Theorem wlkp1 27626

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 28156. (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 27559	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
4		wrdf 13963	. . . . . . 7 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
5		wlkp1.n	. . . . . . . . . 10 ⊢ 𝑁 = (♯‘𝐹)
6	5	eqcomi 2748	. . . . . . . . 9 ⊢ (♯‘𝐹) = 𝑁
7	6	oveq2i 7184	. . . . . . . 8 ⊢ (0..^(♯‘𝐹)) = (0..^𝑁)
8	7	feq2i 6497	. . . . . . 7 ⊢ (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 ↔ 𝐹:(0..^𝑁)⟶dom 𝐼)
9	4, 8	sylib 221	. . . . . 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 6691	. . . . . . 7 ⊢ 𝑁 ∈ V
12	11	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ V)
13		wlkp1.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
14		snidg 4551	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵})
15	13, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
16		wlkp1.e	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))
17		dmsnopg 6046	. . . . . . . 8 ⊢ (𝐸 ∈ (Edg‘𝐺) → dom {⟨𝐵, 𝐸⟩} = {𝐵})
18	16, 17	syl 17	. . . . . . 7 ⊢ (𝜑 → dom {⟨𝐵, 𝐸⟩} = {𝐵})
19	15, 18	eleqtrrd 2837	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ dom {⟨𝐵, 𝐸⟩})
20	12, 19	fsnd 6663	. . . . 5 ⊢ (𝜑 → {⟨𝑁, 𝐵⟩}:{𝑁}⟶dom {⟨𝐵, 𝐸⟩})
21		fzodisjsn 13169	. . . . . 6 ⊢ ((0..^𝑁) ∩ {𝑁}) = ∅
22	21	a1i 11	. . . . 5 ⊢ (𝜑 → ((0..^𝑁) ∩ {𝑁}) = ∅)
23		fun 6541	. . . . 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 27619	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐻) = (𝑁 + 1))
35	34	oveq2d 7189	. . . . . 6 ⊢ (𝜑 → (0..^(♯‘𝐻)) = (0..^(𝑁 + 1)))
36		wlkcl 27560	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0)
37		eleq1 2821	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 𝑁 → ((♯‘𝐹) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
38	37	eqcoms 2747	. . . . . . . . . 10 ⊢ (𝑁 = (♯‘𝐹) → ((♯‘𝐹) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
39		elnn0uz 12368	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0))
40	39	biimpi 219	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘0))
41	38, 40	syl6bi 256	. . . . . . . . 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 13239	. . . . . . 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 2774	. . . . 5 ⊢ (𝜑 → (0..^(♯‘𝐻)) = ((0..^𝑁) ∪ {𝑁}))
47	33	dmeqd 5749	. . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝑆) = dom (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
48		dmun 5754	. . . . . 6 ⊢ dom (𝐼 ∪ {⟨𝐵, 𝐸⟩}) = (dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩})
49	47, 48	eqtrdi 2790	. . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑆) = (dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩}))
50	26, 46, 49	feq123d 6494	. . . 4 ⊢ (𝜑 → (𝐻:(0..^(♯‘𝐻))⟶dom (iEdg‘𝑆) ↔ (𝐹 ∪ {⟨𝑁, 𝐵⟩}):((0..^𝑁) ∪ {𝑁})⟶(dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩})))
51	24, 50	mpbird 260	. . 3 ⊢ (𝜑 → 𝐻:(0..^(♯‘𝐻))⟶dom (iEdg‘𝑆))
52		iswrdb 13964	. . 3 ⊢ (𝐻 ∈ Word dom (iEdg‘𝑆) ↔ 𝐻:(0..^(♯‘𝐻))⟶dom (iEdg‘𝑆))
53	51, 52	sylibr 237	. 2 ⊢ (𝜑 → 𝐻 ∈ Word dom (iEdg‘𝑆))
54	27	wlkp 27561	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
55	1, 54	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
56	5	oveq2i 7184	. . . . . . 7 ⊢ (0...𝑁) = (0...(♯‘𝐹))
