Theorem pfxwlk 35487

Description: A prefix of a walk is a walk. (Contributed by BTernaryTau, 2-Dec-2023.)

Assertion

Ref	Expression
pfxwlk	⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝐿)(Walks‘𝐺)(𝑃 prefix (𝐿 + 1)))

Proof of Theorem pfxwlk

Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2765	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	wlkf 29873	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom (iEdg‘𝐺))
3	2	adantr 485	. . 3 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → 𝐹 ∈ Word dom (iEdg‘𝐺))
4		pfxcl 14705	. . 3 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → (𝐹 prefix 𝐿) ∈ Word dom (iEdg‘𝐺))
5	3, 4	syl 18	. 2 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝐿) ∈ Word dom (iEdg‘𝐺))
6		eqid 2765	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
7	6	wlkp 29875	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
8	7	adantr 485	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
9		elfzuz3 13540	. . . . . . 7 ⊢ (𝐿 ∈ (0...(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝐿))
10		fzss2 13583	. . . . . . 7 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝐿) → (0...𝐿) ⊆ (0...(♯‘𝐹)))
11	9, 10	syl 18	. . . . . 6 ⊢ (𝐿 ∈ (0...(♯‘𝐹)) → (0...𝐿) ⊆ (0...(♯‘𝐹)))
12	11	adantl 486	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0...𝐿) ⊆ (0...(♯‘𝐹)))
13	8, 12	fssresd 6735	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 ↾ (0...𝐿)):(0...𝐿)⟶(Vtx‘𝐺))
14		pfxlen 14711	. . . . . . 7 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 prefix 𝐿)) = 𝐿)
15	2, 14	sylan 591	. . . . . 6 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 prefix 𝐿)) = 𝐿)
16	15	oveq2d 7416	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0...(♯‘(𝐹 prefix 𝐿))) = (0...𝐿))
17	16	feq2d 6679	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → ((𝑃 ↾ (0...𝐿)):(0...(♯‘(𝐹 prefix 𝐿)))⟶(Vtx‘𝐺) ↔ (𝑃 ↾ (0...𝐿)):(0...𝐿)⟶(Vtx‘𝐺)))
18	13, 17	mpbird 260	. . 3 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 ↾ (0...𝐿)):(0...(♯‘(𝐹 prefix 𝐿)))⟶(Vtx‘𝐺))
19	6	wlkpwrd 29876	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ∈ Word (Vtx‘𝐺))
20		fzp1elp1 13596	. . . . . . . 8 ⊢ (𝐿 ∈ (0...(♯‘𝐹)) → (𝐿 + 1) ∈ (0...((♯‘𝐹) + 1)))
21	20	adantl 486	. . . . . . 7 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐿 + 1) ∈ (0...((♯‘𝐹) + 1)))
22		wlklenvp1 29877	. . . . . . . . 9 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝑃) = ((♯‘𝐹) + 1))
23	22	oveq2d 7416	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (0...(♯‘𝑃)) = (0...((♯‘𝐹) + 1)))
24	23	adantr 485	. . . . . . 7 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0...(♯‘𝑃)) = (0...((♯‘𝐹) + 1)))
25	21, 24	eleqtrrd 2868	. . . . . 6 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐿 + 1) ∈ (0...(♯‘𝑃)))
26		pfxres 14707	. . . . . 6 ⊢ ((𝑃 ∈ Word (Vtx‘𝐺) ∧ (𝐿 + 1) ∈ (0...(♯‘𝑃))) → (𝑃 prefix (𝐿 + 1)) = (𝑃 ↾ (0..^(𝐿 + 1))))
27	19, 25, 26	syl2an2r 697	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 prefix (𝐿 + 1)) = (𝑃 ↾ (0..^(𝐿 + 1))))
28		elfzelz 13543	. . . . . . . 8 ⊢ (𝐿 ∈ (0...(♯‘𝐹)) → 𝐿 ∈ ℤ)
29		fzval3 13754	. . . . . . . 8 ⊢ (𝐿 ∈ ℤ → (0...𝐿) = (0..^(𝐿 + 1)))
30	28, 29	syl 18	. . . . . . 7 ⊢ (𝐿 ∈ (0...(♯‘𝐹)) → (0...𝐿) = (0..^(𝐿 + 1)))
31	30	adantl 486	. . . . . 6 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0...𝐿) = (0..^(𝐿 + 1)))
32	31	reseq2d 5969	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 ↾ (0...𝐿)) = (𝑃 ↾ (0..^(𝐿 + 1))))
33	27, 32	eqtr4d 2803	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 prefix (𝐿 + 1)) = (𝑃 ↾ (0...𝐿)))
34	33	feq1d 6677	. . 3 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → ((𝑃 prefix (𝐿 + 1)):(0...(♯‘(𝐹 prefix 𝐿)))⟶(Vtx‘𝐺) ↔ (𝑃 ↾ (0...𝐿)):(0...(♯‘(𝐹 prefix 𝐿)))⟶(Vtx‘𝐺)))
35	18, 34	mpbird 260	. 2 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 prefix (𝐿 + 1)):(0...(♯‘(𝐹 prefix 𝐿)))⟶(Vtx‘𝐺))
36	6, 1	wlkprop 29870	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑥 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑥)) = {(𝑃‘𝑥)}, {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑥)))))
37	36	simp3d 1160	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → ∀𝑥 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑥)) = {(𝑃‘𝑥)}, {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑥))))
