clwwlknonwwlknonb - Metamath Proof Explorer

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Theorem clwwlknonwwlknonb 30197

Description: A word over vertices represents a closed walk of a fixed length 𝑁 on vertex 𝑋 iff the word concatenated with 𝑋 represents a walk of length 𝑁 on 𝑋 and 𝑋. This theorem would not hold for 𝑁 = 0 and 𝑊 = ∅, see clwwlknwwlksnb 30146. (Contributed by AV, 4-Mar-2022.) (Revised by AV, 27-Mar-2022.)

**Hypothesis**
Ref	Expression
clwwlknonwwlknonb.v	⊢ 𝑉 = (Vtx‘𝐺)

**Assertion**
Ref	Expression
clwwlknonwwlknonb	⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑋(𝑁 WWalksNOn 𝐺)𝑋)))

Proof of Theorem clwwlknonwwlknonb Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isclwwlknon 30182	. . 3 ⊢ (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑊‘0) = 𝑋))
2		3anan32 1097	. . . . 5 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = 𝑋) ↔ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = 𝑋) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋))
3		s1eq 14536	. . . . . . . . . . . 12 ⊢ ((𝑊‘0) = 𝑋 → ⟨“(𝑊‘0)”⟩ = ⟨“𝑋”⟩)
4	3	oveq2d 7384	. . . . . . . . . . 11 ⊢ ((𝑊‘0) = 𝑋 → (𝑊 ++ ⟨“(𝑊‘0)”⟩) = (𝑊 ++ ⟨“𝑋”⟩))
5	4	eleq1d 2822	. . . . . . . . . 10 ⊢ ((𝑊‘0) = 𝑋 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺)))
6	5	biimpac 478	. . . . . . . . 9 ⊢ (((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺))
7	6	adantl 481	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺))
8		fvex 6855	. . . . . . . . . . . . . 14 ⊢ (𝑊‘0) ∈ V
9		eleq1 2825	. . . . . . . . . . . . . 14 ⊢ ((𝑊‘0) = 𝑋 → ((𝑊‘0) ∈ V ↔ 𝑋 ∈ V))
10	8, 9	mpbii 233	. . . . . . . . . . . . 13 ⊢ ((𝑊‘0) = 𝑋 → 𝑋 ∈ V)
11		clwwlknonwwlknonb.v	. . . . . . . . . . . . . . . 16 ⊢ 𝑉 = (Vtx‘𝐺)
12		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
13	11, 12	wwlknp 29932	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑋”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑋”⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
14		simprrl 781	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ))) → 𝑊 ∈ Word 𝑉)
15		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑊 ∈ Word 𝑉)
16	15	anim2i 618	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑋 ∈ V ∧ 𝑊 ∈ Word 𝑉))
17	16	ancomd 461	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ V))
18		ccats1alpha 14555	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ V) → ((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ↔ (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉)))
19	17, 18	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → ((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ↔ (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉)))
20		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
21	19, 20	biimtrdi 253	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → ((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 → 𝑋 ∈ 𝑉))
22	21	com12 32	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 → ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑋 ∈ 𝑉))
23	22	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) → ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑋 ∈ 𝑉))
24	23	imp 406	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ))) → 𝑋 ∈ 𝑉)
25		nnnn0 12420	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
26		ccatws1lenp1b 14557	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) ↔ (♯‘𝑊) = 𝑁))
27	25, 26	sylan2 594	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → ((♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) ↔ (♯‘𝑊) = 𝑁))
28	27	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → ((♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) → (♯‘𝑊) = 𝑁))
29	28	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → ((♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) → (♯‘𝑊) = 𝑁))
30	29	com12 32	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) → ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘𝑊) = 𝑁))
31	30	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) → ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘𝑊) = 𝑁))
32	31	imp 406	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ))) → (♯‘𝑊) = 𝑁)
33	32	eqcomd 2743	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ))) → 𝑁 = (♯‘𝑊))
34	14, 24, 33	3jca 1129	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ))) → (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 = (♯‘𝑊)))
35	34	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) → ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 = (♯‘𝑊))))
36	35	3adant3 1133	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑋”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑋”⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 = (♯‘𝑊))))
37	13, 36	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 = (♯‘𝑊))))
38	37	expd 415	. . . . . . . . . . . . 13 ⊢ ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → (𝑋 ∈ V → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 = (♯‘𝑊)))))
39	10, 38	syl5com 31	. . . . . . . . . . . 12 ⊢ ((𝑊‘0) = 𝑋 → ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 = (♯‘𝑊)))))
40	5, 39	sylbid 240	. . . . . . . . . . 11 ⊢ ((𝑊‘0) = 𝑋 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 = (♯‘𝑊)))))
41	40	com13 88	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) → ((𝑊‘0) = 𝑋 → (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 = (♯‘𝑊)))))
42	41	imp32 418	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 = (♯‘𝑊)))
43		ccats1val2 14563	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 = (♯‘𝑊)) → ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = 𝑋)
44	42, 43	syl 17	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = 𝑋)
45		ccat1st1st 14564	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = (𝑊‘0))
46	45	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = (𝑊‘0))
47	4	fveq1d 6844	. . . . . . . . . . . . 13 ⊢ ((𝑊‘0) = 𝑋 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = ((𝑊 ++ ⟨“𝑋”⟩)‘0))
48	47	eqeq1d 2739	. . . . . . . . . . . 12 ⊢ ((𝑊‘0) = 𝑋 → (((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = (𝑊‘0) ↔ ((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0)))
49	48	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → (((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = (𝑊‘0) ↔ ((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0)))
