Theorem wlkres 29703

Description: The restriction ⟨𝐻, 𝑄⟩ of a walk ⟨𝐹, 𝑃⟩ to an initial segment of the walk (of length 𝑁) forms a walk on the subgraph 𝑆 consisting of the edges in the initial segment. Formerly proven directly for Eulerian paths, see eupthres 30244. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 5-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.)

Hypotheses

Ref	Expression
wlkres.v	⊢ 𝑉 = (Vtx‘𝐺)
wlkres.i	⊢ 𝐼 = (iEdg‘𝐺)
wlkres.d	⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)
wlkres.n	⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))
wlkres.s	⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
wlkres.e	⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
wlkres.h	⊢ 𝐻 = (𝐹 prefix 𝑁)
wlkres.q	⊢ 𝑄 = (𝑃 ↾ (0...𝑁))

Assertion

Ref	Expression
wlkres	⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄)

Proof of Theorem wlkres

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

Step	Hyp	Ref	Expression
1		wlkres.d	. . . 4 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)
2		wlkres.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
3	2	wlkf 29647	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
4		pfxwrdsymb 14724	. . . 4 ⊢ (𝐹 ∈ Word dom 𝐼 → (𝐹 prefix 𝑁) ∈ Word (𝐹 “ (0..^𝑁)))
5	1, 3, 4	3syl 18	. . 3 ⊢ (𝜑 → (𝐹 prefix 𝑁) ∈ Word (𝐹 “ (0..^𝑁)))
6		wlkres.h	. . . 4 ⊢ 𝐻 = (𝐹 prefix 𝑁)
7	6	a1i 11	. . 3 ⊢ (𝜑 → 𝐻 = (𝐹 prefix 𝑁))
8		wlkres.e	. . . . . 6 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
9	8	dmeqd 5919	. . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑆) = dom (𝐼 ↾ (𝐹 “ (0..^𝑁))))
10	1, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼)
11		wrdf 14554	. . . . . . 7 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
12		fimass 6757	. . . . . . 7 ⊢ (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 → (𝐹 “ (0..^𝑁)) ⊆ dom 𝐼)
13	10, 11, 12	3syl 18	. . . . . 6 ⊢ (𝜑 → (𝐹 “ (0..^𝑁)) ⊆ dom 𝐼)
14		ssdmres 6033	. . . . . 6 ⊢ ((𝐹 “ (0..^𝑁)) ⊆ dom 𝐼 ↔ dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) = (𝐹 “ (0..^𝑁)))
15	13, 14	sylib 218	. . . . 5 ⊢ (𝜑 → dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) = (𝐹 “ (0..^𝑁)))
16	9, 15	eqtrd 2775	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑆) = (𝐹 “ (0..^𝑁)))
17		wrdeq 14571	. . . 4 ⊢ (dom (iEdg‘𝑆) = (𝐹 “ (0..^𝑁)) → Word dom (iEdg‘𝑆) = Word (𝐹 “ (0..^𝑁)))
18	16, 17	syl 17	. . 3 ⊢ (𝜑 → Word dom (iEdg‘𝑆) = Word (𝐹 “ (0..^𝑁)))
19	5, 7, 18	3eltr4d 2854	. 2 ⊢ (𝜑 → 𝐻 ∈ Word dom (iEdg‘𝑆))
20		wlkres.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
21	20	wlkp 29649	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
22	1, 21	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
23		wlkres.s	. . . . . . 7 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
24	23	feq3d 6724	. . . . . 6 ⊢ (𝜑 → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆) ↔ 𝑃:(0...(♯‘𝐹))⟶𝑉))
25	22, 24	mpbird 257	. . . . 5 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆))
26		fzossfz 13715	. . . . . . 7 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
27		wlkres.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))
28	26, 27	sselid 3993	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝐹)))
29		elfzuz3 13558	. . . . . 6 ⊢ (𝑁 ∈ (0...(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝑁))
30		fzss2 13601	. . . . . 6 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝑁) → (0...𝑁) ⊆ (0...(♯‘𝐹)))
31	28, 29, 30	3syl 18	. . . . 5 ⊢ (𝜑 → (0...𝑁) ⊆ (0...(♯‘𝐹)))
32	25, 31	fssresd 6776	. . . 4 ⊢ (𝜑 → (𝑃 ↾ (0...𝑁)):(0...𝑁)⟶(Vtx‘𝑆))
33	6	fveq2i 6910	. . . . . . 7 ⊢ (♯‘𝐻) = (♯‘(𝐹 prefix 𝑁))
34		pfxlen 14718	. . . . . . . 8 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 prefix 𝑁)) = 𝑁)
35	10, 28, 34	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (♯‘(𝐹 prefix 𝑁)) = 𝑁)
36	33, 35	eqtrid 2787	. . . . . 6 ⊢ (𝜑 → (♯‘𝐻) = 𝑁)
37	36	oveq2d 7447	. . . . 5 ⊢ (𝜑 → (0...(♯‘𝐻)) = (0...𝑁))
38	37	feq2d 6723	. . . 4 ⊢ (𝜑 → ((𝑃 ↾ (0...𝑁)):(0...(♯‘𝐻))⟶(Vtx‘𝑆) ↔ (𝑃 ↾ (0...𝑁)):(0...𝑁)⟶(Vtx‘𝑆)))
