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Theorem pfxwlk 30153

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 2762	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	wlkf 30082	. . . 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 14751	. . 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 2762	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
7	6	wlkp 30084	. . . . . 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 13579	. . . . . . 7 ⊢ (𝐿 ∈ (0...(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ_≥‘𝐿))
10		fzss2 13623	. . . . . . 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 6746	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 ↾ (0...𝐿)):(0...𝐿)⟶(Vtx‘𝐺))
14		pfxlen 14757	. . . . . . 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 7433	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0...(♯‘(𝐹 prefix 𝐿))) = (0...𝐿))
17	16	feq2d 6690	. . . 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 30085	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ∈ Word (Vtx‘𝐺))
20		fzp1elp1 13636	. . . . . . . 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 30086	. . . . . . . . 9 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝑃) = ((♯‘𝐹) + 1))
23	22	oveq2d 7433	. . . . . . . 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 2865	. . . . . 6 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐿 + 1) ∈ (0...(♯‘𝑃)))
26		pfxres 14753	. . . . . 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 13582	. . . . . . . 8 ⊢ (𝐿 ∈ (0...(♯‘𝐹)) → 𝐿 ∈ ℤ)
29		fzval3 13794	. . . . . . . 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 5976	. . . . 5 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 ↾ (0...𝐿)) = (𝑃 ↾ (0..^(𝐿 + 1))))
33	27, 32	eqtr4d 2800	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑃 prefix (𝐿 + 1)) = (𝑃 ↾ (0...𝐿)))
34	33	feq1d 6688	. . 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 30079	. . . . . . 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 7433	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0..^(♯‘(𝐹 prefix 𝐿))) = (0..^𝐿))
41	40	eleq2d 2848	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿))) ↔ 𝑘 ∈ (0..^𝐿)))
42	33	fveq1d 6884	. . . . . . . . . . . 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 13738	. . . . . . . . . . . . . 14 ⊢ (0..^𝐿) ⊆ (0...𝐿)
45	44	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (0..^𝐿) ⊆ (0...𝐿))
46	45	sselda 3934	. . . . . . . . . . . 12 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → 𝑘 ∈ (0...𝐿))
47	46	fvresd 6902	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((𝑃 ↾ (0...𝐿))‘𝑘) = (𝑃‘𝑘))
48	43, 47	eqtr2d 2798	. . . . . . . . . 10 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → (𝑃‘𝑘) = ((𝑃 prefix (𝐿 + 1))‘𝑘))
49	33	fveq1d 6884	. . . . . . . . . . . 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 13824	. . . . . . . . . . . . 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 6902	. . . . . . . . . . 11 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^𝐿)) → ((𝑃 ↾ (0...𝐿))‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)))
54	50, 53	eqtr2d 2798	. . . . . . . . . 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 6902	. . . . . . . . . . . . 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 6886	. . . . . . . . . . . 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 2768	. . . . . . . . . 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 14753	. . . . . . . . . 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 6884	. . . . . . . 8 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → ((𝐹 prefix 𝐿)‘𝑘) = ((𝐹 ↾ (0..^𝐿))‘𝑘))
72	71	fveq2d 6886	. . . . . . 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 2800	. . . . . 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 6886	. . . . . . . . 9 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (ℤ_≥‘(♯‘(𝐹 prefix 𝐿))) = (ℤ_≥‘𝐿))
77	75, 76	eleqtrrd 2865	. . . . . . . 8 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (♯‘𝐹) ∈ (ℤ_≥‘(♯‘(𝐹 prefix 𝐿))))
78		fzoss2 13747	. . . . . . . 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 3934	. . . . . 6 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) ∧ 𝑘 ∈ (0..^(♯‘(𝐹 prefix 𝐿)))) → 𝑘 ∈ (0..^(♯‘𝐹)))
81		wkslem1 30075	. . . . . . 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 3575	. . . . . 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 2779	. . . . . . . 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 4597	. . . . . . . . . 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 2778	. . . . . . 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 4699	. . . . . . . . 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 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 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 3156	. 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 30080	. . . . 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 30081	. . 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)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 if-wif 1078 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3078 Vcvv 3453 ⊆ wss 3902 {csn 4587 {cpr 4589 class class class wbr 5107 dom cdm 5659 ↾ cres 5661 ⟶wf 6533 ‘cfv 6537 (class class class)co 7417 0cc0 11128 1c1 11129 + caddc 11131 ℤcz 12619 ℤ_≥cuz 12891 ...cfz 13565 ..^cfzo 13713 ♯chash 14398 Word cword 14582 prefix cpfx 14744 Vtxcvtx 29461 iEdgciedg 29462 Walkscwlks 30064

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-pm 8833 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-nn 12262 df-n0 12533 df-z 12620 df-uz 12892 df-fz 13566 df-fzo 13714 df-hash 14399 df-word 14583 df-substr 14713 df-pfx 14745 df-wlks 30067

This theorem is used by: swrdwlk 30155

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