Theorem wwlkextwrd 26256

Description: Lemma 0 for wwlkextbij 26261. (Contributed by Alexander van der Vekens, 5-Aug-2018.)

Hypothesis

Ref	Expression
wwlkextbij.d	⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}

Assertion

Ref	Expression
wwlkextwrd	⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)})

Distinct variable groups:   𝑤,𝐸   𝑤,𝑁   𝑤,𝑉   𝑤,𝑊

Allowed substitution hint:   𝐷(𝑤)

Proof of Theorem wwlkextwrd

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlkextbij.d	. 2 ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}
2		3anass 1035	. . . . . 6 ⊢ (((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸) ↔ ((#‘𝑤) = (𝑁 + 2) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)))
3	2	anbi2i 726	. . . . 5 ⊢ ((𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = (𝑁 + 2) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))))
4		anass 679	. . . . 5 ⊢ (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = (𝑁 + 2) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))))
5	3, 4	bitr4i 266	. . . 4 ⊢ ((𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) ↔ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)))
6		wwlknprop 26214	. . . . . . . . . 10 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)))
7		simpl 472	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))) → 𝑁 ∈ ℕ0)
8		simpl 472	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ Word 𝑉)
9	8	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 𝑤 ∈ Word 𝑉)
10		nn0re 11178	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
11		2re 10967	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ ℝ
12	11	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
13		nn0ge0 11195	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
14		2pos 10989	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 < 2
15	14	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 0 < 2)
16	10, 12, 13, 15	addgegt0d 10480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
17	16	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 0 < (𝑁 + 2))
18		breq2 4587	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝑤) = (𝑁 + 2) → (0 < (#‘𝑤) ↔ 0 < (𝑁 + 2)))
19	18	ad2antll 761	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → (0 < (#‘𝑤) ↔ 0 < (𝑁 + 2)))
20	17, 19	mpbird 246	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 0 < (#‘𝑤))
21		hashgt0n0 13017	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 0 < (#‘𝑤)) → 𝑤 ≠ ∅)
22	9, 20, 21	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → 𝑤 ≠ ∅)
23		lswcl 13208	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 𝑤 ≠ ∅) → ( lastS ‘𝑤) ∈ 𝑉)
24	9, 22, 23	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ( lastS ‘𝑤) ∈ 𝑉)
25	24	adantrr 749	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))) → ( lastS ‘𝑤) ∈ 𝑉)
26		swrdcl 13271	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ Word 𝑉 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) ∈ Word 𝑉)
27		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑊 = (𝑤 substr ⟨0, (𝑁 + 1)⟩) → (𝑊 ∈ Word 𝑉 ↔ (𝑤 substr ⟨0, (𝑁 + 1)⟩) ∈ Word 𝑉))
28	26, 27	syl5ibr 235	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑊 = (𝑤 substr ⟨0, (𝑁 + 1)⟩) → (𝑤 ∈ Word 𝑉 → 𝑊 ∈ Word 𝑉))
29	28	eqcoms 2618	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (𝑤 ∈ Word 𝑉 → 𝑊 ∈ Word 𝑉))
30	29	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸) → (𝑤 ∈ Word 𝑉 → 𝑊 ∈ Word 𝑉))
31	30	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ Word 𝑉 → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸) → 𝑊 ∈ Word 𝑉))
32	31	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸) → 𝑊 ∈ Word 𝑉))
33	32	imp 444	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → 𝑊 ∈ Word 𝑉)
34	33	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))) → 𝑊 ∈ Word 𝑉)
35		oveq1 6556	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 = (𝑤 substr ⟨0, (𝑁 + 1)⟩) → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
36	35	eqcoms 2618	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
37	36	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸) → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
38	37	ad2antll 761	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))) → (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
39		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((#‘𝑤) = (𝑁 + 2) → ((#‘𝑤) − 1) = ((𝑁 + 2) − 1))
40	39	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → ((#‘𝑤) − 1) = ((𝑁 + 2) − 1))
41		nn0cn 11179	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
42		2cnd 10970	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℂ)
43		1cnd 9935	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
44	41, 42, 43	addsubassd 10291	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
45		2m1e1 11012	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (2 − 1) = 1
46	45	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ0 → (2 − 1) = 1)
47	46	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + (2 − 1)) = (𝑁 + 1))
48	44, 47	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + 1))
49	40, 48	sylan9eqr 2666	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((#‘𝑤) − 1) = (𝑁 + 1))
50	49	opeq2d 4347	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ⟨0, ((#‘𝑤) − 1)⟩ = ⟨0, (𝑁 + 1)⟩)
51	50	oveq2d 6565	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → (𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) = (𝑤 substr ⟨0, (𝑁 + 1)⟩))
