Theorem pfxwlk 35091

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 2740	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	wlkf 29650	. . . 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 14725	. . 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 2740	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
7	6	wlkp 29652	. . . . . 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 13581	. . . . . . 7 ⊢ (𝐿 ∈ (0...(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝐿))
10		fzss2 13624	. . . . . . 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 6788	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 ↾ (0...𝐿)):(0...𝐿)⟶(Vtx‘𝐺))
14		pfxlen 14731	. . . . . . 7 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 prefix 𝐿)) = 𝐿)
15	2, 14	sylan 579	. . . . . 6 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 prefix 𝐿)) = 𝐿)
16	15	oveq2d 7464	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0...(♯‘(𝐹 prefix 𝐿))) = (0...𝐿))
17	16	feq2d 6733	. . . 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 29653	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ∈ Word (Vtx‘𝐺))
20		fzp1elp1 13637	. . . . . . . 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 29654	. . . . . . . . 9 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝑃) = ((♯‘𝐹) + 1))
23	22	oveq2d 7464	. . . . . . . 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 2847	. . . . . 6 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐿 + 1) ∈ (0...(♯‘𝑃)))
26		pfxres 14727	. . . . . 6 ⊢ ((𝑃 ∈ Word (Vtx‘𝐺) ∧ (𝐿 + 1) ∈ (0...(♯‘𝑃))) → (𝑃 prefix (𝐿 + 1)) = (𝑃 ↾ (0..^(𝐿 + 1))))
27	19, 25, 26	syl2an2r 684	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 prefix (𝐿 + 1)) = (𝑃 ↾ (0..^(𝐿 + 1))))
28		elfzelz 13584	. . . . . . . 8 ⊢ (𝐿 ∈ (0...(♯‘𝐹)) → 𝐿 ∈ ℤ)
29		fzval3 13785	. . . . . . . 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 6009	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 ↾ (0...𝐿)) = (𝑃 ↾ (0..^(𝐿 + 1))))
33	27, 32	eqtr4d 2783	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 prefix (𝐿 + 1)) = (𝑃 ↾ (0...𝐿)))
34	33	feq1d 6732	. . 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 29647	. . . . . . 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 7464	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0..^(♯‘(𝐹 prefix 𝐿))) = (0..^𝐿))
41	40	eleq2d 2830	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿))) ↔ 𝑘 ∈ (0..^𝐿)))
42	33	fveq1d 6922	. . . . . . . . . . . 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 13735	. . . . . . . . . . . . . 14 ⊢ (0..^𝐿) ⊆ (0...𝐿)
45	44	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0..^𝐿) ⊆ (0...𝐿))
46	45	sselda 4008	. . . . . . . . . . . 12 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → 𝑘 ∈ (0...𝐿))
47	46	fvresd 6940	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((𝑃 ↾ (0...𝐿))‘𝑘) = (𝑃‘𝑘))
48	43, 47	eqtr2d 2781	. . . . . . . . . 10 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → (𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘))
49	33	fveq1d 6922	. . . . . . . . . . . 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 13814	. . . . . . . . . . . . 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 6940	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((𝑃 ↾ (0...𝐿))‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)))
54	50, 53	eqtr2d 2781	. . . . . . . . . 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 6940	. . . . . . . . . . . . 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 6924	. . . . . . . . . . . 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 579	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((iEdg‘𝐺)‘((𝐹 ↾ (0..^𝐿))‘𝑘)) = ((iEdg‘𝐺)‘(𝐹‘𝑘)))
64	63	eqcomd 2746	. . . . . . . . . 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 14727	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word dom (iEdg‘𝐺) ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝐿) = (𝐹 ↾ (0..^𝐿)))
70	3, 68, 69	syl2an2r 684	. . . . . . . . 9 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → (𝐹 prefix 𝐿) = (𝐹 ↾ (0..^𝐿)))
71	70	fveq1d 6922	. . . . . . . 8 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → ((𝐹 prefix 𝐿)‘𝑘) = ((𝐹 ↾ (0..^𝐿))‘𝑘))
72	71	fveq2d 6924	. . . . . . 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 2783	. . . . . 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 6924	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (ℤ≥‘(♯‘(𝐹 prefix 𝐿))) = (ℤ≥‘𝐿))
77	75, 76	eleqtrrd 2847	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (♯‘𝐹) ∈ (ℤ≥‘(♯‘(𝐹 prefix 𝐿))))
78		fzoss2 13744	. . . . . . . 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 4008	. . . . . 6 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → 𝑘 ∈ (0..^(♯‘𝐹)))
81		wkslem1 29643	. . . . . . 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 3631	. . . . . 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 2757	. . . . . . . 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 4658	. . . . . . . . . 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 2756	. . . . . . 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 4760	. . . . . . . . 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 4042	. . . . . . 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 1079	. . . . . 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 3152	. 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 29648	. . . . 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 29649	. . 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 1342	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 1063   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ⊆ wss 3976  {csn 4648  {cpr 4650   class class class wbr 5166  dom cdm 5700   ↾ cres 5702  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  0cc0 11184  1c1 11185   + caddc 11187  ℤcz 12639  ℤ_≥cuz 12903  ...cfz 13567  ..^cfzo 13711  ♯chash 14379  Word cword 14562   prefix cpfx 14718  Vtxcvtx 29031  iEdgciedg 29032  Walkscwlks 29632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-hash 14380  df-word 14563  df-substr 14689  df-pfx 14719  df-wlks 29635

This theorem is referenced by:  swrdwlk  35094