Theorem pfxwlk 35637

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 2766	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	wlkf 30001	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom (iEdg‘𝐺))
3	2	adantr 486	. . 3 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → 𝐹 ∈ Word dom (iEdg‘𝐺))
4		pfxcl 14739	. . 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 2766	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
7	6	wlkp 30003	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
8	7	adantr 486	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
9		elfzuz3 13567	. . . . . . 7 ⊢ (𝐿 ∈ (0...(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝐿))
10		fzss2 13611	. . . . . . 7 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝐿) → (0...𝐿) ⊆ (0...(♯‘𝐹)))
11	9, 10	syl 18	. . . . . 6 ⊢ (𝐿 ∈ (0...(♯‘𝐹)) → (0...𝐿) ⊆ (0...(♯‘𝐹)))
12	11	adantl 487	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0...𝐿) ⊆ (0...(♯‘𝐹)))
13	8, 12	fssresd 6752	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 ↾ (0...𝐿)):(0...𝐿)⟶(Vtx‘𝐺))
14		pfxlen 14745	. . . . . . 7 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 prefix 𝐿)) = 𝐿)
15	2, 14	sylan 592	. . . . . 6 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 prefix 𝐿)) = 𝐿)
16	15	oveq2d 7439	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0...(♯‘(𝐹 prefix 𝐿))) = (0...𝐿))
17	16	feq2d 6696	. . . 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 30004	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ∈ Word (Vtx‘𝐺))
20		fzp1elp1 13624	. . . . . . . 8 ⊢ (𝐿 ∈ (0...(♯‘𝐹)) → (𝐿 + 1) ∈ (0...((♯‘𝐹) + 1)))
21	20	adantl 487	. . . . . . 7 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐿 + 1) ∈ (0...((♯‘𝐹) + 1)))
22		wlklenvp1 30005	. . . . . . . . 9 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝑃) = ((♯‘𝐹) + 1))
23	22	oveq2d 7439	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (0...(♯‘𝑃)) = (0...((♯‘𝐹) + 1)))
24	23	adantr 486	. . . . . . 7 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0...(♯‘𝑃)) = (0...((♯‘𝐹) + 1)))
25	21, 24	eleqtrrd 2869	. . . . . 6 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐿 + 1) ∈ (0...(♯‘𝑃)))
26		pfxres 14741	. . . . . 6 ⊢ ((𝑃 ∈ Word (Vtx‘𝐺) ∧ (𝐿 + 1) ∈ (0...(♯‘𝑃))) → (𝑃 prefix (𝐿 + 1)) = (𝑃 ↾ (0..^(𝐿 + 1))))
27	19, 25, 26	syl2an2r 698	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 prefix (𝐿 + 1)) = (𝑃 ↾ (0..^(𝐿 + 1))))
28		elfzelz 13570	. . . . . . . 8 ⊢ (𝐿 ∈ (0...(♯‘𝐹)) → 𝐿 ∈ ℤ)
29		fzval3 13782	. . . . . . . 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 487	. . . . . 6 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0...𝐿) = (0..^(𝐿 + 1)))
32	31	reseq2d 5983	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 ↾ (0...𝐿)) = (𝑃 ↾ (0..^(𝐿 + 1))))
33	27, 32	eqtr4d 2804	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 prefix (𝐿 + 1)) = (𝑃 ↾ (0...𝐿)))
34	33	feq1d 6694	. . 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 29998	. . . . . . 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 1162	. . . . . 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 486	. . . . 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 486	. . . 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 7439	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0..^(♯‘(𝐹 prefix 𝐿))) = (0..^𝐿))
41	40	eleq2d 2852	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿))) ↔ 𝑘 ∈ (0..^𝐿)))
42	33	fveq1d 6890	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → ((𝑃 prefix (𝐿 + 1))‘𝑘) = ((𝑃 ↾ (0...𝐿))‘𝑘))
43	42	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((𝑃 prefix (𝐿 + 1))‘𝑘) = ((𝑃 ↾ (0...𝐿))‘𝑘))
44		fzossfz 13726	. . . . . . . . . . . . . 14 ⊢ (0..^𝐿) ⊆ (0...𝐿)
45	44	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0..^𝐿) ⊆ (0...𝐿))
46	45	sselda 3940	. . . . . . . . . . . 12 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → 𝑘 ∈ (0...𝐿))
47	46	fvresd 6908	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((𝑃 ↾ (0...𝐿))‘𝑘) = (𝑃‘𝑘))
48	43, 47	eqtr2d 2802	. . . . . . . . . 10 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → (𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘))
49	33	fveq1d 6890	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1)) = ((𝑃 ↾ (0...𝐿))‘(𝑘 + 1)))
50	49	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1)) = ((𝑃 ↾ (0...𝐿))‘(𝑘 + 1)))
51		fzofzp1 13812	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0..^𝐿) → (𝑘 + 1) ∈ (0...𝐿))
52	51	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → (𝑘 + 1) ∈ (0...𝐿))
53	52	fvresd 6908	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((𝑃 ↾ (0...𝐿))‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)))