52	51	oveq1d 6564	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩))
53		swrdccatwrd 13320	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ Word 𝑉 ∧ 𝑤 ≠ ∅) → ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
54	9, 22, 53	syl2anc 691	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((𝑤 substr ⟨0, ((#‘𝑤) − 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
55	52, 54	eqtr3d 2646	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))) → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
56	55	adantrr 749	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))) → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) ++ ⟨“( lastS ‘𝑤)”⟩) = 𝑤)
57	38, 56	eqtr2d 2645	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))) → 𝑤 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩))
58		simprrr 801	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))) → {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)
59		wwlknextbi 26253	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ ( lastS ‘𝑤) ∈ 𝑉) ∧ (𝑊 ∈ Word 𝑉 ∧ 𝑤 = (𝑊 ++ ⟨“( lastS ‘𝑤)”⟩) ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ↔ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
60	7, 25, 34, 57, 58, 59	syl23anc 1325	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ↔ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
61	60	exbiri 650	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
62	61	com23 84	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
63	62	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
64	63	adantl 481	. . . . . . . . . 10 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
65	6, 64	mpcom 37	. . . . . . . . 9 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))
66	65	expcomd 453	. . . . . . . 8 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸) → ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
67	66	imp 444	. . . . . . 7 ⊢ ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) → 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))
68		wwlknimp 26215	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸))
69	41, 43, 43	addassd 9941	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + (1 + 1)))
70		1p1e2 11011	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 + 1) = 2
71	70	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → (1 + 1) = 2)
72	71	oveq2d 6565	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + (1 + 1)) = (𝑁 + 2))
73	69, 72	eqtrd 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) + 1) = (𝑁 + 2))
74	73	eqeq2d 2620	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ0 → ((#‘𝑤) = ((𝑁 + 1) + 1) ↔ (#‘𝑤) = (𝑁 + 2)))
75	74	biimpd 218	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ0 → ((#‘𝑤) = ((𝑁 + 1) + 1) → (#‘𝑤) = (𝑁 + 2)))
76	75	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → ((#‘𝑤) = ((𝑁 + 1) + 1) → (#‘𝑤) = (𝑁 + 2)))
77	76	com12 32	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑤) = ((𝑁 + 1) + 1) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (#‘𝑤) = (𝑁 + 2)))
78	77	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (#‘𝑤) = (𝑁 + 2)))
79		simpl 472	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1)) → 𝑤 ∈ Word 𝑉)
80	78, 79	jctild 564	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1)) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
81	80	3adant3 1074	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
82	68, 81	syl 17	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
83	82	com12 32	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
84	83	adantl 481	. . . . . . . . 9 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
85	6, 84	syl 17	. . . . . . . 8 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
86	85	adantr 480	. . . . . . 7 ⊢ ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2))))
87	67, 86	impbid 201	. . . . . 6 ⊢ ((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) → ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ↔ 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))))
88	87	ex 449	. . . . 5 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸) → ((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ↔ 𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))))
89	88	pm5.32rd 670	. . . 4 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = (𝑁 + 2)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) ↔ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))))
90	5, 89	syl5bb 271	. . 3 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)) ↔ (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∧ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸))))
91	90	rabbidva2 3162	. 2 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)} = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)})
92	1, 91	syl5eq 2656	1 ⊢ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)})

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173  ∅c0 3874  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   − cmin 10145  2c2 10947  ℕ₀cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   ++ cconcat 13148  ⟨“cs1 13149   substr csubstr 13150   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-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-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-wwlk 26207  df-wwlkn 26208

This theorem is referenced by:  wwlkextsur  26259  wwlkextbij  26261