Theorem wlkres 27455

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 27997. (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 27399	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
4		pfxwrdsymb 14054	. . . 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 5777	. . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑆) = dom (𝐼 ↾ (𝐹 “ (0..^𝑁))))
10	1, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼)
11		wrdf 13869	. . . . . . 7 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
12		fimass 6558	. . . . . . 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 5879	. . . . . 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 2859	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑆) = (𝐹 “ (0..^𝑁)))
17		wrdeq 13889	. . . 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 2931	. 2 ⊢ (𝜑 → 𝐻 ∈ Word dom (iEdg‘𝑆))
20		wlkres.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
21	20	wlkp 27401	. . . . . . 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 6504	. . . . . 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 13059	. . . . . . 7 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
27		wlkres.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))
28	26, 27	sseldi 3968	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝐹)))
29		elfzuz3 12908	. . . . . 6 ⊢ (𝑁 ∈ (0...(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝑁))
30		fzss2 12950	. . . . . 6 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝑁) → (0...𝑁) ⊆ (0...(♯‘𝐹)))
31	28, 29, 30	3syl 18	. . . . 5 ⊢ (𝜑 → (0...𝑁) ⊆ (0...(♯‘𝐹)))
32	25, 31	fssresd 6548	. . . 4 ⊢ (𝜑 → (𝑃 ↾ (0...𝑁)):(0...𝑁)⟶(Vtx‘𝑆))
33	6	fveq2i 6676	. . . . . . 7 ⊢ (♯‘𝐻) = (♯‘(𝐹 prefix 𝑁))
34		pfxlen 14048	. . . . . . . 8 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 prefix 𝑁)) = 𝑁)
35	10, 28, 34	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → (♯‘(𝐹 prefix 𝑁)) = 𝑁)
36	33, 35	syl5eq 2871	. . . . . 6 ⊢ (𝜑 → (♯‘𝐻) = 𝑁)
37	36	oveq2d 7175	. . . . 5 ⊢ (𝜑 → (0...(♯‘𝐻)) = (0...𝑁))
38	37	feq2d 6503	. . . 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 6508	. . 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 27396	. . . . . 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 483	. . . 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 7175	. . . . . . . . . . 11 ⊢ (𝜑 → (0..^(♯‘𝐻)) = (0..^𝑁))
47	46	eleq2d 2901	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0..^(♯‘𝐻)) ↔ 𝑥 ∈ (0..^𝑁)))
48	40	fveq1i 6674	. . . . . . . . . . . . 13 ⊢ (𝑄‘𝑥) = ((𝑃 ↾ (0...𝑁))‘𝑥)
49		fzossfz 13059	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝑁) ⊆ (0...𝑁)
50	49	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0..^𝑁) ⊆ (0...𝑁))
51	50	sselda 3970	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0...𝑁))
52	51	fvresd 6693	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑃 ↾ (0...𝑁))‘𝑥) = (𝑃‘𝑥))
53	48, 52	syl5req 2872	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → (𝑃‘𝑥) = (𝑄‘𝑥))
54	40	fveq1i 6674	. . . . . . . . . . . . 13 ⊢ (𝑄‘(𝑥 + 1)) = ((𝑃 ↾ (0...𝑁))‘(𝑥 + 1))
55		fzofzp1 13137	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (0..^𝑁) → (𝑥 + 1) ∈ (0...𝑁))
56	55	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (0...𝑁))
57	56	fvresd 6693	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑃 ↾ (0...𝑁))‘(𝑥 + 1)) = (𝑃‘(𝑥 + 1)))
58	54, 57	syl5req 2872	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1)))
59	53, 58	jca 514	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))))
60	59	ex 415	. . . . . . . . . 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 409	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → ((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))))
63	10	ancli 551	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝜑 ∧ 𝐹 ∈ Word dom 𝐼))
64	11	ffund 6521	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ Word dom 𝐼 → Fun 𝐹)
65	64	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) → Fun 𝐹)
66	65	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → Fun 𝐹)
67		fdm 6525	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 → dom 𝐹 = (0..^(♯‘𝐹)))
68		elfzouz2 13055	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝑁))
69		fzoss2 13068	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
70	27, 68, 69	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
71		sseq2 3996	. . . . . . . . . . . . . . . . . . 19 ⊢ (dom 𝐹 = (0..^(♯‘𝐹)) → ((0..^𝑁) ⊆ dom 𝐹 ↔ (0..^𝑁) ⊆ (0..^(♯‘𝐹))))
72	70, 71	syl5ibr 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 410	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) → (0..^𝑁) ⊆ dom 𝐹)
75	74	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → (0..^𝑁) ⊆ dom 𝐹)
