Theorem rusgra0edg 26482

Description: Special case for graphs without edges: There are no walks of length greater than 0. (Contributed by Alexander van der Vekens, 26-Jul-2018.)

Hypotheses

Ref	Expression
rusgranumwlk.w	⊢ 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st ‘𝑐)) = 𝑛})
rusgranumwlk.l	⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊‘𝑛) ∣ ((2nd ‘𝑤)‘0) = 𝑣}))

Assertion

Ref	Expression
rusgra0edg	⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = 0)

Distinct variable groups:   𝐸,𝑐,𝑛   𝑁,𝑐,𝑛   𝑉,𝑐,𝑛   𝑣,𝑁,𝑤   𝑃,𝑛,𝑣,𝑤   𝑣,𝑉   𝑛,𝑊,𝑣,𝑤   𝑤,𝑉,𝑐   𝑣,𝐸,𝑤

Allowed substitution hints:   𝑃(𝑐)   𝐿(𝑤,𝑣,𝑛,𝑐)   𝑊(𝑐)

Proof of Theorem rusgra0edg

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rusisusgra 26458	. . 3 ⊢ (⟨𝑉, 𝐸⟩ RegUSGrph 0 → 𝑉 USGrph 𝐸)
2		id 22	. . 3 ⊢ (𝑃 ∈ 𝑉 → 𝑃 ∈ 𝑉)
3		nnnn0 11176	. . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
4		rusgranumwlk.w	. . . 4 ⊢ 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st ‘𝑐)) = 𝑛})
5		rusgranumwlk.l	. . . 4 ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊‘𝑛) ∣ ((2nd ‘𝑤)‘0) = 𝑣}))
6	4, 5	rusgranumwlklem4 26479	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}))
7	1, 2, 3, 6	syl3an 1360	. 2 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}))
8		df-rab 2905	. . . . 5 ⊢ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} = {𝑤 ∣ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑤‘0) = 𝑃)}
9		usgrav 25867	. . . . . . . . . . . . . 14 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
10	1, 9	syl 17	. . . . . . . . . . . . 13 ⊢ (⟨𝑉, 𝐸⟩ RegUSGrph 0 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
11	10, 3	anim12i 588	. . . . . . . . . . . 12 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑁 ∈ ℕ) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0))
12	11	3adant2 1073	. . . . . . . . . . 11 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0))
13		df-3an 1033	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) ↔ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0))
14	12, 13	sylibr 223	. . . . . . . . . 10 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
15		iswwlkn 26212	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ (𝑤 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))))
16		iswwlk 26211	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑤 ∈ (𝑉 WWalks 𝐸) ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸)))
17	16	3adant3 1074	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑤 ∈ (𝑉 WWalks 𝐸) ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸)))
18	17	anbi1d 737	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝑤 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑤) = (𝑁 + 1)) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))))
19	15, 18	bitrd 267	. . . . . . . . . 10 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))))
20	14, 19	syl 17	. . . . . . . . 9 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))))
21	20	anbi1d 737	. . . . . . . 8 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑤‘0) = 𝑃) ↔ (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1)) ∧ (𝑤‘0) = 𝑃)))
22		oveq1 6556	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑤) = (𝑁 + 1) → ((#‘𝑤) − 1) = ((𝑁 + 1) − 1))
23		nncn 10905	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
24		1cnd 9935	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
25	23, 24	pncand 10272	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) − 1) = 𝑁)
26	25	3ad2ant3 1077	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑁 + 1) − 1) = 𝑁)
27	22, 26	sylan9eqr 2666	. . . . . . . . . . . . . . . 16 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (#‘𝑤) = (𝑁 + 1)) → ((#‘𝑤) − 1) = 𝑁)
28	27	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (#‘𝑤) = (𝑁 + 1)) → (0..^((#‘𝑤) − 1)) = (0..^𝑁))
29	28	raleqdv 3121	. . . . . . . . . . . . . 14 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (#‘𝑤) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^𝑁){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸))
30		rusgrargra 26457	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑉, 𝐸⟩ RegUSGrph 0 → ⟨𝑉, 𝐸⟩ RegGrph 0)
31		0eusgraiff0rgra 26466	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ RegGrph 0 ↔ 𝐸 = ∅))