50	46, 49	syl5ibcom 245	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → ((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0)))
51	50	imp 406	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → ((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0))
52		simprr 773	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → (𝑊‘0) = 𝑋)
53	51, 52	eqtrd 2772	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → ((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋)
54	7, 44, 53	jca31 514	. . . . . . 7 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = 𝑋) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋))
55	54	ex 412	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = 𝑋) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋)))
56		simprl 771	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑊 ∈ Word 𝑉)
57	27	biimpcd 249	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (♯‘𝑊) = 𝑁))
58	57	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (♯‘𝑊) = 𝑁))
59	58	imp 406	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘𝑊) = 𝑁)
60	59	eqcomd 2743	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑁 = (♯‘𝑊))
61	56, 60	jca 511	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑊)))
62	61	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑊))))
63	62	3adant3 1133	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){((𝑊 ++ ⟨“𝑋”⟩)‘𝑖), ((𝑊 ++ ⟨“𝑋”⟩)‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑊))))
64	13, 63	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑊))))
65	64	imp 406	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑊)))
66		eleq1 2825	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 = (♯‘𝑊) → (𝑁 ∈ ℕ ↔ (♯‘𝑊) ∈ ℕ))
67		lbfzo0 13627	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ (0..^(♯‘𝑊)) ↔ (♯‘𝑊) ∈ ℕ)
68	67	biimpri 228	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑊) ∈ ℕ → 0 ∈ (0..^(♯‘𝑊)))
69	66, 68	biimtrdi 253	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘𝑊) → (𝑁 ∈ ℕ → 0 ∈ (0..^(♯‘𝑊))))
70	69	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → (𝑁 = (♯‘𝑊) → 0 ∈ (0..^(♯‘𝑊))))
71	70	ad2antll 730	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑁 = (♯‘𝑊) → 0 ∈ (0..^(♯‘𝑊))))
72	71	anim2d 613	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑊)) → (𝑊 ∈ Word 𝑉 ∧ 0 ∈ (0..^(♯‘𝑊)))))
73	65, 72	mpd 15	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ Word 𝑉 ∧ 0 ∈ (0..^(♯‘𝑊))))
74		ccats1val1 14562	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 ∈ (0..^(♯‘𝑊))) → ((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0))
75	73, 74	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → ((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0))
76	75	eqeq1d 2739	. . . . . . . . . . . 12 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋 ↔ (𝑊‘0) = 𝑋))
77	76	biimpd 229	. . . . . . . . . . 11 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋 → (𝑊‘0) = 𝑋))
78	77	ex 412	. . . . . . . . . 10 ⊢ ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋 → (𝑊‘0) = 𝑋)))
79	78	adantr 480	. . . . . . . . 9 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = 𝑋) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋 → (𝑊‘0) = 𝑋)))
80	79	com3r 87	. . . . . . . 8 ⊢ (((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋 → (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = 𝑋) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊‘0) = 𝑋)))
81	80	impcom 407	. . . . . . 7 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = 𝑋) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊‘0) = 𝑋))
82	5	biimparc 479	. . . . . . . . . 10 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺))
83		simpr 484	. . . . . . . . . 10 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → (𝑊‘0) = 𝑋)
84	82, 83	jca 511	. . . . . . . . 9 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋))
85	84	ex 412	. . . . . . . 8 ⊢ ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → ((𝑊‘0) = 𝑋 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)))
86	85	ad2antrr 727	. . . . . . 7 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = 𝑋) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋) → ((𝑊‘0) = 𝑋 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)))
87	81, 86	syldc 48	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → ((((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = 𝑋) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋) → ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)))
88	55, 87	impbid 212	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) ↔ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = 𝑋) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋)))
89	2, 88	bitr4id 290	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = 𝑋) ↔ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)))
90	11	clwwlknwwlksnb 30146	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺)))
91	90	anbi1d 632	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) ↔ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)))
92	89, 91	bitr4d 282	. . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = 𝑋) ↔ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)))
93	1, 92	bitr4id 290	. 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = 𝑋)))
94		wwlknon 29946	. 2 ⊢ ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑋(𝑁 WWalksNOn 𝐺)𝑋) ↔ ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = 𝑋))
95	93, 94	bitr4di 289	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑋(𝑁 WWalksNOn 𝐺)𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3442 {cpr 4584 ‘cfv 6500 (class class class)co 7368 0cc0 11038 1c1 11039 + caddc 11041 ℕcn 12157 ℕ₀cn0 12413 ..^cfzo 13582 ♯chash 14265 Word cword 14448 ++ cconcat 14505 ⟨“cs1 14531 Vtxcvtx 29085 Edgcedg 29136 WWalksN cwwlksn 29915 WWalksNOn cwwlksnon 29916 ClWWalksN cclwwlkn 30115 ClWWalksNOncclwwlknon 30178

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-dju 9825 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-n0 12414 df-xnn0 12487 df-z 12501 df-uz 12764 df-fz 13436 df-fzo 13583 df-hash 14266 df-word 14449 df-lsw 14498 df-concat 14506 df-s1 14532 df-wwlks 29919 df-wwlksn 29920 df-wwlksnon 29921 df-clwwlk 30073 df-clwwlkn 30116 df-clwwlknon 30179

This theorem is referenced by: (None)

W3C validator