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 29348

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 29297. (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 29333	. . 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 14546	. . . . . . . . . . . 12 ⊢ ((𝑊‘0) = 𝑋 → ⟨“(𝑊‘0)”⟩ = ⟨“𝑋”⟩)
4	3	oveq2d 7421	. . . . . . . . . . 11 ⊢ ((𝑊‘0) = 𝑋 → (𝑊 ++ ⟨“(𝑊‘0)”⟩) = (𝑊 ++ ⟨“𝑋”⟩))
5	4	eleq1d 2818	. . . . . . . . . 10 ⊢ ((𝑊‘0) = 𝑋 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺)))
6	5	biimpac 479	. . . . . . . . 9 ⊢ (((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺))
7	6	adantl 482	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺))
8		fvex 6901	. . . . . . . . . . . . . 14 ⊢ (𝑊‘0) ∈ V
9		eleq1 2821	. . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . 16 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
13	11, 12	wwlknp 29086	. . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ))) → 𝑊 ∈ Word 𝑉)
15		simpl 483	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑊 ∈ Word 𝑉)
16	15	anim2i 617	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑋 ∈ V ∧ 𝑊 ∈ Word 𝑉))
17	16	ancomd 462	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ V))
18		ccats1alpha 14565	. . . . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) → ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑋 ∈ 𝑉))
24	23	imp 407	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ))) → 𝑋 ∈ 𝑉)
25		nnnn0 12475	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
26		ccatws1lenp1b 14567	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ₀) → ((♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1) ↔ (♯‘𝑊) = 𝑁))
27	25, 26	sylan2 593	. . . . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) → ((𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘𝑊) = 𝑁))
32	31	imp 407	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ))) → (♯‘𝑊) = 𝑁)
33	32	eqcomd 2738	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑋 ∈ V ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ))) → 𝑁 = (♯‘𝑊))
34	14, 24, 33	3jca 1128	. . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . 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 419	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → (𝑊 ∈ Word 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 = (♯‘𝑊)))
43		ccats1val2 14573	. . . . . . . . 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 14574	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = (𝑊‘0))
46	45	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = (𝑊‘0))
47	4	fveq1d 6890	. . . . . . . . . . . . 13 ⊢ ((𝑊‘0) = 𝑋 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = ((𝑊 ++ ⟨“𝑋”⟩)‘0))
48	47	eqeq1d 2734	. . . . . . . . . . . 12 ⊢ ((𝑊‘0) = 𝑋 → (((𝑊 ++ ⟨“(𝑊‘0)”⟩)‘0) = (𝑊‘0) ↔ ((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0)))
49	48	adantl 482	. . . . . . . . . . 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 407	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) ∧ ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)) → ((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0))
52		simprr 771	. . . . . . . . 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 515	. . . . . . 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 413	. . . . . 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 769	. . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (♯‘𝑊) = 𝑁))
59	58	imp 407	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘𝑊) = 𝑁)
60	59	eqcomd 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑁 = (♯‘𝑊))
61	56, 60	jca 512	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑊)))
62	61	ex 413	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ Word 𝑉 ∧ (♯‘(𝑊 ++ ⟨“𝑋”⟩)) = (𝑁 + 1)) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑊))))
63	62	3adant3 1132	. . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑊 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝑊)))
66		eleq1 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 = (♯‘𝑊) → (𝑁 ∈ ℕ ↔ (♯‘𝑊) ∈ ℕ))
67		lbfzo0 13668	. . . . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ)) → (𝑁 = (♯‘𝑊) → 0 ∈ (0..^(♯‘𝑊))))
72	71	anim2d 612	. . . . . . . . . . . . . . 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 14572	. . . . . . . . . . . . . 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 2734	. . . . . . . . . . . 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 413	. . . . . . . . . 10 ⊢ ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = 𝑋 → (𝑊‘0) = 𝑋)))
79	78	adantr 481	. . . . . . . . 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 408	. . . . . . 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 480	. . . . . . . . . 10 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺))
83		simpr 485	. . . . . . . . . 10 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → (𝑊‘0) = 𝑋)
84	82, 83	jca 512	. . . . . . . . 9 ⊢ (((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋) → ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋))
85	84	ex 413	. . . . . . . 8 ⊢ ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → ((𝑊‘0) = 𝑋 → ((𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)))
86	85	ad2antrr 724	. . . . . . 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 29297	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺)))
91	90	anbi1d 630	. . . 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 29100	. 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 {cpr 4629 ‘cfv 6540 (class class class)co 7405 0cc0 11106 1c1 11107 + caddc 11109 ℕcn 12208 ℕ₀cn0 12468 ..^cfzo 13623 ♯chash 14286 Word cword 14460 ++ cconcat 14516 ⟨“cs1 14541 Vtxcvtx 28245 Edgcedg 28296 WWalksN cwwlksn 29069 WWalksNOn cwwlksnon 29070 ClWWalksN cclwwlkn 29266 ClWWalksNOncclwwlknon 29329

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-lsw 14509 df-concat 14517 df-s1 14542 df-wwlks 29073 df-wwlksn 29074 df-wwlksnon 29075 df-clwwlk 29224 df-clwwlkn 29267 df-clwwlknon 29330

This theorem is referenced by: (None)

W3C validator