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Theorem wlkres 29194

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 29735. (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 29138	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
4		pfxwrdsymb 14643	. . . 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 5904	. . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑆) = dom (𝐼 ↾ (𝐹 “ (0..^𝑁))))
10	1, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼)
11		wrdf 14473	. . . . . . 7 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
12		fimass 6737	. . . . . . 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 6003	. . . . . 6 ⊢ ((𝐹 “ (0..^𝑁)) ⊆ dom 𝐼 ↔ dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) = (𝐹 “ (0..^𝑁)))
15	13, 14	sylib 217	. . . . 5 ⊢ (𝜑 → dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) = (𝐹 “ (0..^𝑁)))
16	9, 15	eqtrd 2770	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑆) = (𝐹 “ (0..^𝑁)))
17		wrdeq 14490	. . . 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 2846	. 2 ⊢ (𝜑 → 𝐻 ∈ Word dom (iEdg‘𝑆))
20		wlkres.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
21	20	wlkp 29140	. . . . . . 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 6703	. . . . . 6 ⊢ (𝜑 → (𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆) ↔ 𝑃:(0...(♯‘𝐹))⟶𝑉))
25	22, 24	mpbird 256	. . . . 5 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝑆))
26		fzossfz 13655	. . . . . . 7 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
27		wlkres.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))
28	26, 27	sselid 3979	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝐹)))
29		elfzuz3 13502	. . . . . 6 ⊢ (𝑁 ∈ (0...(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ_≥‘𝑁))
30		fzss2 13545	. . . . . 6 ⊢ ((♯‘𝐹) ∈ (ℤ_≥‘𝑁) → (0...𝑁) ⊆ (0...(♯‘𝐹)))
31	28, 29, 30	3syl 18	. . . . 5 ⊢ (𝜑 → (0...𝑁) ⊆ (0...(♯‘𝐹)))
32	25, 31	fssresd 6757	. . . 4 ⊢ (𝜑 → (𝑃 ↾ (0...𝑁)):(0...𝑁)⟶(Vtx‘𝑆))
33	6	fveq2i 6893	. . . . . . 7 ⊢ (♯‘𝐻) = (♯‘(𝐹 prefix 𝑁))
34		pfxlen 14637	. . . . . . . 8 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 prefix 𝑁)) = 𝑁)
35	10, 28, 34	syl2anc 582	. . . . . . 7 ⊢ (𝜑 → (♯‘(𝐹 prefix 𝑁)) = 𝑁)
36	33, 35	eqtrid 2782	. . . . . 6 ⊢ (𝜑 → (♯‘𝐻) = 𝑁)
37	36	oveq2d 7427	. . . . 5 ⊢ (𝜑 → (0...(♯‘𝐻)) = (0...𝑁))
38	37	feq2d 6702	. . . 4 ⊢ (𝜑 → ((𝑃 ↾ (0...𝑁)):(0...(♯‘𝐻))⟶(Vtx‘𝑆) ↔ (𝑃 ↾ (0...𝑁)):(0...𝑁)⟶(Vtx‘𝑆)))
39	32, 38	mpbird 256	. . 3 ⊢ (𝜑 → (𝑃 ↾ (0...𝑁)):(0...(♯‘𝐻))⟶(Vtx‘𝑆))
40		wlkres.q	. . . 4 ⊢ 𝑄 = (𝑃 ↾ (0...𝑁))
41	40	feq1i 6707	. . 3 ⊢ (𝑄:(0...(♯‘𝐻))⟶(Vtx‘𝑆) ↔ (𝑃 ↾ (0...𝑁)):(0...(♯‘𝐻))⟶(Vtx‘𝑆))
42	39, 41	sylibr 233	. 2 ⊢ (𝜑 → 𝑄:(0...(♯‘𝐻))⟶(Vtx‘𝑆))
43	20, 2	wlkprop 29135	. . . . . 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 479	. . . 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 7427	. . . . . . . . . . 11 ⊢ (𝜑 → (0..^(♯‘𝐻)) = (0..^𝑁))
47	46	eleq2d 2817	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0..^(♯‘𝐻)) ↔ 𝑥 ∈ (0..^𝑁)))
48	40	fveq1i 6891	. . . . . . . . . . . . 13 ⊢ (𝑄‘𝑥) = ((𝑃 ↾ (0...𝑁))‘𝑥)
49		fzossfz 13655	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝑁) ⊆ (0...𝑁)
50	49	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0..^𝑁) ⊆ (0...𝑁))
51	50	sselda 3981	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0...𝑁))
52	51	fvresd 6910	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑃 ↾ (0...𝑁))‘𝑥) = (𝑃‘𝑥))
53	48, 52	eqtr2id 2783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → (𝑃‘𝑥) = (𝑄‘𝑥))
54	40	fveq1i 6891	. . . . . . . . . . . . 13 ⊢ (𝑄‘(𝑥 + 1)) = ((𝑃 ↾ (0...𝑁))‘(𝑥 + 1))
55		fzofzp1 13733	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (0..^𝑁) → (𝑥 + 1) ∈ (0...𝑁))
56	55	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (0...𝑁))
57	56	fvresd 6910	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑃 ↾ (0...𝑁))‘(𝑥 + 1)) = (𝑃‘(𝑥 + 1)))
58	54, 57	eqtr2id 2783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1)))
59	53, 58	jca 510	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))))
60	59	ex 411	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0..^𝑁) → ((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1)))))
61	47, 60	sylbid 239	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0..^(♯‘𝐻)) → ((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1)))))
62	61	imp 405	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → ((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))))
63	10	ancli 547	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝜑 ∧ 𝐹 ∈ Word dom 𝐼))
64	11	ffund 6720	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ Word dom 𝐼 → Fun 𝐹)
65	64	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) → Fun 𝐹)
66	65	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → Fun 𝐹)
67		fdm 6725	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 → dom 𝐹 = (0..^(♯‘𝐹)))
68		elfzouz2 13651	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ_≥‘𝑁))
69		fzoss2 13664	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝐹) ∈ (ℤ_≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