38	37	adantr 485	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → ∀𝑥 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑥)) = {(𝑃‘𝑥)}, {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑥))))
39	38	adantr 485	. . . 4 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → ∀𝑥 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑥)) = {(𝑃‘𝑥)}, {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑥))))
40	15	oveq2d 7416	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0..^(♯‘(𝐹 prefix 𝐿))) = (0..^𝐿))
41	40	eleq2d 2851	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿))) ↔ 𝑘 ∈ (0..^𝐿)))
42	33	fveq1d 6873	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → ((𝑃 prefix (𝐿 + 1))‘𝑘) = ((𝑃 ↾ (0...𝐿))‘𝑘))
43	42	adantr 485	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((𝑃 prefix (𝐿 + 1))‘𝑘) = ((𝑃 ↾ (0...𝐿))‘𝑘))
44		fzossfz 13698	. . . . . . . . . . . . . 14 ⊢ (0..^𝐿) ⊆ (0...𝐿)
45	44	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0..^𝐿) ⊆ (0...𝐿))
46	45	sselda 3939	. . . . . . . . . . . 12 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → 𝑘 ∈ (0...𝐿))
47	46	fvresd 6891	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((𝑃 ↾ (0...𝐿))‘𝑘) = (𝑃‘𝑘))
48	43, 47	eqtr2d 2801	. . . . . . . . . 10 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → (𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘))
49	33	fveq1d 6873	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1)) = ((𝑃 ↾ (0...𝐿))‘(𝑘 + 1)))
50	49	adantr 485	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1)) = ((𝑃 ↾ (0...𝐿))‘(𝑘 + 1)))
51		fzofzp1 13784	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0..^𝐿) → (𝑘 + 1) ∈ (0...𝐿))
52	51	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → (𝑘 + 1) ∈ (0...𝐿))
53	52	fvresd 6891	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((𝑃 ↾ (0...𝐿))‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)))
54	50, 53	eqtr2d 2801	. . . . . . . . . 10 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1)))
55	48, 54	jca 520	. . . . . . . . 9 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) ∧ (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))))
56	55	ex 417	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑘 ∈ (0..^𝐿) → ((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) ∧ (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1)))))
57	41, 56	sylbid 243	. . . . . . 7 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿))) → ((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) ∧ (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1)))))
58	57	imp 411	. . . . . 6 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → ((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) ∧ (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))))
59	3	ancli 557	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)))
60		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) ∧ 𝑘 ∈ (0..^𝐿)) → 𝑘 ∈ (0..^𝐿))
61	60	fvresd 6891	. . . . . . . . . . . . 13 ⊢ ((((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) ∧ 𝑘 ∈ (0..^𝐿)) → ((𝐹 ↾ (0..^𝐿))‘𝑘) = (𝐹‘𝑘))
62	61	fveq2d 6875	. . . . . . . . . . . 12 ⊢ ((((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) ∧ 𝑘 ∈ (0..^𝐿)) → ((iEdg‘𝐺)‘((𝐹 ↾ (0..^𝐿))‘𝑘)) = ((iEdg‘𝐺)‘(𝐹‘𝑘)))
63	59, 62	sylan 591	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((iEdg‘𝐺)‘((𝐹 ↾ (0..^𝐿))‘𝑘)) = ((iEdg‘𝐺)‘(𝐹‘𝑘)))
64	63	eqcomd 2771	. . . . . . . . . 10 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘((𝐹 ↾ (0..^𝐿))‘𝑘)))
65	64	ex 417	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑘 ∈ (0..^𝐿) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘((𝐹 ↾ (0..^𝐿))‘𝑘))))
66	41, 65	sylbid 243	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘((𝐹 ↾ (0..^𝐿))‘𝑘))))
67	66	imp 411	. . . . . . 7 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘((𝐹 ↾ (0..^𝐿))‘𝑘)))
68		simplr 780	. . . . . . . . . 10 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → 𝐿 ∈ (0...(♯‘𝐹)))
69		pfxres 14707	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝐿) = (𝐹 ↾ (0..^𝐿)))
70	3, 68, 69	syl2an2r 697	. . . . . . . . 9 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → (𝐹 prefix 𝐿) = (𝐹 ↾ (0..^𝐿)))
71	70	fveq1d 6873	. . . . . . . 8 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → ((𝐹 prefix 𝐿)‘𝑘) = ((𝐹 ↾ (0..^𝐿))‘𝑘))
72	71	fveq2d 6875	. . . . . . 7 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘)) = ((iEdg‘𝐺)‘((𝐹 ↾ (0..^𝐿))‘𝑘)))