39	32, 38	mpbird 257	. . 3 ⊢ (𝜑 → (𝑃 ↾ (0...𝑁)):(0...(♯‘𝐻))⟶(Vtx‘𝑆))
40		wlkres.q	. . . 4 ⊢ 𝑄 = (𝑃 ↾ (0...𝑁))
41	40	feq1i 6728	. . 3 ⊢ (𝑄:(0...(♯‘𝐻))⟶(Vtx‘𝑆) ↔ (𝑃 ↾ (0...𝑁)):(0...(♯‘𝐻))⟶(Vtx‘𝑆))
42	39, 41	sylibr 234	. 2 ⊢ (𝜑 → 𝑄:(0...(♯‘𝐻))⟶(Vtx‘𝑆))
43	20, 2	wlkprop 29644	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))
44	1, 43	syl 17	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))
45	44	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))
46	36	oveq2d 7447	. . . . . . . . . . 11 ⊢ (𝜑 → (0..^(♯‘𝐻)) = (0..^𝑁))
47	46	eleq2d 2825	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0..^(♯‘𝐻)) ↔ 𝑥 ∈ (0..^𝑁)))
48	40	fveq1i 6908	. . . . . . . . . . . . 13 ⊢ (𝑄‘𝑥) = ((𝑃 ↾ (0...𝑁))‘𝑥)
49		fzossfz 13715	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝑁) ⊆ (0...𝑁)
50	49	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0..^𝑁) ⊆ (0...𝑁))
51	50	sselda 3995	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0...𝑁))
52	51	fvresd 6927	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑃 ↾ (0...𝑁))‘𝑥) = (𝑃‘𝑥))
53	48, 52	eqtr2id 2788	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → (𝑃‘𝑥) = (𝑄‘𝑥))
54	40	fveq1i 6908	. . . . . . . . . . . . 13 ⊢ (𝑄‘(𝑥 + 1)) = ((𝑃 ↾ (0...𝑁))‘(𝑥 + 1))
55		fzofzp1 13800	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (0..^𝑁) → (𝑥 + 1) ∈ (0...𝑁))
56	55	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (0...𝑁))
57	56	fvresd 6927	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑃 ↾ (0...𝑁))‘(𝑥 + 1)) = (𝑃‘(𝑥 + 1)))
58	54, 57	eqtr2id 2788	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1)))
59	53, 58	jca 511	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))))
60	59	ex 412	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0..^𝑁) → ((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1)))))
61	47, 60	sylbid 240	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0..^(♯‘𝐻)) → ((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1)))))
62	61	imp 406	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → ((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))))
63	10	ancli 548	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝜑 ∧ 𝐹 ∈ Word dom 𝐼))
64	11	ffund 6741	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ Word dom 𝐼 → Fun 𝐹)
65	64	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) → Fun 𝐹)
66	65	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → Fun 𝐹)
67		fdm 6746	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 → dom 𝐹 = (0..^(♯‘𝐹)))
68		elfzouz2 13711	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝑁))
69		fzoss2 13724	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
70	27, 68, 69	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
71		sseq2 4022	. . . . . . . . . . . . . . . . . . 19 ⊢ (dom 𝐹 = (0..^(♯‘𝐹)) → ((0..^𝑁) ⊆ dom 𝐹 ↔ (0..^𝑁) ⊆ (0..^(♯‘𝐹))))
72	70, 71	imbitrrid 246	. . . . . . . . . . . . . . . . . 18 ⊢ (dom 𝐹 = (0..^(♯‘𝐹)) → (𝜑 → (0..^𝑁) ⊆ dom 𝐹))
73	11, 67, 72	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ Word dom 𝐼 → (𝜑 → (0..^𝑁) ⊆ dom 𝐹))
74	73	impcom 407	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) → (0..^𝑁) ⊆ dom 𝐹)
75	74	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → (0..^𝑁) ⊆ dom 𝐹)
76		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
77	66, 75, 76	resfvresima 7255	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)) = (𝐼‘(𝐹‘𝑥)))
78	63, 77	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)) = (𝐼‘(𝐹‘𝑥)))
79	78	eqcomd 2741	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → (𝐼‘(𝐹‘𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)))
80	79	ex 412	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (0..^𝑁) → (𝐼‘(𝐹‘𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥))))
81	47, 80	sylbid 240	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0..^(♯‘𝐻)) → (𝐼‘(𝐹‘𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥))))
