(𝑋𝑄𝑁))
16	15	ffvelcdmda 7088	. . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑅‘𝑥) ∈ (𝑋𝑄𝑁))
17	14, 16	eqeltrrd 2829	. . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑥 prefix (𝑁 + 1)) ∈ (𝑋𝑄𝑁))
18	17	ralrimiva 3141	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 prefix (𝑁 + 1)) ∈ (𝑋𝑄𝑁))
19	7, 8, 9	numclwwlk2lem1 30173	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ (𝑋𝑄𝑁) → ∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))
20	19	imp 406	. . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑢 ∈ (𝑋𝑄𝑁)) → ∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)))
21	7, 8	numclwwlkovq 30171	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
22	21	eleq2d 2814	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ (𝑋𝑄𝑁) ↔ 𝑢 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}))
23	22	3adant1 1128	. . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ (𝑋𝑄𝑁) ↔ 𝑢 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}))
24		fveq1 6890	. . . . . . . . . 10 ⊢ (𝑤 = 𝑢 → (𝑤‘0) = (𝑢‘0))
25	24	eqeq1d 2729	. . . . . . . . 9 ⊢ (𝑤 = 𝑢 → ((𝑤‘0) = 𝑋 ↔ (𝑢‘0) = 𝑋))
26		fveq2 6891	. . . . . . . . . 10 ⊢ (𝑤 = 𝑢 → (lastS‘𝑤) = (lastS‘𝑢))
27	26	neeq1d 2995	. . . . . . . . 9 ⊢ (𝑤 = 𝑢 → ((lastS‘𝑤) ≠ 𝑋 ↔ (lastS‘𝑢) ≠ 𝑋))
28	25, 27	anbi12d 630	. . . . . . . 8 ⊢ (𝑤 = 𝑢 → (((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋) ↔ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)))
29	28	elrab 3680	. . . . . . 7 ⊢ (𝑢 ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)} ↔ (𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)))
30	23, 29	bitrdi 287	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ (𝑋𝑄𝑁) ↔ (𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋))))
31		wwlknbp1 29642	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ₀ ∧ 𝑢 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑢) = (𝑁 + 1)))
32		3simpc 1148	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑢 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑢) = (𝑁 + 1)) → (𝑢 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑢) = (𝑁 + 1)))
33	31, 32	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (𝑁 WWalksN 𝐺) → (𝑢 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑢) = (𝑁 + 1)))
34	7	wrdeqi 14511	. . . . . . . . . . . . . . . . 17 ⊢ Word 𝑉 = Word (Vtx‘𝐺)
35	34	eleq2i 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ Word 𝑉 ↔ 𝑢 ∈ Word (Vtx‘𝐺))
36	35	anbi1i 623	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ↔ (𝑢 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑢) = (𝑁 + 1)))
37	33, 36	sylibr 233	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (𝑁 WWalksN 𝐺) → (𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)))
38		simpll 766	. . . . . . . . . . . . . . . 16 ⊢ (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑢 ∈ Word 𝑉)
39		nnnn0 12501	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
40		2nn 12307	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 2 ∈ ℕ
41	40	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℕ)
42	41	nnzd 12607	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℤ)
43		nn0pzuz 12911	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℕ₀ ∧ 2 ∈ ℤ) → (𝑁 + 2) ∈ (ℤ_≥‘2))
44	39, 42, 43	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → (𝑁 + 2) ∈ (ℤ_≥‘2))
45	9	numclwwlkovh 30170	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑁 + 2) ∈ (ℤ_≥‘2)) → (𝑋𝐻(𝑁 + 2)) = {𝑤 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))})
46	44, 45	sylan2 592	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝐻(𝑁 + 2)) = {𝑤 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))})
47	46	eleq2d 2814	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↔ 𝑥 ∈ {𝑤 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))}))
48		fveq1 6890	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑥 → (𝑤‘0) = (𝑥‘0))
49	48	eqeq1d 2729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑥 → ((𝑤‘0) = 𝑋 ↔ (𝑥‘0) = 𝑋))
50		fveq1 6890	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑥 → (𝑤‘((𝑁 + 2) − 2)) = (𝑥‘((𝑁 + 2) − 2)))
51	50, 48	neeq12d 2997	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑥 → ((𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0) ↔ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0)))
52	49, 51	anbi12d 630	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = 𝑥 → (((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0)) ↔ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0))))
53	52	elrab 3680	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ {𝑤 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘((𝑁 + 2) − 2)) ≠ (𝑤‘0))} ↔ (𝑥 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0))))
