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Theorem wwlkextproplem3 26271

Description: Lemma 3 for wwlkextprop 26272. (Contributed by Alexander van der Vekens, 1-Aug-2018.)

**Hypotheses**
Ref	Expression
wwlkextprop.x	⊢ 𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))
wwlkextprop.y	⊢ 𝑌 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}

**Assertion**
Ref	Expression
wwlkextproplem3	⊢ ((𝑊 ∈ 𝑋 ∧ (𝑊‘0) = 𝑃 ∧ 𝑁 ∈ ℕ₀) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ 𝑌)

Distinct variable groups: 𝑤,𝐸 𝑤,𝑁 𝑤,𝑃 𝑤,𝑉 𝑤,𝑊 Allowed substitution hints: 𝑋(𝑤) 𝑌(𝑤)

Proof of Theorem wwlkextproplem3 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlknprop 26214	. . . . 5 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ ((𝑁 + 1) ∈ ℕ₀ ∧ 𝑊 ∈ Word 𝑉)))
2		peano2nn0 11210	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
3		iswwlkn 26212	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑁 + 1) ∈ ℕ₀) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ↔ (𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = ((𝑁 + 1) + 1))))
4	2, 3	syl3an3 1353	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ₀) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ↔ (𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = ((𝑁 + 1) + 1))))
5		wwlkprop 26213	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑊 ∈ (𝑉 WWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉))
6		lencl 13179	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ₀)
7		eqcom 2617	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((#‘𝑊) = ((𝑁 + 1) + 1) ↔ ((𝑁 + 1) + 1) = (#‘𝑊))
8		nn0cn 11179	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘𝑊) ∈ ℕ₀ → (#‘𝑊) ∈ ℂ)
9	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (#‘𝑊) ∈ ℂ)
10		1cnd 9935	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 1 ∈ ℂ)
11		nn0cn 11179	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑁 + 1) ∈ ℕ₀ → (𝑁 + 1) ∈ ℂ)
12	2, 11	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℂ)
13	12	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ ℂ)
14		subadd2 10164	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((#‘𝑊) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑁 + 1) ∈ ℂ) → (((#‘𝑊) − 1) = (𝑁 + 1) ↔ ((𝑁 + 1) + 1) = (#‘𝑊)))
15	14	bicomd 212	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((#‘𝑊) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑁 + 1) ∈ ℂ) → (((𝑁 + 1) + 1) = (#‘𝑊) ↔ ((#‘𝑊) − 1) = (𝑁 + 1)))
16	9, 10, 13, 15	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (((𝑁 + 1) + 1) = (#‘𝑊) ↔ ((#‘𝑊) − 1) = (𝑁 + 1)))
17	7, 16	syl5bb 271	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((#‘𝑊) = ((𝑁 + 1) + 1) ↔ ((#‘𝑊) − 1) = (𝑁 + 1)))
18		eqcom 2617	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((#‘𝑊) − 1) = (𝑁 + 1) ↔ (𝑁 + 1) = ((#‘𝑊) − 1))
19	18	biimpi 205	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((#‘𝑊) − 1) = (𝑁 + 1) → (𝑁 + 1) = ((#‘𝑊) − 1))
20	17, 19	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((#‘𝑊) ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 + 1) = ((#‘𝑊) − 1)))
21	20	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((#‘𝑊) ∈ ℕ₀ → (𝑁 ∈ ℕ₀ → ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 + 1) = ((#‘𝑊) − 1))))
22	21	com23 84	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((#‘𝑊) ∈ ℕ₀ → ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ₀ → (𝑁 + 1) = ((#‘𝑊) − 1))))
23	6, 22	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑊 ∈ Word 𝑉 → ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ₀ → (𝑁 + 1) = ((#‘𝑊) − 1))))
24	23	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) → ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ₀ → (𝑁 + 1) = ((#‘𝑊) − 1))))
25	5, 24	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑊 ∈ (𝑉 WWalks 𝐸) → ((#‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ₀ → (𝑁 + 1) = ((#‘𝑊) − 1))))
26	25	imp 444	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ₀ → (𝑁 + 1) = ((#‘𝑊) − 1)))
27	26	com12 32	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → ((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 + 1) = ((#‘𝑊) − 1)))
28	27	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ₀) → ((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 + 1) = ((#‘𝑊) − 1)))
29	28	impcom 445	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ₀)) → (𝑁 + 1) = ((#‘𝑊) − 1))
30	29	opeq2d 4347	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ₀)) → ⟨0, (𝑁 + 1)⟩ = ⟨0, ((#‘𝑊) − 1)⟩)
31	30	oveq2d 6565	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ₀)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) = (𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩))
32		simpll 786	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 𝑊 ∈ (𝑉 WWalks 𝐸))
33		nn0ge0 11195	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ 𝑁)
34		2re 10967	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 2 ∈ ℝ
35	34	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ₀ → 2 ∈ ℝ)