70	27, 68, 69	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
71		sseq2 4007	. . . . . . . . . . . . . . . . . . 19 ⊢ (dom 𝐹 = (0..^(♯‘𝐹)) → ((0..^𝑁) ⊆ dom 𝐹 ↔ (0..^𝑁) ⊆ (0..^(♯‘𝐹))))
72	70, 71	imbitrrid 245	. . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) → (0..^𝑁) ⊆ dom 𝐹)
75	74	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → (0..^𝑁) ⊆ dom 𝐹)
76		simpr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
77	66, 75, 76	resfvresima 7238	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)) = (𝐼‘(𝐹‘𝑥)))
78	63, 77	sylan 578	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)) = (𝐼‘(𝐹‘𝑥)))
79	78	eqcomd 2736	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^𝑁)) → (𝐼‘(𝐹‘𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)))
80	79	ex 411	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (0..^𝑁) → (𝐼‘(𝐹‘𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥))))
81	47, 80	sylbid 239	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0..^(♯‘𝐻)) → (𝐼‘(𝐹‘𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥))))
82	81	imp 405	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (𝐼‘(𝐹‘𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)))
83	8	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
84	6	fveq1i 6891	. . . . . . . . . . 11 ⊢ (𝐻‘𝑥) = ((𝐹 prefix 𝑁)‘𝑥)
85	10	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → 𝐹 ∈ Word dom 𝐼)
86	28	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → 𝑁 ∈ (0...(♯‘𝐹)))
87		pfxres 14633	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝑁) = (𝐹 ↾ (0..^𝑁)))
88	85, 86, 87	syl2anc 582	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (𝐹 prefix 𝑁) = (𝐹 ↾ (0..^𝑁)))
89	88	fveq1d 6892	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → ((𝐹 prefix 𝑁)‘𝑥) = ((𝐹 ↾ (0..^𝑁))‘𝑥))
90	84, 89	eqtrid 2782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (𝐻‘𝑥) = ((𝐹 ↾ (0..^𝑁))‘𝑥))
91	83, 90	fveq12d 6897	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → ((iEdg‘𝑆)‘(𝐻‘𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)))
92	82, 91	eqtr4d 2773	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → (𝐼‘(𝐹‘𝑥)) = ((iEdg‘𝑆)‘(𝐻‘𝑥)))
93	62, 92	jca 510	. . . . . . 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 6894	. . . . . . . . . . 11 ⊢ (𝜑 → (ℤ_≥‘(♯‘𝐻)) = (ℤ_≥‘𝑁))
96	94, 95	eleqtrrd 2834	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐹) ∈ (ℤ_≥‘(♯‘𝐻)))
97		fzoss2 13664	. . . . . . . . . 10 ⊢ ((♯‘𝐹) ∈ (ℤ_≥‘(♯‘𝐻)) → (0..^(♯‘𝐻)) ⊆ (0..^(♯‘𝐹)))
98	96, 97	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (0..^(♯‘𝐻)) ⊆ (0..^(♯‘𝐹)))
99	98	sselda 3981	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(♯‘𝐻))) → 𝑥 ∈ (0..^(♯‘𝐹)))
100		wkslem1 29131	. . . . . . . . 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 3607	. . . . . . . 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 2747	. . . . . . . . . 10 ⊢ (((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) → ((𝑃‘𝑥) = (𝑃‘(𝑥 + 1)) ↔ (𝑄‘𝑥) = (𝑄‘(𝑥 + 1))))
104	103	adantr 479	. . . . . . . . 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 483	. . . . . . . . . 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 4637	. . . . . . . . . . . 12 ⊢ ((𝑃‘𝑥) = (𝑄‘𝑥) → {(𝑃‘𝑥)} = {(𝑄‘𝑥)})
107	106	adantr 479	. . . . . . . . . . 11 ⊢ (((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) → {(𝑃‘𝑥)} = {(𝑄‘𝑥)})
108	107	adantr 479	. . . . . . . . . 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 2746	. . . . . . . . 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 4738	. . . . . . . . . . 11 ⊢ (((𝑃‘𝑥) = (𝑄‘𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) → {(𝑃‘𝑥), (𝑃‘(𝑥 + 1))} = {(𝑄‘𝑥), (𝑄‘(𝑥 + 1))})
111	110	adantr 479	. . . . . . . . . 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 4014	. . . . . . . . 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 1076	. . . . . . . 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 228	. . . . . . 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 1133	. . . 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 29193	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
121		eqid 2730	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
122		eqid 2730	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
123	121, 122	iswlkg 29137	. . 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 1340	1 ⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 if-wif 1059 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ∀wral 3059 Vcvv 3472 ⊆ wss 3947 {csn 4627 {cpr 4629 class class class wbr 5147 dom cdm 5675 ↾ cres 5677 “ cima 5678 Fun wfun 6536 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 0cc0 11112 1c1 11113 + caddc 11115 ℤ_≥cuz 12826 ...cfz 13488 ..^cfzo 13631 ♯chash 14294 Word cword 14468 prefix cpfx 14624 Vtxcvtx 28523 iEdgciedg 28524 Walkscwlks 29120

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-ifp 1060 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-hash 14295 df-word 14469 df-substr 14595 df-pfx 14625 df-wlks 29123

This theorem is referenced by: trlres 29224 eupthres 29735

W3C validator