76		simpr 487	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
77	66, 75, 76	resfvresima 7000	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)) = (𝐼‘(𝐹‘𝑥)))
78	63, 77	sylan 582	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)) = (𝐼‘(𝐹‘𝑥)))
79	78	eqcomd 2830	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → (𝐼‘(𝐹‘𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)))
80	79	ex 415	. . . . . . . . . . 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 409	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (𝐼‘(𝐹‘𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)))
83	8	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
84	6	fveq1i 6674	. . . . . . . . . . 11 ⊢ (𝐻‘𝑥) = ((𝐹 prefix 𝑁)‘𝑥)
85	10	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → 𝐹 ∈ Word dom 𝐼)
86	28	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → 𝑁 ∈ (0...(♯‘𝐹)))
87		pfxres 14044	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝑁) = (𝐹 ↾ (0..^𝑁)))
88	85, 86, 87	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (𝐹 prefix 𝑁) = (𝐹 ↾ (0..^𝑁)))
89	88	fveq1d 6675	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → ((𝐹 prefix 𝑁)‘𝑥) = ((𝐹 ↾ (0..^𝑁))‘𝑥))
90	84, 89	syl5eq 2871	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (𝐻‘𝑥) = ((𝐹 ↾ (0..^𝑁))‘𝑥))
91	83, 90	fveq12d 6680	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → ((iEdg‘𝑆)‘(𝐻‘𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)))
92	82, 91	eqtr4d 2862	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (𝐼‘(𝐹‘𝑥)) = ((iEdg‘𝑆)‘(𝐻‘𝑥)))
93	62, 92	jca 514	. . . . . . 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 6677	. . . . . . . . . . 11 ⊢ (𝜑 → (ℤ≥‘(♯‘𝐻)) = (ℤ≥‘𝑁))
96	94, 95	eleqtrrd 2919	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐹) ∈ (ℤ≥‘(♯‘𝐻)))
97		fzoss2 13068	. . . . . . . . . 10 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘(♯‘𝐻)) → (0..^(♯‘𝐻)) ⊆ (0..^(♯‘𝐹)))
98	96, 97	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (0..^(♯‘𝐻)) ⊆ (0..^(♯‘𝐹)))
99	98	sselda 3970	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → 𝑥 ∈ (0..^(♯‘𝐹)))
100		wkslem1 27392	. . . . . . . . 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 3621	. . . . . . . 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 2838	. . . . . . . . . 10 ⊢ (((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) → ((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)) ↔ (𝑄‘𝑥) = (𝑄‘(𝑥 + 1))))
104	103	adantr 483	. . . . . . . . 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 487	. . . . . . . . . 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 4580	. . . . . . . . . . . 12 ⊢ ((𝑃‘𝑥) = (𝑄‘𝑥) → {(𝑃‘𝑥)} = {(𝑄‘𝑥)})
107	106	adantr 483	. . . . . . . . . . 11 ⊢ (((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) → {(𝑃‘𝑥)} = {(𝑄‘𝑥)})
108	107	adantr 483	. . . . . . . . . 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 2840	. . . . . . . . 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 4674	. . . . . . . . . . 11 ⊢ (((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) → {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = {(𝑄‘𝑥), (𝑄‘(𝑥 + 1))})
111	110	adantr 483	. . . . . . . . . 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 4003	. . . . . . . . 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 1072	. . . . . . . 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 1131	. . . 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 3185	. 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 27454	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
121		eqid 2824	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
122		eqid 2824	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
123	121, 122	iswlkg 27398	. . 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 1338	1 ⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  if-wif 1057   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  ∀wral 3141  Vcvv 3497   ⊆ wss 3939  {csn 4570  {cpr 4572   class class class wbr 5069  dom cdm 5558   ↾ cres 5560   “ cima 5561  Fun wfun 6352  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159  0cc0 10540  1c1 10541   + caddc 10543  ℤ_≥cuz 12246  ...cfz 12895  ..^cfzo 13036  ♯chash 13693  Word cword 13864   prefix cpfx 14035  Vtxcvtx 26784  iEdgciedg 26785  Walkscwlks 27381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-map 8411  df-pm 8412  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-card 9371  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-n0 11901  df-z 11985  df-uz 12247  df-fz 12896  df-fzo 13037  df-hash 13694  df-word 13865  df-substr 14006  df-pfx 14036  df-wlks 27384

This theorem is referenced by:  trlres  27485  eupthres  27997