36		nn0re 11178	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
37	35, 36	addge02d 10495	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ₀ → (0 ≤ 𝑁 ↔ 2 ≤ (𝑁 + 2)))
38	33, 37	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ₀ → 2 ≤ (𝑁 + 2))
39		nn0cn 11179	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
40		1cnd 9935	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ₀ → 1 ∈ ℂ)
41	39, 40, 40	addassd 9941	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1)))
42		1p1e2 11011	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (1 + 1) = 2
43	42	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ₀ → (1 + 1) = 2)
44	43	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + (1 + 1)) = (𝑁 + 2))
45	41, 44	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) + 1) = (𝑁 + 2))
46	38, 45	breqtrrd 4611	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ₀ → 2 ≤ ((𝑁 + 1) + 1))
47	46	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 2 ≤ ((𝑁 + 1) + 1))
48		breq2 4587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑊) = ((𝑁 + 1) + 1) → (2 ≤ (#‘𝑊) ↔ 2 ≤ ((𝑁 + 1) + 1)))
49	48	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (2 ≤ (#‘𝑊) ↔ 2 ≤ ((𝑁 + 1) + 1)))
50	47, 49	mpbird 246	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 2 ≤ (#‘𝑊))
51	32, 50	jca 553	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑊 ∈ (𝑉 WWalks 𝐸) ∧ 2 ≤ (#‘𝑊)))
52	51	3ad2antr3 1221	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ₀)) → (𝑊 ∈ (𝑉 WWalks 𝐸) ∧ 2 ≤ (#‘𝑊)))
53		wwlkm1edg 26263	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ 2 ≤ (#‘𝑊)) → (𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) ∈ (𝑉 WWalks 𝐸))
54	52, 53	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ₀)) → (𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) ∈ (𝑉 WWalks 𝐸))
55	31, 54	eqeltrd 2688	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ₀)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑉 WWalks 𝐸))
56	55	expcom 450	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ₀) → ((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑉 WWalks 𝐸)))
57	4, 56	sylbid 229	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ₀) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑉 WWalks 𝐸)))
58	57	3expia 1259	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑁 ∈ ℕ₀ → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑉 WWalks 𝐸))))
59	58	com23 84	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑁 ∈ ℕ₀ → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑉 WWalks 𝐸))))
60	59	imp 444	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → (𝑁 ∈ ℕ₀ → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑉 WWalks 𝐸)))
61	60	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) ∧ (𝑊‘0) = 𝑃) → (𝑁 ∈ ℕ₀ → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑉 WWalks 𝐸)))
62	61	imp 444	. . . . . . . . 9 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) ∧ (𝑊‘0) = 𝑃) ∧ 𝑁 ∈ ℕ₀) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑉 WWalks 𝐸))
63		wwlknimp 26215	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))
64		simpll 786	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 𝑊 ∈ Word 𝑉)
65		peano2nn0 11210	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 + 1) ∈ ℕ₀ → ((𝑁 + 1) + 1) ∈ ℕ₀)
66	2, 65	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) + 1) ∈ ℕ₀)
67		peano2re 10088	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
68	36, 67	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℝ)
69	68	lep1d 10834	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ≤ ((𝑁 + 1) + 1))
70		elfz2nn0 12300	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 + 1) ∈ (0...((𝑁 + 1) + 1)) ↔ ((𝑁 + 1) ∈ ℕ₀ ∧ ((𝑁 + 1) + 1) ∈ ℕ₀ ∧ (𝑁 + 1) ≤ ((𝑁 + 1) + 1)))
71	2, 66, 69, 70	syl3anbrc 1239	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ (0...((𝑁 + 1) + 1)))
72	71	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ (0...((𝑁 + 1) + 1)))
73		oveq2 6557	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑊) = ((𝑁 + 1) + 1) → (0...(#‘𝑊)) = (0...((𝑁 + 1) + 1)))
74	73	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → (0...(#‘𝑊)) = (0...((𝑁 + 1) + 1)))
75	72, 74	eleqtrrd 2691	. . . . . . . . . . . . . . . . 17 ⊢ (((#‘𝑊) = ((𝑁 + 1) + 1) ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ (0...(#‘𝑊)))
76	75	adantll 746	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ (0...(#‘𝑊)))
77	64, 76	jca 553	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (0...(#‘𝑊))))
78	77	ex 449	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ₀ → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (0...(#‘𝑊)))))
79	78	3adant3 1074	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → (𝑁 ∈ ℕ₀ → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (0...(#‘𝑊)))))
80	63, 79	syl 17	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑁 ∈ ℕ₀ → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (0...(#‘𝑊)))))