54	47, 53	bitrdi 287	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↔ (𝑥 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0)))))
55	54	3adant1 1128	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↔ (𝑥 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0)))))
56	55	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↔ (𝑥 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0)))))
57	7	clwwlknbp 29832	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ((𝑁 + 2) ClWWalksN 𝐺) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = (𝑁 + 2)))
58		lencl 14507	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑢 ∈ Word 𝑉 → (♯‘𝑢) ∈ ℕ₀)
59		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉)) → 𝑥 ∈ Word 𝑉)
60		df-2 12297	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ 2 = (1 + 1)
61	60	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑁 ∈ ℕ → 2 = (1 + 1))
62	61	oveq2d 7430	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑁 ∈ ℕ → (𝑁 + 2) = (𝑁 + (1 + 1)))
63		nncn 12242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
64		1cnd 11231	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
65	63, 64, 64	addassd 11258	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1)))
66	62, 65	eqtr4d 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑁 ∈ ℕ → (𝑁 + 2) = ((𝑁 + 1) + 1))
67	66	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (𝑁 + 2) = ((𝑁 + 1) + 1))
68	67	eqeq2d 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → ((♯‘𝑥) = (𝑁 + 2) ↔ (♯‘𝑥) = ((𝑁 + 1) + 1)))
69	68	biimpcd 248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((♯‘𝑥) = (𝑁 + 2) → ((((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (♯‘𝑥) = ((𝑁 + 1) + 1)))
70	69	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → ((((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) → (♯‘𝑥) = ((𝑁 + 1) + 1)))
71	70	impcom 407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉)) → (♯‘𝑥) = ((𝑁 + 1) + 1))
72		oveq1 7421	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((♯‘𝑢) = (𝑁 + 1) → ((♯‘𝑢) + 1) = ((𝑁 + 1) + 1))
73	72	ad3antlr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉)) → ((♯‘𝑢) + 1) = ((𝑁 + 1) + 1))
74	71, 73	eqtr4d 2770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉)) → (♯‘𝑥) = ((♯‘𝑢) + 1))
75	59, 74	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ 𝑁 ∈ ℕ) ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉)) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1)))
76	75	exp31 419	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((♯‘𝑢) ∈ ℕ₀ ∧ (♯‘𝑢) = (𝑁 + 1)) → (𝑁 ∈ ℕ → (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1)))))
77	58, 76	sylan 579	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) → (𝑁 ∈ ℕ → (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1)))))
78	77	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → ((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) → (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1)))))
79	78	3ad2ant3 1133	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) → (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1)))))
80	79	impcom 407	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))))
81	80	com12 32	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑥 ∈ Word 𝑉) → (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))))
82	81	ancoms 458	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = (𝑁 + 2)) → (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))))
83	57, 82	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ((𝑁 + 2) ClWWalksN 𝐺) → (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))))
84	83	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0))) → (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))))
85	84	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ((𝑥 ∈ ((𝑁 + 2) ClWWalksN 𝐺) ∧ ((𝑥‘0) = 𝑋 ∧ (𝑥‘((𝑁 + 2) − 2)) ≠ (𝑥‘0))) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))))
86	56, 85	sylbid 239	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))))
87	86	ralrimiv 3140	. . . . . . . . . . . . . . . 16 ⊢ (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1)))
88	38, 87	jca 511	. . . . . . . . . . . . . . 15 ⊢ (((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑢 ∈ Word 𝑉 ∧ ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))))
89	88	ex 412	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ Word 𝑉 ∧ (♯‘𝑢) = (𝑁 + 1)) → ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ Word 𝑉 ∧ ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1)))))