73	67, 72	eqtr4d 2803	. . . . . 6 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘)))
74	58, 73	jca 520	. . . . 5 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → (((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) ∧ (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘))))
75	9	adantl 486	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (♯‘𝐹) ∈ (ℤ≥‘𝐿))
76	15	fveq2d 6875	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (ℤ≥‘(♯‘(𝐹 prefix 𝐿))) = (ℤ≥‘𝐿))
77	75, 76	eleqtrrd 2868	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (♯‘𝐹) ∈ (ℤ≥‘(♯‘(𝐹 prefix 𝐿))))
78		fzoss2 13707	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘(♯‘(𝐹 prefix 𝐿))) → (0..^(♯‘(𝐹 prefix 𝐿))) ⊆ (0..^(♯‘𝐹)))
79	77, 78	syl 18	. . . . . . 7 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0..^(♯‘(𝐹 prefix 𝐿))) ⊆ (0..^(♯‘𝐹)))
80	79	sselda 3939	. . . . . 6 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → 𝑘 ∈ (0..^(♯‘𝐹)))
81		wkslem1 29866	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (if-((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑥)) = {(𝑃‘𝑥)}, {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑥))) ↔ if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))
82	81	rspcv 3580	. . . . . 6 ⊢ (𝑘 ∈ (0..^(♯‘𝐹)) → (∀𝑥 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑥)) = {(𝑃‘𝑥)}, {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑥))) → if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))
83	80, 82	syl 18	. . . . 5 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → (∀𝑥 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑥)) = {(𝑃‘𝑥)}, {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑥))) → if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))
84		eqeq12 2782	. . . . . . . 8 ⊢ (((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) ∧ (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))) → ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ↔ ((𝑃 prefix (𝐿 + 1))‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))))
85	84	adantr 485	. . . . . . 7 ⊢ ((((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) ∧ (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘))) → ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ↔ ((𝑃 prefix (𝐿 + 1))‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))))
86		simpr 489	. . . . . . . 8 ⊢ ((((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) ∧ (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘))) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘)))
87		sneq 4595	. . . . . . . . . 10 ⊢ ((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) → {(𝑃‘𝑘)} = {((𝑃 prefix (𝐿 + 1))‘𝑘)})
88	87	adantr 485	. . . . . . . . 9 ⊢ (((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) ∧ (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))) → {(𝑃‘𝑘)} = {((𝑃 prefix (𝐿 + 1))‘𝑘)})
89	88	adantr 485	. . . . . . . 8 ⊢ ((((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) ∧ (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘))) → {(𝑃‘𝑘)} = {((𝑃 prefix (𝐿 + 1))‘𝑘)})
90	86, 89	eqeq12d 2781	. . . . . . 7 ⊢ ((((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) ∧ (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘))) → (((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)} ↔ ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘)) = {((𝑃 prefix (𝐿 + 1))‘𝑘)}))
91		preq12 4697	. . . . . . . . 9 ⊢ (((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) ∧ (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))) → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {((𝑃 prefix (𝐿 + 1))‘𝑘), ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))})
92	91	adantr 485	. . . . . . . 8 ⊢ ((((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) ∧ (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘))) → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {((𝑃 prefix (𝐿 + 1))‘𝑘), ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))})
93	92, 86	sseq12d 3972	. . . . . . 7 ⊢ ((((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) ∧ (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘))) → ({(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘)) ↔ {((𝑃 prefix (𝐿 + 1))‘𝑘), ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘))))
