Theorem wlkres 29815

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 30363. (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 29761	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
4		pfxwrdsymb 14700	. . . 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 5879	. . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑆) = dom (𝐼 ↾ (𝐹 “ (0..^𝑁))))
10	1, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼)
11		wrdf 14528	. . . . . . 7 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
12		fimass 6708	. . . . . . 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 5997	. . . . . 6 ⊢ ((𝐹 “ (0..^𝑁)) ⊆ dom 𝐼 ↔ dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) = (𝐹 “ (0..^𝑁)))
15	13, 14	sylib 220	. . . . 5 ⊢ (𝜑 → dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) = (𝐹 “ (0..^𝑁)))
16	9, 15	eqtrd 2796	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑆) = (𝐹 “ (0..^𝑁)))
17		wrdeq 14546	. . . 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 2876	. 2 ⊢ (𝜑 → 𝐻 ∈ Word dom (iEdg‘𝑆))
20		wlkres.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
21	20	wlkp 29763	. . . . . . 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 6672	. . . . . 6 ⊢ (𝜑 → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆) ↔ 𝑃:(0...(♯‘𝐹))⟶𝑉))
25	22, 24	mpbird 259	. . . . 5 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆))
26		fzossfz 13681	. . . . . . 7 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
27		wlkres.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))
28	26, 27	sselid 3934	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝐹)))
29		elfzuz3 13523	. . . . . 6 ⊢ (𝑁 ∈ (0...(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝑁))
30		fzss2 13566	. . . . . 6 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝑁) → (0...𝑁) ⊆ (0...(♯‘𝐹)))
31	28, 29, 30	3syl 18	. . . . 5 ⊢ (𝜑 → (0...𝑁) ⊆ (0...(♯‘𝐹)))
32	25, 31	fssresd 6727	. . . 4 ⊢ (𝜑 → (𝑃 ↾ (0...𝑁)):(0...𝑁)⟶(Vtx‘𝑆))
33	6	fveq2i 6866	. . . . . . 7 ⊢ (♯‘𝐻) = (♯‘(𝐹 prefix 𝑁))
34		pfxlen 14694	. . . . . . . 8 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 prefix 𝑁)) = 𝑁)
35	10, 28, 34	syl2anc 593	. . . . . . 7 ⊢ (𝜑 → (♯‘(𝐹 prefix 𝑁)) = 𝑁)
36	33, 35	eqtrid 2808	. . . . . 6 ⊢ (𝜑 → (♯‘𝐻) = 𝑁)
37	36	oveq2d 7408	. . . . 5 ⊢ (𝜑 → (0...(♯‘𝐻)) = (0...𝑁))
38	37	feq2d 6671	. . . 4 ⊢ (𝜑 → ((𝑃 ↾ (0...𝑁)):(0...(♯‘𝐻))⟶(Vtx‘𝑆) ↔ (𝑃 ↾ (0...𝑁)):(0...𝑁)⟶(Vtx‘𝑆)))
39	32, 38	mpbird 259	. . 3 ⊢ (𝜑 → (𝑃 ↾ (0...𝑁)):(0...(♯‘𝐻))⟶(Vtx‘𝑆))
40		wlkres.q	. . . 4 ⊢ 𝑄 = (𝑃 ↾ (0...𝑁))
41	40	feq1i 6678	. . 3 ⊢ (𝑄:(0...(♯‘𝐻))⟶(Vtx‘𝑆) ↔ (𝑃 ↾ (0...𝑁)):(0...(♯‘𝐻))⟶(Vtx‘𝑆))
42	39, 41	sylibr 236	. 2 ⊢ (𝜑 → 𝑄:(0...(♯‘𝐻))⟶(Vtx‘𝑆))
43	20, 2	wlkprop 29758	. . . . . 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 484	. . . 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 7408	. . . . . . . . . . 11 ⊢ (𝜑 → (0..^(♯‘𝐻)) = (0..^𝑁))
47	46	eleq2d 2847	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0..^(♯‘𝐻)) ↔ 𝑥 ∈ (0..^𝑁)))
48	40	fveq1i 6864	. . . . . . . . . . . . 13 ⊢ (𝑄‘𝑥) = ((𝑃 ↾ (0...𝑁))‘𝑥)
49		fzossfz 13681	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝑁) ⊆ (0...𝑁)
50	49	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0..^𝑁) ⊆ (0...𝑁))
51	50	sselda 3936	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0...𝑁))
52	51	fvresd 6883	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑃 ↾ (0...𝑁))‘𝑥) = (𝑃‘𝑥))
53	48, 52	eqtr2id 2809	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → (𝑃‘𝑥) = (𝑄‘𝑥))
54	40	fveq1i 6864	. . . . . . . . . . . . 13 ⊢ (𝑄‘(𝑥 + 1)) = ((𝑃 ↾ (0...𝑁))‘(𝑥 + 1))
55		fzofzp1 13767	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (0..^𝑁) → (𝑥 + 1) ∈ (0...𝑁))
56	55	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (0...𝑁))
57	56	fvresd 6883	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑃 ↾ (0...𝑁))‘(𝑥 + 1)) = (𝑃‘(𝑥 + 1)))
58	54, 57	eqtr2id 2809	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1)))
59	53, 58	jca 519	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))))
60	59	ex 416	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0..^𝑁) → ((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1)))))
61	47, 60	sylbid 242	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0..^(♯‘𝐻)) → ((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1)))))
62	61	imp 410	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → ((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))))
63	10	ancli 556	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝜑 ∧ 𝐹 ∈ Word dom 𝐼))
64	11	ffund 6692	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ Word dom 𝐼 → Fun 𝐹)
65	64	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) → Fun 𝐹)
66	65	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → Fun 𝐹)
67		fdm 6697	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 → dom 𝐹 = (0..^(♯‘𝐹)))
68		elfzouz2 13677	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝑁))
69		fzoss2 13690	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
70	27, 68, 69	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
71		sseq2 3962	. . . . . . . . . . . . . . . . . . 19 ⊢ (dom 𝐹 = (0..^(♯‘𝐹)) → ((0..^𝑁) ⊆ dom 𝐹 ↔ (0..^𝑁) ⊆ (0..^(♯‘𝐹))))
