Theorem wlkp1 26634

Description: Append one path segment (edge) 𝐸 from vertex (𝑃‘𝑁) to a vertex 𝐶 to a walk ⟨𝐹, 𝑃⟩ to become a walk ⟨𝐻, 𝑄⟩ of the supergraph 𝑆 obtained by adding the new edge to the graph 𝐺. Formerly proven directly for Eulerian paths (for pseudographs), see eupthp1 27194. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 6-Mar-2021.) (Proof shortened by AV, 18-Apr-2021.)

Hypotheses

Ref	Expression
wlkp1.v	⊢ 𝑉 = (Vtx‘𝐺)
wlkp1.i	⊢ 𝐼 = (iEdg‘𝐺)
wlkp1.f	⊢ (𝜑 → Fun 𝐼)
wlkp1.a	⊢ (𝜑 → 𝐼 ∈ Fin)
wlkp1.b	⊢ (𝜑 → 𝐵 ∈ V)
wlkp1.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
wlkp1.d	⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)
wlkp1.w	⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)
wlkp1.n	⊢ 𝑁 = (#‘𝐹)
wlkp1.e	⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))
wlkp1.x	⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸)
wlkp1.u	⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
wlkp1.h	⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})
wlkp1.q	⊢ 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})
wlkp1.s	⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
wlkp1.l	⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶})

Assertion

Ref	Expression
wlkp1	⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄)

