Theorem pfxwlk 35318

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 2736	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	wlkf 29688	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom (iEdg‘𝐺))
3	2	adantr 480	. . 3 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → 𝐹 ∈ Word dom (iEdg‘𝐺))
4		pfxcl 14601	. . 3 ⊢ (𝐹 ∈ Word dom (iEdg‘𝐺) → (𝐹 prefix 𝐿) ∈ Word dom (iEdg‘𝐺))
5	3, 4	syl 17	. 2 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝐿) ∈ Word dom (iEdg‘𝐺))
6		eqid 2736	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
7	6	wlkp 29690	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
8	7	adantr 480	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
9		elfzuz3 13437	. . . . . . 7 ⊢ (𝐿 ∈ (0...(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝐿))
10		fzss2 13480	. . . . . . 7 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝐿) → (0...𝐿) ⊆ (0...(♯‘𝐹)))
11	9, 10	syl 17	. . . . . 6 ⊢ (𝐿 ∈ (0...(♯‘𝐹)) → (0...𝐿) ⊆ (0...(♯‘𝐹)))
12	11	adantl 481	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0...𝐿) ⊆ (0...(♯‘𝐹)))
13	8, 12	fssresd 6701	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 ↾ (0...𝐿)):(0...𝐿)⟶(Vtx‘𝐺))
14		pfxlen 14607	. . . . . . 7 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 prefix 𝐿)) = 𝐿)
15	2, 14	sylan 580	. . . . . 6 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 prefix 𝐿)) = 𝐿)
16	15	oveq2d 7374	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0...(♯‘(𝐹 prefix 𝐿))) = (0...𝐿))
17	16	feq2d 6646	. . . 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 257	. . 3 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 ↾ (0...𝐿)):(0...(♯‘(𝐹 prefix 𝐿)))⟶(Vtx‘𝐺))
19	6	wlkpwrd 29691	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ∈ Word (Vtx‘𝐺))
20		fzp1elp1 13493	. . . . . . . 8 ⊢ (𝐿 ∈ (0...(♯‘𝐹)) → (𝐿 + 1) ∈ (0...((♯‘𝐹) + 1)))
21	20	adantl 481	. . . . . . 7 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐿 + 1) ∈ (0...((♯‘𝐹) + 1)))
22		wlklenvp1 29692	. . . . . . . . 9 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝑃) = ((♯‘𝐹) + 1))
23	22	oveq2d 7374	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (0...(♯‘𝑃)) = (0...((♯‘𝐹) + 1)))
24	23	adantr 480	. . . . . . 7 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0...(♯‘𝑃)) = (0...((♯‘𝐹) + 1)))
25	21, 24	eleqtrrd 2839	. . . . . 6 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐿 + 1) ∈ (0...(♯‘𝑃)))
26		pfxres 14603	. . . . . 6 ⊢ ((𝑃 ∈ Word (Vtx‘𝐺) ∧ (𝐿 + 1) ∈ (0...(♯‘𝑃))) → (𝑃 prefix (𝐿 + 1)) = (𝑃 ↾ (0..^(𝐿 + 1))))
27	19, 25, 26	syl2an2r 685	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 prefix (𝐿 + 1)) = (𝑃 ↾ (0..^(𝐿 + 1))))
28		elfzelz 13440	. . . . . . . 8 ⊢ (𝐿 ∈ (0...(♯‘𝐹)) → 𝐿 ∈ ℤ)
29		fzval3 13650	. . . . . . . 8 ⊢ (𝐿 ∈ ℤ → (0...𝐿) = (0..^(𝐿 + 1)))
30	28, 29	syl 17	. . . . . . 7 ⊢ (𝐿 ∈ (0...(♯‘𝐹)) → (0...𝐿) = (0..^(𝐿 + 1)))
31	30	adantl 481	. . . . . 6 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0...𝐿) = (0..^(𝐿 + 1)))
32	31	reseq2d 5938	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 ↾ (0...𝐿)) = (𝑃 ↾ (0..^(𝐿 + 1))))
33	27, 32	eqtr4d 2774	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 prefix (𝐿 + 1)) = (𝑃 ↾ (0...𝐿)))
34	33	feq1d 6644	. . 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 257	. 2 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 prefix (𝐿 + 1)):(0...(♯‘(𝐹 prefix 𝐿)))⟶(Vtx‘𝐺))
36	6, 1	wlkprop 29685	. . . . . . 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 1144	. . . . . 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 480	. . . . 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 480	. . . 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 7374	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0..^(♯‘(𝐹 prefix 𝐿))) = (0..^𝐿))
41	40	eleq2d 2822	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿))) ↔ 𝑘 ∈ (0..^𝐿)))
42	33	fveq1d 6836	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → ((𝑃 prefix (𝐿 + 1))‘𝑘) = ((𝑃 ↾ (0...𝐿))‘𝑘))
43	42	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((𝑃 prefix (𝐿 + 1))‘𝑘) = ((𝑃 ↾ (0...𝐿))‘𝑘))
44		fzossfz 13594	. . . . . . . . . . . . . 14 ⊢ (0..^𝐿) ⊆ (0...𝐿)
45	44	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0..^𝐿) ⊆ (0...𝐿))
46	45	sselda 3933	. . . . . . . . . . . 12 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → 𝑘 ∈ (0...𝐿))
47	46	fvresd 6854	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((𝑃 ↾ (0...𝐿))‘𝑘) = (𝑃‘𝑘))
48	43, 47	eqtr2d 2772	. . . . . . . . . 10 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → (𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘))
49	33	fveq1d 6836	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1)) = ((𝑃 ↾ (0...𝐿))‘(𝑘 + 1)))
50	49	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1)) = ((𝑃 ↾ (0...𝐿))‘(𝑘 + 1)))
51		fzofzp1 13680	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0..^𝐿) → (𝑘 + 1) ∈ (0...𝐿))
52	51	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → (𝑘 + 1) ∈ (0...𝐿))
