Theorem wlkp1 29660

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 30198. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 6-Mar-2021.) (Proof shortened by AV, 18-Apr-2021.) (Revised by AV, 8-Apr-2024.)

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 ∈ 𝑊)
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 29595	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
4		wrdf 14427	. . . . . . 7 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
5		wlkp1.n	. . . . . . . . . 10 ⊢ 𝑁 = (♯‘𝐹)
6	5	eqcomi 2742	. . . . . . . . 9 ⊢ (♯‘𝐹) = 𝑁
7	6	oveq2i 7363	. . . . . . . 8 ⊢ (0..^(♯‘𝐹)) = (0..^𝑁)
8	7	feq2i 6648	. . . . . . 7 ⊢ (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 ↔ 𝐹:(0..^𝑁)⟶dom 𝐼)
9	4, 8	sylib 218	. . . . . 6 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^𝑁)⟶dom 𝐼)
10	1, 3, 9	3syl 18	. . . . 5 ⊢ (𝜑 → 𝐹:(0..^𝑁)⟶dom 𝐼)
11	5	fvexi 6842	. . . . . . 7 ⊢ 𝑁 ∈ V
12	11	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ V)
13		wlkp1.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
14		snidg 4612	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵})
15	13, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
16		wlkp1.e	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))
17		dmsnopg 6165	. . . . . . . 8 ⊢ (𝐸 ∈ (Edg‘𝐺) → dom {⟨𝐵, 𝐸⟩} = {𝐵})
18	16, 17	syl 17	. . . . . . 7 ⊢ (𝜑 → dom {⟨𝐵, 𝐸⟩} = {𝐵})
19	15, 18	eleqtrrd 2836	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ dom {⟨𝐵, 𝐸⟩})
20	12, 19	fsnd 6812	. . . . 5 ⊢ (𝜑 → {⟨𝑁, 𝐵⟩}:{𝑁}⟶dom {⟨𝐵, 𝐸⟩})
21		fzodisjsn 13599	. . . . . 6 ⊢ ((0..^𝑁) ∩ {𝑁}) = ∅
22	21	a1i 11	. . . . 5 ⊢ (𝜑 → ((0..^𝑁) ∩ {𝑁}) = ∅)
23		fun 6690	. . . . 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⟩}))
24	10, 20, 22, 23	syl21anc 837	. . . 4 ⊢ (𝜑 → (𝐹 ∪ {⟨𝑁, 𝐵⟩}):((0..^𝑁) ∪ {𝑁})⟶(dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩}))
25		wlkp1.h	. . . . . 6 ⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})
26	25	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩}))
27		wlkp1.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
28		wlkp1.f	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐼)
29		wlkp1.a	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ Fin)
30		wlkp1.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
31		wlkp1.d	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)
32		wlkp1.x	. . . . . . . 8 ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸)
33		wlkp1.u	. . . . . . . 8 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
34	27, 2, 28, 29, 13, 30, 31, 1, 5, 16, 32, 33, 25	wlkp1lem2 29653	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐻) = (𝑁 + 1))
35	34	oveq2d 7368	. . . . . 6 ⊢ (𝜑 → (0..^(♯‘𝐻)) = (0..^(𝑁 + 1)))
36		wlkcl 29596	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0)
37		eleq1 2821	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 𝑁 → ((♯‘𝐹) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
38	37	eqcoms 2741	. . . . . . . . . 10 ⊢ (𝑁 = (♯‘𝐹) → ((♯‘𝐹) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
39		elnn0uz 12779	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0))
40	39	biimpi 216	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘0))
41	38, 40	biimtrdi 253	. . . . . . . . 9 ⊢ (𝑁 = (♯‘𝐹) → ((♯‘𝐹) ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘0)))
42	5, 41	ax-mp 5	. . . . . . . 8 ⊢ ((♯‘𝐹) ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘0))
43	1, 36, 42	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
44		fzosplitsn 13678	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
45	43, 44	syl 17	. . . . . 6 ⊢ (𝜑 → (0..^(𝑁 + 1)) = ((0..^𝑁) ∪ {𝑁}))
46	35, 45	eqtrd 2768	. . . . 5 ⊢ (𝜑 → (0..^(♯‘𝐻)) = ((0..^𝑁) ∪ {𝑁}))
47	33	dmeqd 5849	. . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝑆) = dom (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
48		dmun 5854	. . . . . 6 ⊢ dom (𝐼 ∪ {⟨𝐵, 𝐸⟩}) = (dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩})
49	47, 48	eqtrdi 2784	. . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑆) = (dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩}))