54	50, 53	eqtr2d 2802	. . . . . . . . . 10 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → (𝑃‘(𝑘 + 1)) = ((𝑃 prefix (𝐿 + 1))‘(𝑘 + 1)))
55	48, 54	jca 521	. . . . . . . . 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 418	. . . . . . . 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 412	. . . . . 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 558	. . . . . . . . . . . 12 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)))
60		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) ∧ 𝑘 ∈ (0..^𝐿)) → 𝑘 ∈ (0..^𝐿))
61	60	fvresd 6908	. . . . . . . . . . . . 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 6892	. . . . . . . . . . . 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 592	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((iEdg‘𝐺)‘((𝐹 ↾ (0..^𝐿))‘𝑘)) = ((iEdg‘𝐺)‘(𝐹‘𝑘)))
64	63	eqcomd 2772	. . . . . . . . . 10 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((iEdg‘𝐺)‘(𝐹‘𝑘)) = ((iEdg‘𝐺)‘((𝐹 ↾ (0..^𝐿))‘𝑘)))
65	64	ex 418	. . . . . . . . 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 412	. . . . . . 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 781	. . . . . . . . . 10 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → 𝐿 ∈ (0...(♯‘𝐹)))
69		pfxres 14741	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝐿) = (𝐹 ↾ (0..^𝐿)))
70	3, 68, 69	syl2an2r 698	. . . . . . . . 9 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → (𝐹 prefix 𝐿) = (𝐹 ↾ (0..^𝐿)))
71	70	fveq1d 6890	. . . . . . . 8 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → ((𝐹 prefix 𝐿)‘𝑘) = ((𝐹 ↾ (0..^𝐿))‘𝑘))
72	71	fveq2d 6892	. . . . . . 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 2804	. . . . . 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 521	. . . . 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 487	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (♯‘𝐹) ∈ (ℤ≥‘𝐿))
76	15	fveq2d 6892	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (ℤ≥‘(♯‘(𝐹 prefix 𝐿))) = (ℤ≥‘𝐿))
77	75, 76	eleqtrrd 2869	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (♯‘𝐹) ∈ (ℤ≥‘(♯‘(𝐹 prefix 𝐿))))
78		fzoss2 13735	. . . . . . . 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 3940	. . . . . 6 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → 𝑘 ∈ (0..^(♯‘𝐹)))
81		wkslem1 29994	. . . . . . 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 2783	. . . . . . . 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 486	. . . . . . 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 490	. . . . . . . 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 4604	. . . . . . . . . 10 ⊢ ((𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘) → {(𝑃‘𝑘)} = {((𝑃 prefix (𝐿 + 1))‘𝑘)})
88	87	adantr 486	. . . . . . . . 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 486	. . . . . . . 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 2782	. . . . . . 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 4706	. . . . . . . . 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 486	. . . . . . . 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 3973	. . . . . . 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 1095	. . . . . 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 3160	. 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 29999	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
100	99	simp1d 1160	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐺 ∈ V)
101	100	adantr 486	. . 3 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → 𝐺 ∈ V)
102	6, 1	iswlkg 30000	. . 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 1361	1 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝐿)(Walks‘𝐺)(𝑃 prefix (𝐿 + 1)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  if-wif 1078   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082  Vcvv 3458   ⊆ wss 3908  {csn 4594  {cpr 4596   class class class wbr 5114  dom cdm 5666   ↾ cres 5668  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423  0cc0 11118  1c1 11119   + caddc 11121  ℤcz 12609  ℤ_≥cuz 12880  ...cfz 13553  ..^cfzo 13701  ♯chash 14386  Word cword 14570   prefix cpfx 14732  Vtxcvtx 29383  iEdgciedg 29384  Walkscwlks 29983

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8703  df-map 8835  df-pm 8836  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-card 9944  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-n0 12523  df-z 12610  df-uz 12881  df-fz 13554  df-fzo 13702  df-hash 14387  df-word 14571  df-substr 14701  df-pfx 14733  df-wlks 29986

This theorem is used by:  swrdwlk  35640