90	37, 89	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ (𝑁 WWalksN 𝐺) → ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ Word 𝑉 ∧ ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1)))))
91	90	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) → ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ Word 𝑉 ∧ ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1)))))
92	91	imp 406	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑢 ∈ Word 𝑉 ∧ ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))))
93		nfcv 2898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣𝑋
94		nfmpo1 7494	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑣(𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})
95	9, 94	nfcxfr 2896	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣𝐻
96		nfcv 2898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣(𝑁 + 2)
97	93, 95, 96	nfov 7444	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣(𝑋𝐻(𝑁 + 2))
98	97	reuccatpfxs1 14721	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ Word 𝑉 ∧ ∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑢) + 1))) → (∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (♯‘𝑢))))
99	92, 98	syl 17	. . . . . . . . . 10 ⊢ (((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (♯‘𝑢))))
100	99	imp 406	. . . . . . . . 9 ⊢ ((((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (♯‘𝑢)))
101	31	simp3d 1142	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (𝑁 WWalksN 𝐺) → (♯‘𝑢) = (𝑁 + 1))
102	101	eqcomd 2733	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ (𝑁 WWalksN 𝐺) → (𝑁 + 1) = (♯‘𝑢))
103	102	ad4antr 731	. . . . . . . . . . . 12 ⊢ (((((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑁 + 1) = (♯‘𝑢))
104	103	oveq2d 7430	. . . . . . . . . . 11 ⊢ (((((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (♯‘𝑢)))
105	104	eqeq2d 2738	. . . . . . . . . 10 ⊢ (((((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))) ∧ 𝑥 ∈ (𝑋𝐻(𝑁 + 2))) → (𝑢 = (𝑥 prefix (𝑁 + 1)) ↔ 𝑢 = (𝑥 prefix (♯‘𝑢))))
106	105	reubidva 3387	. . . . . . . . 9 ⊢ ((((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))) → (∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (𝑁 + 1)) ↔ ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (♯‘𝑢))))
107	100, 106	mpbird 257	. . . . . . . 8 ⊢ ((((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) ∧ (𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) ∧ ∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (𝑁 + 1)))
108	107	exp31 419	. . . . . . 7 ⊢ ((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) → ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (𝑁 + 1)))))
109	108	com12 32	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑢 ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑢‘0) = 𝑋 ∧ (lastS‘𝑢) ≠ 𝑋)) → (∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (𝑁 + 1)))))
110	30, 109	sylbid 239	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑢 ∈ (𝑋𝑄𝑁) → (∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (𝑁 + 1)))))
111	110	imp 406	. . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑢 ∈ (𝑋𝑄𝑁)) → (∃!𝑣 ∈ 𝑉 (𝑢 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (𝑁 + 1))))
112	20, 111	mpd 15	. . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑢 ∈ (𝑋𝑄𝑁)) → ∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (𝑁 + 1)))
113	112	ralrimiva 3141	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ∀𝑢 ∈ (𝑋𝑄𝑁)∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (𝑁 + 1)))
114	10	f1ompt 7115	. 2 ⊢ (𝑅:(𝑋𝐻(𝑁 + 2))–1-1-onto→(𝑋𝑄𝑁) ↔ (∀𝑥 ∈ (𝑋𝐻(𝑁 + 2))(𝑥 prefix (𝑁 + 1)) ∈ (𝑋𝑄𝑁) ∧ ∀𝑢 ∈ (𝑋𝑄𝑁)∃!𝑥 ∈ (𝑋𝐻(𝑁 + 2))𝑢 = (𝑥 prefix (𝑁 + 1))))
115	18, 113, 114	sylanbrc 582	1 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))–1-1-onto→(𝑋𝑄𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∀wral 3056 ∃!wreu 3369 {crab 3427 ↦ cmpt 5225 –1-1-onto→wf1o 6541 ‘cfv 6542 (class class class)co 7414 ∈ cmpo 7416 0cc0 11130 1c1 11131 + caddc 11133 − cmin 11466 ℕcn 12234 2c2 12289 ℕ₀cn0 12494 ℤcz 12580 ℤ_≥cuz 12844 ♯chash 14313 Word cword 14488 lastSclsw 14536 ++ cconcat 14544 ⟨“cs1 14569 prefix cpfx 14644 Vtxcvtx 28796 WWalksN cwwlksn 29624 ClWWalksN cclwwlkn 29821 ClWWalksNOncclwwlknon 29884 FriendGraph cfrgr 30055

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-n0 12495 df-xnn0 12567 df-z 12581 df-uz 12845 df-rp 12999 df-fz 13509 df-fzo 13652 df-hash 14314 df-word 14489 df-lsw 14537 df-concat 14545 df-s1 14570 df-substr 14615 df-pfx 14645 df-wwlks 29628 df-wwlksn 29629 df-clwwlk 29779 df-clwwlkn 29822 df-clwwlknon 29885 df-frgr 30056

This theorem is referenced by: numclwwlk2lem3 30177

W3C validator