Proof of Theorem wlkp1

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkp1.w	. . . . . 6 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)
2		wlkp1.i	. . . . . . 7 ⊢ 𝐼 = (iEdg‘𝐺)
3	2	wlkf 26566	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
4		wrdf 13342	. . . . . . 7 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(#‘𝐹))⟶dom 𝐼)
5		wlkp1.n	. . . . . . . . . 10 ⊢ 𝑁 = (#‘𝐹)
6	5	eqcomi 2660	. . . . . . . . 9 ⊢ (#‘𝐹) = 𝑁
7	6	oveq2i 6701	. . . . . . . 8 ⊢ (0..^(#‘𝐹)) = (0..^𝑁)
8	7	feq2i 6075	. . . . . . 7 ⊢ (𝐹:(0..^(#‘𝐹))⟶dom 𝐼 ↔ 𝐹:(0..^𝑁)⟶dom 𝐼)
9	4, 8	sylib 208	. . . . . 6 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^𝑁)⟶dom 𝐼)
10	1, 3, 9	3syl 18	. . . . 5 ⊢ (𝜑 → 𝐹:(0..^𝑁)⟶dom 𝐼)
11		fvex 6239	. . . . . . . 8 ⊢ (#‘𝐹) ∈ V
12	5, 11	eqeltri 2726	. . . . . . 7 ⊢ 𝑁 ∈ V
13	12	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ V)
14		wlkp1.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V)
15		snidg 4239	. . . . . . . 8 ⊢ (𝐵 ∈ V → 𝐵 ∈ {𝐵})
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
17		wlkp1.e	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))
18		dmsnopg 5642	. . . . . . . 8 ⊢ (𝐸 ∈ (Edg‘𝐺) → dom {⟨𝐵, 𝐸⟩} = {𝐵})
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → dom {⟨𝐵, 𝐸⟩} = {𝐵})
20	16, 19	eleqtrrd 2733	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ dom {⟨𝐵, 𝐸⟩})
21	13, 20	fsnd 6217	. . . . 5 ⊢ (𝜑 → {⟨𝑁, 𝐵⟩}:{𝑁}⟶dom {⟨𝐵, 𝐸⟩})
22		fzodisjsn 12545	. . . . . 6 ⊢ ((0..^𝑁) ∩ {𝑁}) = ∅
23	22	a1i 11	. . . . 5 ⊢ (𝜑 → ((0..^𝑁) ∩ {𝑁}) = ∅)
24		fun 6104	. . . . 5 ⊢ (((𝐹:(0..^𝑁)⟶dom 𝐼 ∧ {⟨𝑁, 𝐵⟩}:{𝑁}⟶dom {⟨𝐵, 𝐸⟩}) ∧ ((0..^𝑁) ∩ {𝑁}) = ∅) → (𝐹 ∪ {⟨𝑁, 𝐵⟩}):((0..^𝑁) ∪ {𝑁})⟶(dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩}))
25	10, 21, 23, 24	syl21anc 1365	. . . 4 ⊢ (𝜑 → (𝐹 ∪ {⟨𝑁, 𝐵⟩}):((0..^𝑁) ∪ {𝑁})⟶(dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩}))
26		wlkp1.h	. . . . . 6 ⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})
27	26	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩}))
28		wlkp1.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
29		wlkp1.f	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐼)
30		wlkp1.a	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ Fin)
31		wlkp1.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
32		wlkp1.d	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)
33		wlkp1.x	. . . . . . . 8 ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸)
34		wlkp1.u	. . . . . . . 8 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
35	28, 2, 29, 30, 14, 31, 32, 1, 5, 17, 33, 34, 26	wlkp1lem2 26627	. . . . . . 7 ⊢ (𝜑 → (#‘𝐻) = (𝑁 + 1))
36	35	oveq2d 6706	. . . . . 6 ⊢ (𝜑 → (0..^(#‘𝐻)) = (0..^(𝑁 + 1)))
37		wlkcl 26567	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (#‘𝐹) ∈ ℕ0)
38		eleq1 2718	. . . . . . . . . . 11 ⊢ ((#‘𝐹) = 𝑁 → ((#‘𝐹) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
39	38	eqcoms 2659	. . . . . . . . . 10 ⊢ (𝑁 = (#‘𝐹) → ((#‘𝐹) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
40		elnn0uz 11763	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0))
41	40	biimpi 206	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘0))
42	39, 41	syl6bi 243	. . . . . . . . 9 ⊢ (𝑁 = (#‘𝐹) → ((#‘𝐹) ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘0)))
43	5, 42	ax-mp 5	. . . . . . . 8 ⊢ ((#‘𝐹) ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘0))
44	1, 37, 43	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
45		fzosplitsn 12616	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
46	44, 45	syl 17	. . . . . 6 ⊢ (𝜑 → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
47	36, 46	eqtrd 2685	. . . . 5 ⊢ (𝜑 → (0..^(#‘𝐻)) = ((0..^𝑁) ∪ {𝑁}))
48	34	dmeqd 5358	. . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝑆) = dom (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
49		dmun 5363	. . . . . 6 ⊢ dom (𝐼 ∪ {⟨𝐵, 𝐸⟩}) = (dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩})
50	48, 49	syl6eq 2701	. . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑆) = (dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩}))