94	85, 90, 93	ifpbi123d 1093	. . . . . 6 ⊢ ((((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) ∧ (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘))) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ↔ if-(((𝑃 prefix (𝐿 + 1))‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1)), ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘)) = {((𝑃 prefix (𝐿 + 1))‘𝑘)}, {((𝑃 prefix (𝐿 + 1))‘𝑘), ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘)))))
95	94	biimpd 232	. . . . 5 ⊢ ((((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) ∧ (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))) ∧ ((iEdg‘𝐺)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘))) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑘))) → if-(((𝑃 prefix (𝐿 + 1))‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1)), ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘)) = {((𝑃 prefix (𝐿 + 1))‘𝑘)}, {((𝑃 prefix (𝐿 + 1))‘𝑘), ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘)))))
96	74, 83, 95	sylsyld 62	. . . 4 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → (∀𝑥 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)), ((iEdg‘𝐺)‘(𝐹‘𝑥)) = {(𝑃‘𝑥)}, {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐹‘𝑥))) → if-(((𝑃 prefix (𝐿 + 1))‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1)), ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘)) = {((𝑃 prefix (𝐿 + 1))‘𝑘)}, {((𝑃 prefix (𝐿 + 1))‘𝑘), ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘)))))
97	39, 96	mpd 16	. . 3 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → if-(((𝑃 prefix (𝐿 + 1))‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1)), ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘)) = {((𝑃 prefix (𝐿 + 1))‘𝑘)}, {((𝑃 prefix (𝐿 + 1))‘𝑘), ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘))))
98	97	ralrimiva 3157	. 2 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → ∀𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))if-(((𝑃 prefix (𝐿 + 1))‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1)), ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘)) = {((𝑃 prefix (𝐿 + 1))‘𝑘)}, {((𝑃 prefix (𝐿 + 1))‘𝑘), ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘))))
99		wlkv 29871	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
100	99	simp1d 1158	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐺 ∈ V)
101	100	adantr 485	. . 3 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → 𝐺 ∈ V)
102	6, 1	iswlkg 29872	. . 3 ⊢ (𝐺 ∈ V → ((𝐹 prefix 𝐿)(Walks‘𝐺)(𝑃 prefix (𝐿 + 1)) ↔ ((𝐹 prefix 𝐿) ∈ Word dom (iEdg‘𝐺) ∧ (𝑃 prefix (𝐿 + 1)):(0...(♯‘(𝐹 prefix 𝐿)))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))if-(((𝑃 prefix (𝐿 + 1))‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1)), ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘)) = {((𝑃 prefix (𝐿 + 1))‘𝑘)}, {((𝑃 prefix (𝐿 + 1))‘𝑘), ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘))))))
103	101, 102	syl 18	. 2 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → ((𝐹 prefix 𝐿)(Walks‘𝐺)(𝑃 prefix (𝐿 + 1)) ↔ ((𝐹 prefix 𝐿) ∈ Word dom (iEdg‘𝐺) ∧ (𝑃 prefix (𝐿 + 1)):(0...(♯‘(𝐹 prefix 𝐿)))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))if-(((𝑃 prefix (𝐿 + 1))‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1)), ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘)) = {((𝑃 prefix (𝐿 + 1))‘𝑘)}, {((𝑃 prefix (𝐿 + 1))‘𝑘), ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘((𝐹 prefix 𝐿)‘𝑘))))))
104	5, 35, 98, 103	mpbir3and 1359	1 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝐿)(Walks‘𝐺)(𝑃 prefix (𝐿 + 1)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  if-wif 1076   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  Vcvv 3457   ⊆ wss 3907  {csn 4585  {cpr 4587   class class class wbr 5105  dom cdm 5652   ↾ cres 5654  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  0cc0 11088  1c1 11089   + caddc 11091  ℤcz 12582  ℤ_≥cuz 12853  ...cfz 13526  ..^cfzo 13673  ♯chash 14357  Word cword 14540   prefix cpfx 14698  Vtxcvtx 29255  iEdgciedg 29256  Walkscwlks 29855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1077  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-map 8814  df-pm 8815  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-n0 12496  df-z 12583  df-uz 12854  df-fz 13527  df-fzo 13674  df-hash 14358  df-word 14541  df-substr 14669  df-pfx 14699  df-wlks 29858

This theorem is referenced by:  swrdwlk  35490