32		rneq 5272	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐸 = ∅ → ran 𝐸 = ran ∅)
33		rn0 5298	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ran ∅ = ∅
34	32, 33	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐸 = ∅ → ran 𝐸 = ∅)
35	34	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐸 = ∅ → ({(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ∅))
36		noel 3878	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ¬ {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ∅
37	36	bifal 1488	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ∅ ↔ ⊥)
38	35, 37	syl6bb 275	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐸 = ∅ → ({(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ⊥))
39	38	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐸 = ∅ ∧ (𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ({(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ⊥))
40	39	ralbidv 2969	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 = ∅ ∧ (𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (∀𝑖 ∈ (0..^𝑁){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^𝑁)⊥))
41		fal 1482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ¬ ⊥
42	41	ralf0 4030	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑖 ∈ (0..^𝑁)⊥ ↔ (0..^𝑁) = ∅)
43	42	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 = ∅ ∧ (𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (∀𝑖 ∈ (0..^𝑁)⊥ ↔ (0..^𝑁) = ∅))
44		0nn0 11184	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ ℕ0
45	44	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 0 ∈ ℕ0)
46		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ)
47		nngt0 10926	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
48	45, 46, 47	3jca 1235	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → (0 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 0 < 𝑁))
49	48	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐸 = ∅ ∧ (𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (0 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 0 < 𝑁))
50		elfzo0 12376	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0 ∈ (0..^𝑁) ↔ (0 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 0 < 𝑁))
51	49, 50	sylibr 223	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐸 = ∅ ∧ (𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → 0 ∈ (0..^𝑁))
52		fzon0 12356	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0..^𝑁) ≠ ∅ ↔ 0 ∈ (0..^𝑁))
53	51, 52	sylibr 223	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐸 = ∅ ∧ (𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (0..^𝑁) ≠ ∅)
54	53	neneqd 2787	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐸 = ∅ ∧ (𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ¬ (0..^𝑁) = ∅)
55		nbfal 1486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ (0..^𝑁) = ∅ ↔ ((0..^𝑁) = ∅ ↔ ⊥))
56	54, 55	sylib 207	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 = ∅ ∧ (𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ((0..^𝑁) = ∅ ↔ ⊥))
57	40, 43, 56	3bitrd 293	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 = ∅ ∧ (𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (∀𝑖 ∈ (0..^𝑁){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ⊥))
58	57	ex 449	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸 = ∅ → ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (∀𝑖 ∈ (0..^𝑁){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ⊥)))
59	31, 58	syl6bi 242	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ RegGrph 0 → ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (∀𝑖 ∈ (0..^𝑁){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ⊥))))
60	1, 30, 59	sylc 63	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑉, 𝐸⟩ RegUSGrph 0 → ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (∀𝑖 ∈ (0..^𝑁){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ⊥)))
61	60	3impib 1254	. . . . . . . . . . . . . . 15 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (∀𝑖 ∈ (0..^𝑁){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ⊥))
62	61	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (#‘𝑤) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^𝑁){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ⊥))
