Theorem wlkp1 26578

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 27076. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 6-Mar-2021.) (Prove 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 26510	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
4		wrdf 13310	. . . . . . 7 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(#‘𝐹))⟶dom 𝐼)
5		wlkp1.n	. . . . . . . . . 10 ⊢ 𝑁 = (#‘𝐹)
6	5	eqcomi 2631	. . . . . . . . 9 ⊢ (#‘𝐹) = 𝑁
7	6	oveq2i 6661	. . . . . . . 8 ⊢ (0..^(#‘𝐹)) = (0..^𝑁)
8	7	feq2i 6037	. . . . . . 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 6201	. . . . . . . 8 ⊢ (#‘𝐹) ∈ V
12	5, 11	eqeltri 2697	. . . . . . 7 ⊢ 𝑁 ∈ V
13	12	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ V)
14		wlkp1.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V)
15		snidg 4206	. . . . . . . 8 ⊢ (𝐵 ∈ V → 𝐵 ∈ {𝐵})
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
17		wlkp1.e	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))
18		dmsnopg 5606	. . . . . . . 8 ⊢ (𝐸 ∈ (Edg‘𝐺) → dom {⟨𝐵, 𝐸⟩} = {𝐵})
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → dom {⟨𝐵, 𝐸⟩} = {𝐵})
20	16, 19	eleqtrrd 2704	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ dom {⟨𝐵, 𝐸⟩})
21	13, 20	fsnd 6179	. . . . 5 ⊢ (𝜑 → {⟨𝑁, 𝐵⟩}:{𝑁}⟶dom {⟨𝐵, 𝐸⟩})
22		fzodisjsn 12505	. . . . . 6 ⊢ ((0..^𝑁) ∩ {𝑁}) = ∅
23	22	a1i 11	. . . . 5 ⊢ (𝜑 → ((0..^𝑁) ∩ {𝑁}) = ∅)
24		fun 6066	. . . . 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 1325	. . . 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 26571	. . . . . . 7 ⊢ (𝜑 → (#‘𝐻) = (𝑁 + 1))
36	35	oveq2d 6666	. . . . . 6 ⊢ (𝜑 → (0..^(#‘𝐻)) = (0..^(𝑁 + 1)))
37		wlkcl 26511	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (#‘𝐹) ∈ ℕ0)
38		eleq1 2689	. . . . . . . . . . 11 ⊢ ((#‘𝐹) = 𝑁 → ((#‘𝐹) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
39	38	eqcoms 2630	. . . . . . . . . 10 ⊢ (𝑁 = (#‘𝐹) → ((#‘𝐹) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
40		elnn0uz 11725	. . . . . . . . . . 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 12576	. . . . . . 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 2656	. . . . 5 ⊢ (𝜑 → (0..^(#‘𝐻)) = ((0..^𝑁) ∪ {𝑁}))
48	34	dmeqd 5326	. . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝑆) = dom (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
49		dmun 5331	. . . . . 6 ⊢ dom (𝐼 ∪ {⟨𝐵, 𝐸⟩}) = (dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩})
50	48, 49	syl6eq 2672	. . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑆) = (dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩}))
51	27, 47, 50	feq123d 6034	. . . 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 13311	. . 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 26512	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(#‘𝐹))⟶𝑉)
56	1, 55	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃:(0...(#‘𝐹))⟶𝑉)
57	5	oveq2i 6661	. . . . . . 7 ⊢ (0...𝑁) = (0...(#‘𝐹))
58	57	feq2i 6037	. . . . . 6 ⊢ (𝑃:(0...𝑁)⟶𝑉 ↔ 𝑃:(0...(#‘𝐹))⟶𝑉)
59	56, 58	sylibr 224	. . . . 5 ⊢ (𝜑 → 𝑃:(0...𝑁)⟶𝑉)
60		ovexd 6680	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ V)
61	60, 31	fsnd 6179	. . . . 5 ⊢ (𝜑 → {⟨(𝑁 + 1), 𝐶⟩}:{(𝑁 + 1)}⟶𝑉)
62		fzp1disj 12399	. . . . . 6 ⊢ ((0...𝑁) ∩ {(𝑁 + 1)}) = ∅
63	62	a1i 11	. . . . 5 ⊢ (𝜑 → ((0...𝑁) ∩ {(𝑁 + 1)}) = ∅)
64		fun 6066	. . . . 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 1325	. . . 4 ⊢ (𝜑 → (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):((0...𝑁) ∪ {(𝑁 + 1)})⟶(𝑉 ∪ 𝑉))
66		fzsuc 12388	. . . . . 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 3756	. . . . . . 7 ⊢ (𝑉 ∪ 𝑉) = 𝑉
69	68	eqcomi 2631	. . . . . 6 ⊢ 𝑉 = (𝑉 ∪ 𝑉)
70	69	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑉 = (𝑉 ∪ 𝑉))
71	67, 70	feq23d 6040	. . . 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 6666	. . . 4 ⊢ (𝜑 → (0...(#‘𝐻)) = (0...(𝑁 + 1)))
76		wlkp1.s	. . . 4 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
77	74, 75, 76	feq123d 6034	. . 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 26577	. 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 26573	. . 3 ⊢ (𝜑 → (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
82		eqid 2622	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
83		eqid 2622	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
84	82, 83	iswlk 26506	. . 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 1245	1 ⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  if-wif 1012   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  {cpr 4179  ⟨cop 4183   class class class wbr 4653  dom cdm 5114  Fun wfun 5882  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  0cc0 9936  1c1 9937   + caddc 9939  ℕ₀cn0 11292  ℤ_≥cuz 11687  ...cfz 12326  ..^cfzo 12465  #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939  Walkscwlks 26492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-wlks 26495

This theorem is referenced by:  eupthp1  27076