clwwlknonwwlknonb - Metamath Proof Explorer

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

Theorem clwwlknonwwlknonb 30712

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 30661. (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 30697	. . 3 ⊢ (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑊‘0) = 𝑋))
2		3anan32 1113	. . . . 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 14760	. . . . . . . . . . . 12 ⊢ ((𝑊‘0) = 𝑋 → ⟨“(𝑊‘0)”⟩ = ⟨“𝑋”⟩)
4	3	oveq2d 7446	. . . . . . . . . . 11 ⊢ ((𝑊‘0) = 𝑋 → (𝑊 ++ ⟨“(𝑊‘0)”⟩) = (𝑊 ++ ⟨“𝑋”⟩))
5	4	eleq1d 2849	. . . . . . . . . 10 ⊢ ((𝑊‘0) = 𝑋 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺)))
6	5	biimpac 484	. . . . . . . . 9 ⊢ (((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺))
7	6	adantl 487	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺))
8		fvex 6909	. . . . . . . . . . . . . 14 ⊢ (𝑊‘0) ∈ V
9		eleq1 2852	. . . . . . . . . . . . . 14 ⊢ ((𝑊‘0) = 𝑋 → ((𝑊‘0) ∈ V ↔ 𝑋 ∈ V))
10	8, 9	mpbii 236	. . . . . . . . . . . . 13 ⊢ ((𝑊‘0) = 𝑋 → 𝑋 ∈ V)
11		clwwlknonwwlknonb.v	. . . . . . . . . . . . . . . 16 ⊢ 𝑉 = (Vtx‘𝐺)
12		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
13	11, 12	wwlknp 30447	. . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ))) → 𝑊 ∈ Word 𝑉)
15		simpl 488	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑊 ∈ Word 𝑉)
16	15	anim2i 629	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑋 ∈ V ∧ 𝑊 ∈ Word 𝑉))
17	16	ancomd 467	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ V))
18		ccats1alpha 14781	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ V) → ((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ↔ (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉)))
19	17, 18	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → ((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ↔ (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉)))
20		simpr 490	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
21	19, 20	biimtrdi 256	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → ((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 → 𝑋 ∈ 𝑉))
22	21	com12 33	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 → ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑋 ∈ 𝑉))
23	22	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) → ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑋 ∈ 𝑉))
24	23	imp 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ))) → 𝑋 ∈ 𝑉)
25		nnnn0 12626	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
26		ccatws1lenp1b 14783	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) ↔ (♯‘𝑊) = 𝑁))
27	25, 26	sylan2 605	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → ((♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) ↔ (♯‘𝑊) = 𝑁))
28	27	biimpd 232	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → ((♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) → (♯‘𝑊) = 𝑁))
29	28	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → ((♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) → (♯‘𝑊) = 𝑁))
30	29	com12 33	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) → ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘𝑊) = 𝑁))
31	30	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) → ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘𝑊) = 𝑁))
32	31	imp 412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ))) → (♯‘𝑊) = 𝑁)
33	32	eqcomd 2769	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ))) → 𝑁 = (♯‘𝑊))
34	14, 24, 33	3jca 1146	. . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . 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 1150	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 = (♯‘𝑊))))
38	37	expd 421	. . . . . . . . . . . . 13 ⊢ ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → (𝑋 ∈ V → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 = (♯‘𝑊)))))
39	10, 38	syl5com 32	. . . . . . . . . . . 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 243	. . . . . . . . . . 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 89	. . . . . . . . . 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 424	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 = (♯‘𝑊)))
43		ccats1val2 14789	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 = (♯‘𝑊)) → ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = 𝑋)
44	42, 43	syl 18	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = 𝑋)
45		ccat1st1st 14790	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = (𝑊‘0))
46	45	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = (𝑊‘0))
47	4	fveq1d 6898	. . . . . . . . . . . . 13 ⊢ ((𝑊‘0) = 𝑋 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = ((𝑊 ++ ⟨“𝑋”⟩)‘0))
48	47	eqeq1d 2765	. . . . . . . . . . . 12 ⊢ ((𝑊‘0) = 𝑋 → (((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = (𝑊‘0) ↔ ((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0)))
49	48	adantl 487	. . . . . . . . . . 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 248	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → ((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0)))
51	50	imp 412	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → ((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0))
52		simprr 785	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → (𝑊‘0) = 𝑋)
53	51, 52	eqtrd 2798	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → ((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋)
54	7, 44, 53	jca31 524	. . . . . . 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 418	. . . . . 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 783	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑊 ∈ Word 𝑉)
57	27	biimpcd 252	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (♯‘𝑊) = 𝑁))
58	57	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (♯‘𝑊) = 𝑁))
59	58	imp 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘𝑊) = 𝑁)
60	59	eqcomd 2769	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑁 = (♯‘𝑊))
61	56, 60	jca 521	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑊)))
62	61	ex 418	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑊))))
63	62	3adant3 1150	. . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑊))))
65	64	imp 412	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑊)))
66		eleq1 2852	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 = (♯‘𝑊) → (𝑁 ∈ ℕ ↔ (♯‘𝑊) ∈ ℕ))
67		lbfzo0 13847	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ (0..^(♯‘𝑊)) ↔ (♯‘𝑊) ∈ ℕ)
68	67	biimpri 231	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑊) ∈ ℕ → 0 ∈ (0..^(♯‘𝑊)))
69	66, 68	biimtrdi 256	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘𝑊) → (𝑁 ∈ ℕ → 0 ∈ (0..^(♯‘𝑊))))
70	69	com12 33	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → (𝑁 = (♯‘𝑊) → 0 ∈ (0..^(♯‘𝑊))))
71	70	ad2antll 742	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑁 = (♯‘𝑊) → 0 ∈ (0..^(♯‘𝑊))))
72	71	anim2d 624	. . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ Word 𝑉 ∧ 0 ∈ (0..^(♯‘𝑊))))
74		ccats1val1 14788	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 ∈ (0..^(♯‘𝑊))) → ((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0))
75	73, 74	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → ((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0))
76	75	eqeq1d 2765	. . . . . . . . . . . 12 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋 ↔ (𝑊‘0) = 𝑋))
77	76	biimpd 232	. . . . . . . . . . 11 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋 → (𝑊‘0) = 𝑋))
78	77	ex 418	. . . . . . . . . 10 ⊢ ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋 → (𝑊‘0) = 𝑋)))
79	78	adantr 486	. . . . . . . . 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 88	. . . . . . . 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 413	. . . . . . 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 485	. . . . . . . . . 10 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺))
83		simpr 490	. . . . . . . . . 10 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → (𝑊‘0) = 𝑋)
84	82, 83	jca 521	. . . . . . . . 9 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋))
85	84	ex 418	. . . . . . . 8 ⊢ ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → ((𝑊‘0) = 𝑋 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)))
86	85	ad2antrr 739	. . . . . . 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 49	. . . . . 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 215	. . . . 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 293	. . . 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 30661	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺)))
91	90	anbi1d 643	. . . 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 285	. . 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 293	. 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 30461	. 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 292	1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑋(𝑁 WWalksNOn 𝐺)𝑋)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2146 ∀wral 3080 Vcvv 3454 {cpr 4590 ‘cfv 6547 (class class class)co 7430 0cc0 11213 1c1 11214 + caddc 11216 ℕcn 12348 ℕ₀cn0 12619 ..^cfzo 13801 ♯chash 14487 Word cword 14671 ++ cconcat 14728 ⟨“cs1 14755 Vtxcvtx 29589 Edgcedg 29640 WWalksN cwwlksn 30430 WWalksNOn cwwlksnon 30431 ClWWalksN cclwwlkn 30630 ClWWalksNOncclwwlknon 30693