82	81	imp 406	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (𝐼‘(𝐹‘𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)))
83	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
84	6	fveq1i 6908	. . . . . . . . . . 11 ⊢ (𝐻‘𝑥) = ((𝐹 prefix 𝑁)‘𝑥)
85	10	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → 𝐹 ∈ Word dom 𝐼)
86	28	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → 𝑁 ∈ (0...(♯‘𝐹)))
87		pfxres 14714	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝑁) = (𝐹 ↾ (0..^𝑁)))
88	85, 86, 87	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (𝐹 prefix 𝑁) = (𝐹 ↾ (0..^𝑁)))
89	88	fveq1d 6909	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → ((𝐹 prefix 𝑁)‘𝑥) = ((𝐹 ↾ (0..^𝑁))‘𝑥))
90	84, 89	eqtrid 2787	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (𝐻‘𝑥) = ((𝐹 ↾ (0..^𝑁))‘𝑥))
91	83, 90	fveq12d 6914	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → ((iEdg‘𝑆)‘(𝐻‘𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)))
92	82, 91	eqtr4d 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (𝐼‘(𝐹‘𝑥)) = ((iEdg‘𝑆)‘(𝐻‘𝑥)))
93	62, 92	jca 511	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) ∧ (𝐼‘(𝐹‘𝑥)) = ((iEdg‘𝑆)‘(𝐻‘𝑥))))
94	27, 68	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐹) ∈ (ℤ≥‘𝑁))
95	36	fveq2d 6911	. . . . . . . . . . 11 ⊢ (𝜑 → (ℤ≥‘(♯‘𝐻)) = (ℤ≥‘𝑁))
96	94, 95	eleqtrrd 2842	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐹) ∈ (ℤ≥‘(♯‘𝐻)))
97		fzoss2 13724	. . . . . . . . . 10 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘(♯‘𝐻)) → (0..^(♯‘𝐻)) ⊆ (0..^(♯‘𝐹)))
98	96, 97	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (0..^(♯‘𝐻)) ⊆ (0..^(♯‘𝐹)))
99	98	sselda 3995	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → 𝑥 ∈ (0..^(♯‘𝐹)))
100		wkslem1 29640	. . . . . . . . 9 ⊢ (𝑘 = 𝑥 → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) ↔ if-((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)), (𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥)}, {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ⊆ (𝐼‘(𝐹‘𝑥)))))
101	100	rspcv 3618	. . . . . . . 8 ⊢ (𝑥 ∈ (0..^(♯‘𝐹)) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → if-((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)), (𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥)}, {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ⊆ (𝐼‘(𝐹‘𝑥)))))
102	99, 101	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → if-((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)), (𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥)}, {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ⊆ (𝐼‘(𝐹‘𝑥)))))
103		eqeq12 2752	. . . . . . . . . 10 ⊢ (((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) → ((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)) ↔ (𝑄‘𝑥) = (𝑄‘(𝑥 + 1))))
104	103	adantr 480	. . . . . . . . 9 ⊢ ((((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) ∧ (𝐼‘(𝐹‘𝑥)) = ((iEdg‘𝑆)‘(𝐻‘𝑥))) → ((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)) ↔ (𝑄‘𝑥) = (𝑄‘(𝑥 + 1))))
105		simpr 484	. . . . . . . . . 10 ⊢ ((((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) ∧ (𝐼‘(𝐹‘𝑥)) = ((iEdg‘𝑆)‘(𝐻‘𝑥))) → (𝐼‘(𝐹‘𝑥)) = ((iEdg‘𝑆)‘(𝐻‘𝑥)))
106		sneq 4641	. . . . . . . . . . . 12 ⊢ ((𝑃‘𝑥) = (𝑄‘𝑥) → {(𝑃‘𝑥)} = {(𝑄‘𝑥)})
107	106	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) → {(𝑃‘𝑥)} = {(𝑄‘𝑥)})
108	107	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) ∧ (𝐼‘(𝐹‘𝑥)) = ((iEdg‘𝑆)‘(𝐻‘𝑥))) → {(𝑃‘𝑥)} = {(𝑄‘𝑥)})
109	105, 108	eqeq12d 2751	. . . . . . . . 9 ⊢ ((((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) ∧ (𝐼‘(𝐹‘𝑥)) = ((iEdg‘𝑆)‘(𝐻‘𝑥))) → ((𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥)} ↔ ((iEdg‘𝑆)‘(𝐻‘𝑥)) = {(𝑄‘𝑥)}))
110		preq12 4740	. . . . . . . . . . 11 ⊢ (((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) → {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = {(𝑄‘𝑥), (𝑄‘(𝑥 + 1))})
111	110	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) ∧ (𝐼‘(𝐹‘𝑥)) = ((iEdg‘𝑆)‘(𝐻‘𝑥))) → {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = {(𝑄‘𝑥), (𝑄‘(𝑥 + 1))})