51	27, 47, 50	feq123d 6072	. . . 4 ⊢ (𝜑 → (𝐻:(0..^(#‘𝐻))⟶dom (iEdg‘𝑆) ↔ (𝐹 ∪ {⟨𝑁, 𝐵⟩}):((0..^𝑁) ∪ {𝑁})⟶(dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩})))
52	25, 51	mpbird 247	. . 3 ⊢ (𝜑 → 𝐻:(0..^(#‘𝐻))⟶dom (iEdg‘𝑆))
53		iswrdb 13343	. . 3 ⊢ (𝐻 ∈ Word dom (iEdg‘𝑆) ↔ 𝐻:(0..^(#‘𝐻))⟶dom (iEdg‘𝑆))
54	52, 53	sylibr 224	. 2 ⊢ (𝜑 → 𝐻 ∈ Word dom (iEdg‘𝑆))
55	28	wlkp 26568	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(#‘𝐹))⟶𝑉)
56	1, 55	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃:(0...(#‘𝐹))⟶𝑉)
57	5	oveq2i 6701	. . . . . . 7 ⊢ (0...𝑁) = (0...(#‘𝐹))
58	57	feq2i 6075	. . . . . 6 ⊢ (𝑃:(0...𝑁)⟶𝑉 ↔ 𝑃:(0...(#‘𝐹))⟶𝑉)
59	56, 58	sylibr 224	. . . . 5 ⊢ (𝜑 → 𝑃:(0...𝑁)⟶𝑉)
60		ovexd 6720	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ V)
61	60, 31	fsnd 6217	. . . . 5 ⊢ (𝜑 → {⟨(𝑁 + 1), 𝐶⟩}:{(𝑁 + 1)}⟶𝑉)
62		fzp1disj 12437	. . . . . 6 ⊢ ((0...𝑁) ∩ {(𝑁 + 1)}) = ∅
63	62	a1i 11	. . . . 5 ⊢ (𝜑 → ((0...𝑁) ∩ {(𝑁 + 1)}) = ∅)
64		fun 6104	. . . . 5 ⊢ (((𝑃:(0...𝑁)⟶𝑉 ∧ {⟨(𝑁 + 1), 𝐶⟩}:{(𝑁 + 1)}⟶𝑉) ∧ ((0...𝑁) ∩ {(𝑁 + 1)}) = ∅) → (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):((0...𝑁) ∪ {(𝑁 + 1)})⟶(𝑉 ∪ 𝑉))
65	59, 61, 63, 64	syl21anc 1365	. . . 4 ⊢ (𝜑 → (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):((0...𝑁) ∪ {(𝑁 + 1)})⟶(𝑉 ∪ 𝑉))
66		fzsuc 12426	. . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0...(𝑁 + 1)) = ((0...𝑁) ∪ {(𝑁 + 1)}))
67	44, 66	syl 17	. . . . 5 ⊢ (𝜑 → (0...(𝑁 + 1)) = ((0...𝑁) ∪ {(𝑁 + 1)}))
68		unidm 3789	. . . . . . 7 ⊢ (𝑉 ∪ 𝑉) = 𝑉
69	68	eqcomi 2660	. . . . . 6 ⊢ 𝑉 = (𝑉 ∪ 𝑉)
70	69	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑉 = (𝑉 ∪ 𝑉))
71	67, 70	feq23d 6078	. . . 4 ⊢ (𝜑 → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):(0...(𝑁 + 1))⟶𝑉 ↔ (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):((0...𝑁) ∪ {(𝑁 + 1)})⟶(𝑉 ∪ 𝑉)))
72	65, 71	mpbird 247	. . 3 ⊢ (𝜑 → (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):(0...(𝑁 + 1))⟶𝑉)
73		wlkp1.q	. . . . 5 ⊢ 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})
74	73	a1i 11	. . . 4 ⊢ (𝜑 → 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}))
75	35	oveq2d 6706	. . . 4 ⊢ (𝜑 → (0...(#‘𝐻)) = (0...(𝑁 + 1)))
76		wlkp1.s	. . . 4 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
77	74, 75, 76	feq123d 6072	. . 3 ⊢ (𝜑 → (𝑄:(0...(#‘𝐻))⟶(Vtx‘𝑆) ↔ (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):(0...(𝑁 + 1))⟶𝑉))
78	72, 77	mpbird 247	. 2 ⊢ (𝜑 → 𝑄:(0...(#‘𝐻))⟶(Vtx‘𝑆))
79		wlkp1.l	. . 3 ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶})
80	28, 2, 29, 30, 14, 31, 32, 1, 5, 17, 33, 34, 26, 73, 76, 79	wlkp1lem8 26633	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))))
81	28, 2, 29, 30, 14, 31, 32, 1, 5, 17, 33, 34, 26, 73, 76	wlkp1lem4 26629	. . 3 ⊢ (𝜑 → (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
82		eqid 2651	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
83		eqid 2651	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
84	82, 83	iswlk 26562	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V) → (𝐻(Walks‘𝑆)𝑄 ↔ (𝐻 ∈ Word dom (iEdg‘𝑆) ∧ 𝑄:(0...(#‘𝐻))⟶(Vtx‘𝑆) ∧ ∀𝑘 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))))))
85	81, 84	syl 17	. 2 ⊢ (𝜑 → (𝐻(Walks‘𝑆)𝑄 ↔ (𝐻 ∈ Word dom (iEdg‘𝑆) ∧ 𝑄:(0...(#‘𝐻))⟶(Vtx‘𝑆) ∧ ∀𝑘 ∈ (0..^(#‘𝐻))if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘))))))
86	54, 78, 80, 85	mpbir3and 1264	1 ⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383  if-wif 1032   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  Vcvv 3231   ∪ cun 3605   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  {csn 4210  {cpr 4212  ⟨cop 4216   class class class wbr 4685  dom cdm 5143  Fun wfun 5920  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  Fincfn 7997  0cc0 9974  1c1 9975   + caddc 9977  ℕ₀cn0 11330  ℤ_≥cuz 11725  ...cfz 12364  ..^cfzo 12504  #chash 13157  Word cword 13323  Vtxcvtx 25919  iEdgciedg 25920  Edgcedg 25984  Walkscwlks 26548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-wlks 26551

This theorem is referenced by:  eupthp1  27194