clwwlknonwwlknonb - Metamath Proof Explorer

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Theorem clwwlknonwwlknonb 29624

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 29573. (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 29609	. . 3 ⊢ (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑊‘0) = 𝑋))
2		3anan32 1095	. . . . 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 14556	. . . . . . . . . . . 12 ⊢ ((𝑊‘0) = 𝑋 → ⟨“(𝑊‘0)”⟩ = ⟨“𝑋”⟩)
4	3	oveq2d 7429	. . . . . . . . . . 11 ⊢ ((𝑊‘0) = 𝑋 → (𝑊 ++ ⟨“(𝑊‘0)”⟩) = (𝑊 ++ ⟨“𝑋”⟩))
5	4	eleq1d 2816	. . . . . . . . . 10 ⊢ ((𝑊‘0) = 𝑋 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺)))
6	5	biimpac 477	. . . . . . . . 9 ⊢ (((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺))
7	6	adantl 480	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺))
8		fvex 6905	. . . . . . . . . . . . . 14 ⊢ (𝑊‘0) ∈ V
9		eleq1 2819	. . . . . . . . . . . . . 14 ⊢ ((𝑊‘0) = 𝑋 → ((𝑊‘0) ∈ V ↔ 𝑋 ∈ V))
10	8, 9	mpbii 232	. . . . . . . . . . . . 13 ⊢ ((𝑊‘0) = 𝑋 → 𝑋 ∈ V)
11		clwwlknonwwlknonb.v	. . . . . . . . . . . . . . . 16 ⊢ 𝑉 = (Vtx‘𝐺)
12		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
13	11, 12	wwlknp 29362	. . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ))) → 𝑊 ∈ Word 𝑉)
15		simpl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑊 ∈ Word 𝑉)
16	15	anim2i 615	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑋 ∈ V ∧ 𝑊 ∈ Word 𝑉))
17	16	ancomd 460	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ V))
18		ccats1alpha 14575	. . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
21	19, 20	syl6bi 252	. . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) → ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑋 ∈ 𝑉))
24	23	imp 405	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ))) → 𝑋 ∈ 𝑉)
25		nnnn0 12485	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
26		ccatws1lenp1b 14577	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) ↔ (♯‘𝑊) = 𝑁))
27	25, 26	sylan2 591	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → ((♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) ↔ (♯‘𝑊) = 𝑁))
28	27	biimpd 228	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → ((♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) → (♯‘𝑊) = 𝑁))
29	28	adantl 480	. . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) → ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘𝑊) = 𝑁))
32	31	imp 405	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ))) → (♯‘𝑊) = 𝑁)
33	32	eqcomd 2736	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ))) → 𝑁 = (♯‘𝑊))
34	14, 24, 33	3jca 1126	. . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . 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 1130	. . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . 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 239	. . . . . . . . . . 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 417	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 = (♯‘𝑊)))
43		ccats1val2 14583	. . . . . . . . 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 14584	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = (𝑊‘0))
46	45	adantr 479	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = (𝑊‘0))
47	4	fveq1d 6894	. . . . . . . . . . . . 13 ⊢ ((𝑊‘0) = 𝑋 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = ((𝑊 ++ ⟨“𝑋”⟩)‘0))
48	47	eqeq1d 2732	. . . . . . . . . . . 12 ⊢ ((𝑊‘0) = 𝑋 → (((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = (𝑊‘0) ↔ ((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0)))
49	48	adantl 480	. . . . . . . . . . 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 244	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → ((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0)))
51	50	imp 405	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → ((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0))
52		simprr 769	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → (𝑊‘0) = 𝑋)
53	51, 52	eqtrd 2770	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → ((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋)
54	7, 44, 53	jca31 513	. . . . . . 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 411	. . . . . 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 767	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑊 ∈ Word 𝑉)
57	27	biimpcd 248	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (♯‘𝑊) = 𝑁))
58	57	adantl 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (♯‘𝑊) = 𝑁))
59	58	imp 405	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘𝑊) = 𝑁)
60	59	eqcomd 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑁 = (♯‘𝑊))
61	56, 60	jca 510	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑊)))
62	61	ex 411	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑊))))
63	62	3adant3 1130	. . . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑊)))
66		eleq1 2819	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 = (♯‘𝑊) → (𝑁 ∈ ℕ ↔ (♯‘𝑊) ∈ ℕ))
67		lbfzo0 13678	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ (0..^(♯‘𝑊)) ↔ (♯‘𝑊) ∈ ℕ)
68	67	biimpri 227	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑊) ∈ ℕ → 0 ∈ (0..^(♯‘𝑊)))
69	66, 68	syl6bi 252	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘𝑊) → (𝑁 ∈ ℕ → 0 ∈ (0..^(♯‘𝑊))))
70	69	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → (𝑁 = (♯‘𝑊) → 0 ∈ (0..^(♯‘𝑊))))
71	70	ad2antll 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑁 = (♯‘𝑊) → 0 ∈ (0..^(♯‘𝑊))))
72	71	anim2d 610	. . . . . . . . . . . . . . 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 14582	. . . . . . . . . . . . . 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 2732	. . . . . . . . . . . 12 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋 ↔ (𝑊‘0) = 𝑋))
77	76	biimpd 228	. . . . . . . . . . 11 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋 → (𝑊‘0) = 𝑋))
78	77	ex 411	. . . . . . . . . 10 ⊢ ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋 → (𝑊‘0) = 𝑋)))
79	78	adantr 479	. . . . . . . . 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 406	. . . . . . 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 478	. . . . . . . . . 10 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺))
83		simpr 483	. . . . . . . . . 10 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → (𝑊‘0) = 𝑋)
84	82, 83	jca 510	. . . . . . . . 9 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋))
85	84	ex 411	. . . . . . . 8 ⊢ ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → ((𝑊‘0) = 𝑋 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)))
86	85	ad2antrr 722	. . . . . . 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 211	. . . . 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 289	. . . 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 29573	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺)))
91	90	anbi1d 628	. . . 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 281	. . 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 289	. 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 29376	. 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 288	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 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ∀wral 3059 Vcvv 3472 {cpr 4631 ‘cfv 6544 (class class class)co 7413 0cc0 11114 1c1 11115 + caddc 11117 ℕcn 12218 ℕ₀cn0 12478 ..^cfzo 13633 ♯chash 14296 Word cword 14470 ++ cconcat 14526 ⟨“cs1 14551 Vtxcvtx 28521 Edgcedg 28572 WWalksN cwwlksn 29345 WWalksNOn cwwlksnon 29346 ClWWalksN cclwwlkn 29542 ClWWalksNOncclwwlknon 29605

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-oadd 8474 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-dju 9900 df-card 9938 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-nn 12219 df-2 12281 df-n0 12479 df-xnn0 12551 df-z 12565 df-uz 12829 df-fz 13491 df-fzo 13634 df-hash 14297 df-word 14471 df-lsw 14519 df-concat 14527 df-s1 14552 df-wwlks 29349 df-wwlksn 29350 df-wwlksnon 29351 df-clwwlk 29500 df-clwwlkn 29543 df-clwwlknon 29606

This theorem is referenced by: (None)

W3C validator