57	56	feq2i 6497	. . . . . 6 ⊢ (𝑃:(0...𝑁)⟶𝑉 ↔ 𝑃:(0...(♯‘𝐹))⟶𝑉)
58	55, 57	sylibr 237	. . . . 5 ⊢ (𝜑 → 𝑃:(0...𝑁)⟶𝑉)
59		ovexd 7208	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ V)
60	59, 30	fsnd 6663	. . . . 5 ⊢ (𝜑 → {⟨(𝑁 + 1), 𝐶⟩}:{(𝑁 + 1)}⟶𝑉)
61		fzp1disj 13060	. . . . . 6 ⊢ ((0...𝑁) ∩ {(𝑁 + 1)}) = ∅
62	61	a1i 11	. . . . 5 ⊢ (𝜑 → ((0...𝑁) ∩ {(𝑁 + 1)}) = ∅)
63		fun 6541	. . . . 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 13048	. . . . . 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 4043	. . . . . . 7 ⊢ (𝑉 ∪ 𝑉) = 𝑉
68	67	eqcomi 2748	. . . . . 6 ⊢ 𝑉 = (𝑉 ∪ 𝑉)
69	68	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑉 = (𝑉 ∪ 𝑉))
70	66, 69	feq23d 6500	. . . 4 ⊢ (𝜑 → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):(0...(𝑁 + 1))⟶𝑉 ↔ (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):((0...𝑁) ∪ {(𝑁 + 1)})⟶(𝑉 ∪ 𝑉)))
71	64, 70	mpbird 260	. . 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 7189	. . . 4 ⊢ (𝜑 → (0...(♯‘𝐻)) = (0...(𝑁 + 1)))
75		wlkp1.s	. . . 4 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
76	73, 74, 75	feq123d 6494	. . 3 ⊢ (𝜑 → (𝑄:(0...(♯‘𝐻))⟶(Vtx‘𝑆) ↔ (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):(0...(𝑁 + 1))⟶𝑉))
77	71, 76	mpbird 260	. 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 27625	. 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 27621	. . 3 ⊢ (𝜑 → (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
81		eqid 2739	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
82		eqid 2739	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
83	81, 82	iswlk 27555	. . 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 209   ∧ wa 399  if-wif 1062   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  ∀wral 3054  Vcvv 3399   ∪ cun 3842   ∩ cin 3843   ⊆ wss 3844  ∅c0 4212  {csn 4517  {cpr 4519  ⟨cop 4523   class class class wbr 5031  dom cdm 5526  Fun wfun 6334  ⟶wf 6336  ‘cfv 6340  (class class class)co 7173  Fincfn 8558  0cc0 10618  1c1 10619   + caddc 10621  ℕ₀cn0 11979  ℤ_≥cuz 12327  ...cfz 12984  ..^cfzo 13127  ♯chash 13785  Word cword 13958  Vtxcvtx 26944  iEdgciedg 26945  Edgcedg 26995  Walkscwlks 27541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482  ax-cnex 10674  ax-resscn 10675  ax-1cn 10676  ax-icn 10677  ax-addcl 10678  ax-addrcl 10679  ax-mulcl 10680  ax-mulrcl 10681  ax-mulcom 10682  ax-addass 10683  ax-mulass 10684  ax-distr 10685  ax-i2m1 10686  ax-1ne0 10687  ax-1rid 10688  ax-rnegex 10689  ax-rrecex 10690  ax-cnre 10691  ax-pre-lttri 10692  ax-pre-lttrn 10693  ax-pre-ltadd 10694  ax-pre-mulgt0 10695

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-int 4838  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7130  df-ov 7176  df-oprab 7177  df-mpo 7178  df-om 7603  df-1st 7717  df-2nd 7718  df-wrecs 7979  df-recs 8040  df-rdg 8078  df-1o 8134  df-oadd 8138  df-er 8323  df-map 8442  df-pm 8443  df-en 8559  df-dom 8560  df-sdom 8561  df-fin 8562  df-dju 9406  df-card 9444  df-pnf 10758  df-mnf 10759  df-xr 10760  df-ltxr 10761  df-le 10762  df-sub 10953  df-neg 10954  df-nn 11720  df-n0 11980  df-z 12066  df-uz 12328  df-fz 12985  df-fzo 13128  df-hash 13786  df-word 13959  df-wlks 27544

This theorem is referenced by:  eupthp1  28156