112	111, 105	sseq12d 4029	. . . . . . . . 9 ⊢ ((((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) ∧ (𝐼‘(𝐹‘𝑥)) = ((iEdg‘𝑆)‘(𝐻‘𝑥))) → ({(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ⊆ (𝐼‘(𝐹‘𝑥)) ↔ {(𝑄‘𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑥))))
113	104, 109, 112	ifpbi123d 1078	. . . . . . . 8 ⊢ ((((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) ∧ (𝐼‘(𝐹‘𝑥)) = ((iEdg‘𝑆)‘(𝐻‘𝑥))) → (if-((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)), (𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥)}, {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ⊆ (𝐼‘(𝐹‘𝑥))) ↔ if-((𝑄‘𝑥) = (𝑄‘(𝑥 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑥)) = {(𝑄‘𝑥)}, {(𝑄‘𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑥)))))
114	113	biimpd 229	. . . . . . 7 ⊢ ((((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) ∧ (𝐼‘(𝐹‘𝑥)) = ((iEdg‘𝑆)‘(𝐻‘𝑥))) → (if-((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)), (𝐼‘(𝐹‘𝑥)) = {(𝑃‘𝑥)}, {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} ⊆ (𝐼‘(𝐹‘𝑥))) → if-((𝑄‘𝑥) = (𝑄‘(𝑥 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑥)) = {(𝑄‘𝑥)}, {(𝑄‘𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑥)))))
115	93, 102, 114	sylsyld 61	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → if-((𝑄‘𝑥) = (𝑄‘(𝑥 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑥)) = {(𝑄‘𝑥)}, {(𝑄‘𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑥)))))
116	115	com12 32	. . . . 5 ⊢ (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → if-((𝑄‘𝑥) = (𝑄‘(𝑥 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑥)) = {(𝑄‘𝑥)}, {(𝑄‘𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑥)))))
117	116	3ad2ant3 1134	. . . 4 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) → ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → if-((𝑄‘𝑥) = (𝑄‘(𝑥 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑥)) = {(𝑄‘𝑥)}, {(𝑄‘𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑥)))))
118	45, 117	mpcom 38	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → if-((𝑄‘𝑥) = (𝑄‘(𝑥 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑥)) = {(𝑄‘𝑥)}, {(𝑄‘𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑥))))
119	118	ralrimiva 3144	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (0..^(♯‘𝐻))if-((𝑄‘𝑥) = (𝑄‘(𝑥 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑥)) = {(𝑄‘𝑥)}, {(𝑄‘𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑥))))
120	20, 2, 1, 27, 23	wlkreslem 29702	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
121		eqid 2735	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
122		eqid 2735	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
123	121, 122	iswlkg 29646	. . 3 ⊢ (𝑆 ∈ V → (𝐻(Walks‘𝑆)𝑄 ↔ (𝐻 ∈ Word dom (iEdg‘𝑆) ∧ 𝑄:(0...(♯‘𝐻))⟶(Vtx‘𝑆) ∧ ∀𝑥 ∈ (0..^(♯‘𝐻))if-((𝑄‘𝑥) = (𝑄‘(𝑥 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑥)) = {(𝑄‘𝑥)}, {(𝑄‘𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑥))))))
124	120, 123	syl 17	. 2 ⊢ (𝜑 → (𝐻(Walks‘𝑆)𝑄 ↔ (𝐻 ∈ Word dom (iEdg‘𝑆) ∧ 𝑄:(0...(♯‘𝐻))⟶(Vtx‘𝑆) ∧ ∀𝑥 ∈ (0..^(♯‘𝐻))if-((𝑄‘𝑥) = (𝑄‘(𝑥 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑥)) = {(𝑄‘𝑥)}, {(𝑄‘𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑥))))))
125	19, 42, 119, 124	mpbir3and 1341	1 ⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  if-wif 1062   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478   ⊆ wss 3963  {csn 4631  {cpr 4633   class class class wbr 5148  dom cdm 5689   ↾ cres 5691   “ cima 5692  Fun wfun 6557  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154   + caddc 11156  ℤ_≥cuz 12876  ...cfz 13544  ..^cfzo 13691  ♯chash 14366  Word cword 14549   prefix cpfx 14705  Vtxcvtx 29028  iEdgciedg 29029  Walkscwlks 29629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

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 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-substr 14676  df-pfx 14706  df-wlks 29632

This theorem is referenced by:  trlres  29733  eupthres  30244