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-cnex 11269 ax-resscn 11270 ax-1cn 11271 ax-icn 11272 ax-addcl 11273 ax-addrcl 11274 ax-mulcl 11275 ax-mulrcl 11276 ax-mulcom 11277 ax-addass 11278 ax-mulass 11279 ax-distr 11280 ax-i2m1 11281 ax-1ne0 11282 ax-1rid 11283 ax-rnegex 11284 ax-rrecex 11285 ax-cnre 11286 ax-pre-lttri 11287 ax-pre-lttrn 11288 ax-pre-ltadd 11289 ax-pre-mulgt0 11290

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6311 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7888 df-1st 8011 df-2nd 8012 df-frecs 8313 df-wrecs 8344 df-recs 8393 df-rdg 8432 df-1o 8490 df-2o 8491 df-oadd 8494 df-er 8731 df-map 8863 df-en 8988 df-dom 8989 df-sdom 8990 df-fin 8991 df-dju 9995 df-card 10033 df-pnf 11358 df-mnf 11359 df-xr 11360 df-ltxr 11361 df-le 11362 df-sub 11556 df-neg 11557 df-nn 12349 df-2 12418 df-n0 12620 df-xnn0 12693 df-z 12707 df-uz 12979 df-fz 13653 df-fzo 13802 df-hash 14488 df-word 14672 df-lsw 14721 df-concat 14729 df-s1 14756 df-wwlks 30434 df-wwlksn 30435 df-wwlksnon 30436 df-clwwlk 30588 df-clwwlkn 30631 df-clwwlknon 30694

This theorem is used by: (None)

W3C validator