72	70, 71	imbitrrid 248	. . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) → (0..^𝑁) ⊆ dom 𝐹)
75	74	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → (0..^𝑁) ⊆ dom 𝐹)
76		simpr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
77	66, 75, 76	resfvresima 7215	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)) = (𝐼‘(𝐹‘𝑥)))
78	63, 77	sylan 589	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)) = (𝐼‘(𝐹‘𝑥)))
79	78	eqcomd 2767	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → (𝐼‘(𝐹‘𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)))
80	79	ex 416	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (0..^𝑁) → (𝐼‘(𝐹‘𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥))))
81	47, 80	sylbid 242	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0..^(♯‘𝐻)) → (𝐼‘(𝐹‘𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥))))
82	81	imp 410	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (𝐼‘(𝐹‘𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)))
83	8	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
84	6	fveq1i 6864	. . . . . . . . . . 11 ⊢ (𝐻‘𝑥) = ((𝐹 prefix 𝑁)‘𝑥)
85	10	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → 𝐹 ∈ Word dom 𝐼)
86	28	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → 𝑁 ∈ (0...(♯‘𝐹)))
87		pfxres 14690	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝑁) = (𝐹 ↾ (0..^𝑁)))
88	85, 86, 87	syl2anc 593	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (𝐹 prefix 𝑁) = (𝐹 ↾ (0..^𝑁)))
89	88	fveq1d 6865	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → ((𝐹 prefix 𝑁)‘𝑥) = ((𝐹 ↾ (0..^𝑁))‘𝑥))
90	84, 89	eqtrid 2808	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (𝐻‘𝑥) = ((𝐹 ↾ (0..^𝑁))‘𝑥))
91	83, 90	fveq12d 6870	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → ((iEdg‘𝑆)‘(𝐻‘𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)))
92	82, 91	eqtr4d 2799	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (𝐼‘(𝐹‘𝑥)) = ((iEdg‘𝑆)‘(𝐻‘𝑥)))
93	62, 92	jca 519	. . . . . . 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 6867	. . . . . . . . . . 11 ⊢ (𝜑 → (ℤ≥‘(♯‘𝐻)) = (ℤ≥‘𝑁))
96	94, 95	eleqtrrd 2864	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐹) ∈ (ℤ≥‘(♯‘𝐻)))
97		fzoss2 13690	. . . . . . . . . 10 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘(♯‘𝐻)) → (0..^(♯‘𝐻)) ⊆ (0..^(♯‘𝐹)))
98	96, 97	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (0..^(♯‘𝐻)) ⊆ (0..^(♯‘𝐹)))
99	98	sselda 3936	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → 𝑥 ∈ (0..^(♯‘𝐹)))
100		wkslem1 29754	. . . . . . . . 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 3577	. . . . . . . 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 2778	. . . . . . . . . 10 ⊢ (((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) → ((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)) ↔ (𝑄‘𝑥) = (𝑄‘(𝑥 + 1))))
104	103	adantr 484	. . . . . . . . 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 488	. . . . . . . . . 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 4591	. . . . . . . . . . . 12 ⊢ ((𝑃‘𝑥) = (𝑄‘𝑥) → {(𝑃‘𝑥)} = {(𝑄‘𝑥)})
107	106	adantr 484	. . . . . . . . . . 11 ⊢ (((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) → {(𝑃‘𝑥)} = {(𝑄‘𝑥)})
108	107	adantr 484	. . . . . . . . . 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 2777	. . . . . . . . 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 4693	. . . . . . . . . . 11 ⊢ (((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) → {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = {(𝑄‘𝑥), (𝑄‘(𝑥 + 1))})
111	110	adantr 484	. . . . . . . . . 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 3969	. . . . . . . . 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 1089	. . . . . . . 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 231	. . . . . . 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 1147	. . . 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 3153	. 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 29814	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
121		eqid 2761	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
122		eqid 2761	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
123	121, 122	iswlkg 29760	. . 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 1355	1 ⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  if-wif 1073   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   ⊆ wss 3904  {csn 4581  {cpr 4583   class class class wbr 5099  dom cdm 5645   ↾ cres 5647   “ cima 5648  Fun wfun 6511  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392  0cc0 11070  1c1 11071   + caddc 11073  ℤ_≥cuz 12836  ...cfz 13509  ..^cfzo 13656  ♯chash 14340  Word cword 14523   prefix cpfx 14681  Vtxcvtx 29143  iEdgciedg 29144  Walkscwlks 29743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ifp 1074  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8673  df-map 8805  df-pm 8806  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-n0 12479  df-z 12566  df-uz 12837  df-fz 13510  df-fzo 13657  df-hash 14341  df-word 14524  df-substr 14652  df-pfx 14682  df-wlks 29746

This theorem is referenced by:  trlres  29845  eupthres  30363