53	52	fvresd 6854	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((𝑃 ↾ (0...𝐿))‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)))
54	50, 53	eqtr2d 2772	. . . . . . . . . 10 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1)))
55	48, 54	jca 511	. . . . . . . . 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 412	. . . . . . . 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 240	. . . . . . 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 406	. . . . . 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 548	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)))
60		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) ∧ 𝑘 ∈ (0..^𝐿)) → 𝑘 ∈ (0..^𝐿))
61	60	fvresd 6854	. . . . . . . . . . . . 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 6838	. . . . . . . . . . . 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 580	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((iEdg‘𝐺)‘((𝐹 ↾ (0..^𝐿))‘𝑘)) = ((iEdg‘𝐺)‘(𝐹‘𝑘)))
64	63	eqcomd 2742	. . . . . . . . . 10 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘((𝐹 ↾ (0..^𝐿))‘𝑘)))
65	64	ex 412	. . . . . . . . 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 240	. . . . . . . 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 406	. . . . . . 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 768	. . . . . . . . . 10 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → 𝐿 ∈ (0...(♯‘𝐹)))
69		pfxres 14603	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝐿) = (𝐹 ↾ (0..^𝐿)))
70	3, 68, 69	syl2an2r 685	. . . . . . . . 9 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → (𝐹 prefix 𝐿) = (𝐹 ↾ (0..^𝐿)))
71	70	fveq1d 6836	. . . . . . . 8 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → ((𝐹 prefix 𝐿)‘𝑘) = ((𝐹 ↾ (0..^𝐿))‘𝑘))
72	71	fveq2d 6838	. . . . . . 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 2774	. . . . . 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 511	. . . . 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 481	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (♯‘𝐹) ∈ (ℤ≥‘𝐿))
76	15	fveq2d 6838	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (ℤ≥‘(♯‘(𝐹 prefix 𝐿))) = (ℤ≥‘𝐿))
77	75, 76	eleqtrrd 2839	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (♯‘𝐹) ∈ (ℤ≥‘(♯‘(𝐹 prefix 𝐿))))
78		fzoss2 13603	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘(♯‘(𝐹 prefix 𝐿))) → (0..^(♯‘(𝐹 prefix 𝐿))) ⊆ (0..^(♯‘𝐹)))
79	77, 78	syl 17	. . . . . . 7 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0..^(♯‘(𝐹 prefix 𝐿))) ⊆ (0..^(♯‘𝐹)))
80	79	sselda 3933	. . . . . 6 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → 𝑘 ∈ (0..^(♯‘𝐹)))
81		wkslem1 29681	. . . . . . 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 3572	. . . . . 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 17	. . . . 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 2753	. . . . . . . 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 480	. . . . . . 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 484	. . . . . . . 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 4590	. . . . . . . . . 10 ⊢ ((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) → {(𝑃‘𝑘)} = {((𝑃 prefix (𝐿 + 1))‘𝑘)})
88	87	adantr 480	. . . . . . . . 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 480	. . . . . . . 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 2752	. . . . . . 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 4692	. . . . . . . . 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 480	. . . . . . . 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 3967	. . . . . . 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 1078	. . . . . 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 229	. . . . 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 61	. . . 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 15	. . 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 3128	. 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 29686	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
100	99	simp1d 1142	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐺 ∈ V)
101	100	adantr 480	. . 3 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → 𝐺 ∈ V)
102	6, 1	iswlkg 29687	. . 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 17	. 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 1343	1 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝐿)(Walks‘𝐺)(𝑃 prefix (𝐿 + 1)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  if-wif 1062   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051  Vcvv 3440   ⊆ wss 3901  {csn 4580  {cpr 4582   class class class wbr 5098  dom cdm 5624   ↾ cres 5626  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  0cc0 11026  1c1 11027   + caddc 11029  ℤcz 12488  ℤ_≥cuz 12751  ...cfz 13423  ..^cfzo 13570  ♯chash 14253  Word cword 14436   prefix cpfx 14594  Vtxcvtx 29069  iEdgciedg 29070  Walkscwlks 29670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424  df-fzo 13571  df-hash 14254  df-word 14437  df-substr 14565  df-pfx 14595  df-wlks 29673

This theorem is referenced by:  swrdwlk  35321