63	29, 62	bitrd 267	. . . . . . . . . . . . 13 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (#‘𝑤) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ⊥))
64	63	3anbi3d 1397	. . . . . . . . . . . 12 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (#‘𝑤) = (𝑁 + 1)) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ⊥)))
65		df-3an 1033	. . . . . . . . . . . . 13 ⊢ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ⊥) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉) ∧ ⊥))
66		ancom 465	. . . . . . . . . . . . 13 ⊢ (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉) ∧ ⊥) ↔ (⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)))
67	65, 66	bitri 263	. . . . . . . . . . . 12 ⊢ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ⊥) ↔ (⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)))
68	64, 67	syl6bb 275	. . . . . . . . . . 11 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ (#‘𝑤) = (𝑁 + 1)) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉))))
69	68	ex 449	. . . . . . . . . 10 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((#‘𝑤) = (𝑁 + 1) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)))))
70	69	pm5.32rd 670	. . . . . . . . 9 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1)) ↔ ((⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)) ∧ (#‘𝑤) = (𝑁 + 1))))
71	70	anbi1d 737	. . . . . . . 8 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1)) ∧ (𝑤‘0) = 𝑃) ↔ (((⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)) ∧ (#‘𝑤) = (𝑁 + 1)) ∧ (𝑤‘0) = 𝑃)))
72		anass 679	. . . . . . . . . 10 ⊢ ((((⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)) ∧ (#‘𝑤) = (𝑁 + 1)) ∧ (𝑤‘0) = 𝑃) ↔ ((⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)) ∧ ((#‘𝑤) = (𝑁 + 1) ∧ (𝑤‘0) = 𝑃)))
73		anass 679	. . . . . . . . . . 11 ⊢ (((⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)) ∧ ((#‘𝑤) = (𝑁 + 1) ∧ (𝑤‘0) = 𝑃)) ↔ (⊥ ∧ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉) ∧ ((#‘𝑤) = (𝑁 + 1) ∧ (𝑤‘0) = 𝑃))))
74	41	intnanr 952	. . . . . . . . . . . 12 ⊢ ¬ (⊥ ∧ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉) ∧ ((#‘𝑤) = (𝑁 + 1) ∧ (𝑤‘0) = 𝑃)))
75	74	bifal 1488	. . . . . . . . . . 11 ⊢ ((⊥ ∧ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉) ∧ ((#‘𝑤) = (𝑁 + 1) ∧ (𝑤‘0) = 𝑃))) ↔ ⊥)
76	73, 75	bitri 263	. . . . . . . . . 10 ⊢ (((⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)) ∧ ((#‘𝑤) = (𝑁 + 1) ∧ (𝑤‘0) = 𝑃)) ↔ ⊥)
77	72, 76	bitri 263	. . . . . . . . 9 ⊢ ((((⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)) ∧ (#‘𝑤) = (𝑁 + 1)) ∧ (𝑤‘0) = 𝑃) ↔ ⊥)
78	77	a1i 11	. . . . . . . 8 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((((⊥ ∧ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉)) ∧ (#‘𝑤) = (𝑁 + 1)) ∧ (𝑤‘0) = 𝑃) ↔ ⊥))
79	21, 71, 78	3bitrd 293	. . . . . . 7 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑤‘0) = 𝑃) ↔ ⊥))
80	79	abbidv 2728	. . . . . 6 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → {𝑤 ∣ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑤‘0) = 𝑃)} = {𝑤 ∣ ⊥})
81	41	abf 3930	. . . . . 6 ⊢ {𝑤 ∣ ⊥} = ∅
82	80, 81	syl6eq 2660	. . . . 5 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → {𝑤 ∣ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑤‘0) = 𝑃)} = ∅)
83	8, 82	syl5eq 2656	. . . 4 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} = ∅)
84	83	fveq2d 6107	. . 3 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (#‘∅))
85		hash0 13019	. . 3 ⊢ (#‘∅) = 0
86	84, 85	syl6eq 2660	. 2 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = 0)
87	7, 86	eqtrd 2644	1 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = 0)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ⊥wfal 1480   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173  ∅c0 3874  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1^st c1st 7057  2^nd c2nd 7058  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   − cmin 10145  ℕcn 10897  ℕ₀cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   USGrph cusg 25859   Walks cwalk 26026   WWalks cwwlk 26205   WWalksN cwwlkn 26206   RegGrph crgra 26449   RegUSGrph crusgra 26450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-wwlk 26207  df-wwlkn 26208  df-vdgr 26421  df-rgra 26451  df-rusgra 26452

This theorem is referenced by: (None)