50	26, 46, 49	feq123d 6645	. . . 4 ⊢ (𝜑 → (𝐻:(0..^(♯‘𝐻))⟶dom (iEdg‘𝑆) ↔ (𝐹 ∪ {⟨𝑁, 𝐵⟩}):((0..^𝑁) ∪ {𝑁})⟶(dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩})))
51	24, 50	mpbird 257	. . 3 ⊢ (𝜑 → 𝐻:(0..^(♯‘𝐻))⟶dom (iEdg‘𝑆))
52		iswrdb 14429	. . 3 ⊢ (𝐻 ∈ Word dom (iEdg‘𝑆) ↔ 𝐻:(0..^(♯‘𝐻))⟶dom (iEdg‘𝑆))
53	51, 52	sylibr 234	. 2 ⊢ (𝜑 → 𝐻 ∈ Word dom (iEdg‘𝑆))
54	27	wlkp 29597	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
55	1, 54	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
56	5	oveq2i 7363	. . . . . . 7 ⊢ (0...𝑁) = (0...(♯‘𝐹))
57	56	feq2i 6648	. . . . . 6 ⊢ (𝑃:(0...𝑁)⟶𝑉 ↔ 𝑃:(0...(♯‘𝐹))⟶𝑉)
58	55, 57	sylibr 234	. . . . 5 ⊢ (𝜑 → 𝑃:(0...𝑁)⟶𝑉)
59		ovexd 7387	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ V)
60	59, 30	fsnd 6812	. . . . 5 ⊢ (𝜑 → {⟨(𝑁 + 1), 𝐶⟩}:{(𝑁 + 1)}⟶𝑉)
61		fzp1disj 13485	. . . . . 6 ⊢ ((0...𝑁) ∩ {(𝑁 + 1)}) = ∅
62	61	a1i 11	. . . . 5 ⊢ (𝜑 → ((0...𝑁) ∩ {(𝑁 + 1)}) = ∅)
63		fun 6690	. . . . 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))
64	58, 60, 62, 63	syl21anc 837	. . . 4 ⊢ (𝜑 → (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):((0...𝑁) ∪ {(𝑁 + 1)})⟶(𝑉 ∪ 𝑉))
65		fzsuc 13473	. . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0...(𝑁 + 1)) = ((0...𝑁) ∪ {(𝑁 + 1)}))
66	43, 65	syl 17	. . . . 5 ⊢ (𝜑 → (0...(𝑁 + 1)) = ((0...𝑁) ∪ {(𝑁 + 1)}))
67		unidm 4106	. . . . . . 7 ⊢ (𝑉 ∪ 𝑉) = 𝑉
68	67	eqcomi 2742	. . . . . 6 ⊢ 𝑉 = (𝑉 ∪ 𝑉)
69	68	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑉 = (𝑉 ∪ 𝑉))
70	66, 69	feq23d 6651	. . . 4 ⊢ (𝜑 → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):(0...(𝑁 + 1))⟶𝑉 ↔ (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):((0...𝑁) ∪ {(𝑁 + 1)})⟶(𝑉 ∪ 𝑉)))
71	64, 70	mpbird 257	. . 3 ⊢ (𝜑 → (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):(0...(𝑁 + 1))⟶𝑉)
72		wlkp1.q	. . . . 5 ⊢ 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})
73	72	a1i 11	. . . 4 ⊢ (𝜑 → 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}))
74	34	oveq2d 7368	. . . 4 ⊢ (𝜑 → (0...(♯‘𝐻)) = (0...(𝑁 + 1)))
75		wlkp1.s	. . . 4 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
76	73, 74, 75	feq123d 6645	. . 3 ⊢ (𝜑 → (𝑄:(0...(♯‘𝐻))⟶(Vtx‘𝑆) ↔ (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):(0...(𝑁 + 1))⟶𝑉))
77	71, 76	mpbird 257	. 2 ⊢ (𝜑 → 𝑄:(0...(♯‘𝐻))⟶(Vtx‘𝑆))
78		wlkp1.l	. . 3 ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶})
79	27, 2, 28, 29, 13, 30, 31, 1, 5, 16, 32, 33, 25, 72, 75, 78	wlkp1lem8 29659	. 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))))
80	27, 2, 28, 29, 13, 30, 31, 1, 5, 16, 32, 33, 25, 72, 75	wlkp1lem4 29655	. . 3 ⊢ (𝜑 → (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
81		eqid 2733	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
82		eqid 2733	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
83	81, 82	iswlk 29591	. . 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))))))
84	80, 83	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))))))
85	53, 77, 79, 84	mpbir3and 1343	1 ⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  if-wif 1062   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048  Vcvv 3437   ∪ cun 3896   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  {csn 4575  {cpr 4577  ⟨cop 4581   class class class wbr 5093  dom cdm 5619  Fun wfun 6480  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352  Fincfn 8875  0cc0 11013  1c1 11014   + caddc 11016  ℕ₀cn0 12388  ℤ_≥cuz 12738  ...cfz 13409  ..^cfzo 13556  ♯chash 14239  Word cword 14422  Vtxcvtx 28976  iEdgciedg 28977  Edgcedg 29027  Walkscwlks 29577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

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 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-oadd 8395  df-er 8628  df-map 8758  df-pm 8759  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-dju 9801  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-n0 12389  df-z 12476  df-uz 12739  df-fz 13410  df-fzo 13557  df-hash 14240  df-word 14423  df-wlks 29580

This theorem is referenced by:  eupthp1  30198