81	80	ad2antlr 759	. . . . . . . . . . 11 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) ∧ (𝑊‘0) = 𝑃) → (𝑁 ∈ ℕ₀ → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (0...(#‘𝑊)))))
82	81	imp 444	. . . . . . . . . 10 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) ∧ (𝑊‘0) = 𝑃) ∧ 𝑁 ∈ ℕ₀) → (𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (0...(#‘𝑊))))
83		swrd0len 13274	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑁 + 1) ∈ (0...(#‘𝑊))) → (#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑁 + 1))
84	82, 83	syl 17	. . . . . . . . 9 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) ∧ (𝑊‘0) = 𝑃) ∧ 𝑁 ∈ ℕ₀) → (#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑁 + 1))
85	62, 84	jca 553	. . . . . . . 8 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) ∧ (𝑊‘0) = 𝑃) ∧ 𝑁 ∈ ℕ₀) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑉 WWalks 𝐸) ∧ (#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑁 + 1)))
86		iswwlkn 26212	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ₀) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑉 WWalks 𝐸) ∧ (#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑁 + 1))))
87	86	3expia 1259	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑁 ∈ ℕ₀ → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑉 WWalks 𝐸) ∧ (#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑁 + 1)))))
88	87	adantr 480	. . . . . . . . . 10 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → (𝑁 ∈ ℕ₀ → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑉 WWalks 𝐸) ∧ (#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑁 + 1)))))
89	88	adantr 480	. . . . . . . . 9 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) ∧ (𝑊‘0) = 𝑃) → (𝑁 ∈ ℕ₀ → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑉 WWalks 𝐸) ∧ (#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑁 + 1)))))
90	89	imp 444	. . . . . . . 8 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) ∧ (𝑊‘0) = 𝑃) ∧ 𝑁 ∈ ℕ₀) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑉 WWalks 𝐸) ∧ (#‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)) = (𝑁 + 1))))
91	85, 90	mpbird 246	. . . . . . 7 ⊢ (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) ∧ (𝑊‘0) = 𝑃) ∧ 𝑁 ∈ ℕ₀) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁))
92	91	exp41 636	. . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → ((𝑊‘0) = 𝑃 → (𝑁 ∈ ℕ₀ → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))))
93	92	adantr 480	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ ((𝑁 + 1) ∈ ℕ₀ ∧ 𝑊 ∈ Word 𝑉)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → ((𝑊‘0) = 𝑃 → (𝑁 ∈ ℕ₀ → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))))
94	1, 93	mpcom 37	. . . 4 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → ((𝑊‘0) = 𝑃 → (𝑁 ∈ ℕ₀ → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁))))
95		wwlkextprop.x	. . . 4 ⊢ 𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))
96	94, 95	eleq2s 2706	. . 3 ⊢ (𝑊 ∈ 𝑋 → ((𝑊‘0) = 𝑃 → (𝑁 ∈ ℕ₀ → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁))))
97	96	3imp 1249	. 2 ⊢ ((𝑊 ∈ 𝑋 ∧ (𝑊‘0) = 𝑃 ∧ 𝑁 ∈ ℕ₀) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁))
98	95	wwlkextproplem1 26269	. . . 4 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑁 ∈ ℕ₀) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑊‘0))
99	98	3adant2 1073	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ (𝑊‘0) = 𝑃 ∧ 𝑁 ∈ ℕ₀) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑊‘0))
100		simp2 1055	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ (𝑊‘0) = 𝑃 ∧ 𝑁 ∈ ℕ₀) → (𝑊‘0) = 𝑃)
101	99, 100	eqtrd 2644	. 2 ⊢ ((𝑊 ∈ 𝑋 ∧ (𝑊‘0) = 𝑃 ∧ 𝑁 ∈ ℕ₀) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃)
102		fveq1 6102	. . . 4 ⊢ (𝑤 = (𝑊 substr ⟨0, (𝑁 + 1)⟩) → (𝑤‘0) = ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0))
103	102	eqeq1d 2612	. . 3 ⊢ (𝑤 = (𝑊 substr ⟨0, (𝑁 + 1)⟩) → ((𝑤‘0) = 𝑃 ↔ ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃))
104		wwlkextprop.y	. . 3 ⊢ 𝑌 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}
105	103, 104	elrab2 3333	. 2 ⊢ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ 𝑌 ↔ ((𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃))
106	97, 101, 105	sylanbrc 695	1 ⊢ ((𝑊 ∈ 𝑋 ∧ (𝑊‘0) = 𝑃 ∧ 𝑁 ∈ ℕ₀) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ 𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 Vcvv 3173 {cpr 4127 ⟨cop 4131 class class class wbr 4583 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 ≤ cle 9954 − cmin 10145 2c2 10947 ℕ₀cn0 11169 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 substr csubstr 13150 WWalks cwwlk 26205 WWalksN cwwlkn 26206

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-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-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-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-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-substr 13158 df-wwlk 26207 df-wwlkn 26208

This theorem is referenced by: wwlkextprop 26272

W3C validator