Theorem 1wlkp1 40890

Description: Append one path segment (edge) 𝐸 from vertex (𝑃‘𝑁) to a vertex 𝐶 to a 1-walk ⟨𝐹, 𝑃⟩ to become a 1-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 41384. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 6-Mar-2021.) (Prove shortened by AV, 18-Apr-2021.)

Hypotheses

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

Assertion

Ref	Expression
1wlkp1	⊢ (𝜑 → 𝐻(1Walks‘𝑆)𝑄)

Proof of Theorem 1wlkp1

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1wlkp1.w	. . . . . 6 ⊢ (𝜑 → 𝐹(1Walks‘𝐺)𝑃)
2		1wlkp1.i	. . . . . . 7 ⊢ 𝐼 = (iEdg‘𝐺)
3	2	1wlkf 40819	. . . . . 6 ⊢ (𝐹(1Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
4		wrdf 13165	. . . . . . 7 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(#‘𝐹))⟶dom 𝐼)
5		1wlkp1.n	. . . . . . . . . 10 ⊢ 𝑁 = (#‘𝐹)
6	5	eqcomi 2619	. . . . . . . . 9 ⊢ (#‘𝐹) = 𝑁
7	6	oveq2i 6560	. . . . . . . 8 ⊢ (0..^(#‘𝐹)) = (0..^𝑁)
8	7	feq2i 5950	. . . . . . 7 ⊢ (𝐹:(0..^(#‘𝐹))⟶dom 𝐼 ↔ 𝐹:(0..^𝑁)⟶dom 𝐼)
9	4, 8	sylib 207	. . . . . 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 6113	. . . . . . . 8 ⊢ (#‘𝐹) ∈ V
12	5, 11	eqeltri 2684	. . . . . . 7 ⊢ 𝑁 ∈ V
13	12	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ V)
14		1wlkp1.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V)
15		snidg 4153	. . . . . . . 8 ⊢ (𝐵 ∈ V → 𝐵 ∈ {𝐵})
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
17		1wlkp1.e	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))
18		dmsnopg 5524	. . . . . . . 8 ⊢ (𝐸 ∈ (Edg‘𝐺) → dom {⟨𝐵, 𝐸⟩} = {𝐵})
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → dom {⟨𝐵, 𝐸⟩} = {𝐵})
20	16, 19	eleqtrrd 2691	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ dom {⟨𝐵, 𝐸⟩})
21	13, 20	fsnd 6091	. . . . 5 ⊢ (𝜑 → {⟨𝑁, 𝐵⟩}:{𝑁}⟶dom {⟨𝐵, 𝐸⟩})
22		fzodisjsn 12374	. . . . . 6 ⊢ ((0..^𝑁) ∩ {𝑁}) = ∅
23	22	a1i 11	. . . . 5 ⊢ (𝜑 → ((0..^𝑁) ∩ {𝑁}) = ∅)
24		fun 5979	. . . . 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 1317	. . . 4 ⊢ (𝜑 → (𝐹 ∪ {⟨𝑁, 𝐵⟩}):((0..^𝑁) ∪ {𝑁})⟶(dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩}))
26		1wlkp1.h	. . . . . 6 ⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})
27	26	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩}))
28		1wlkp1.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
29		1wlkp1.f	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐼)
30		1wlkp1.a	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ Fin)
31		1wlkp1.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
32		1wlkp1.d	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)
33		1wlkp1.x	. . . . . . . 8 ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸)
34		1wlkp1.u	. . . . . . . 8 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
35	28, 2, 29, 30, 14, 31, 32, 1, 5, 17, 33, 34, 26	1wlkp1lem2 40883	. . . . . . 7 ⊢ (𝜑 → (#‘𝐻) = (𝑁 + 1))
36	35	oveq2d 6565	. . . . . 6 ⊢ (𝜑 → (0..^(#‘𝐻)) = (0..^(𝑁 + 1)))
37		1wlkcl 40820	. . . . . . . 8 ⊢ (𝐹(1Walks‘𝐺)𝑃 → (#‘𝐹) ∈ ℕ0)
38		eleq1 2676	. . . . . . . . . . 11 ⊢ ((#‘𝐹) = 𝑁 → ((#‘𝐹) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
39	38	eqcoms 2618	. . . . . . . . . 10 ⊢ (𝑁 = (#‘𝐹) → ((#‘𝐹) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
40		elnn0uz 11601	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (ℤ≥‘0))
41	40	biimpi 205	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (ℤ≥‘0))
42	39, 41	syl6bi 242	. . . . . . . . 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 12442	. . . . . . 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 2644	. . . . 5 ⊢ (𝜑 → (0..^(#‘𝐻)) = ((0..^𝑁) ∪ {𝑁}))
48	34	dmeqd 5248	. . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝑆) = dom (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
49		dmun 5253	. . . . . 6 ⊢ dom (𝐼 ∪ {⟨𝐵, 𝐸⟩}) = (dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩})
50	48, 49	syl6eq 2660	. . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑆) = (dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩}))
51	27, 47, 50	feq123d 5947	. . . 4 ⊢ (𝜑 → (𝐻:(0..^(#‘𝐻))⟶dom (iEdg‘𝑆) ↔ (𝐹 ∪ {⟨𝑁, 𝐵⟩}):((0..^𝑁) ∪ {𝑁})⟶(dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩})))
52	25, 51	mpbird 246	. . 3 ⊢ (𝜑 → 𝐻:(0..^(#‘𝐻))⟶dom (iEdg‘𝑆))
53		iswrdb 13166	. . 3 ⊢ (𝐻 ∈ Word dom (iEdg‘𝑆) ↔ 𝐻:(0..^(#‘𝐻))⟶dom (iEdg‘𝑆))
54	52, 53	sylibr 223	. 2 ⊢ (𝜑 → 𝐻 ∈ Word dom (iEdg‘𝑆))
55	28	1wlkp 40821	. . . . . . 7 ⊢ (𝐹(1Walks‘𝐺)𝑃 → 𝑃:(0...(#‘𝐹))⟶𝑉)
56	1, 55	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃:(0...(#‘𝐹))⟶𝑉)
57	5	oveq2i 6560	. . . . . . 7 ⊢ (0...𝑁) = (0...(#‘𝐹))
58	57	feq2i 5950	. . . . . 6 ⊢ (𝑃:(0...𝑁)⟶𝑉 ↔ 𝑃:(0...(#‘𝐹))⟶𝑉)
59	56, 58	sylibr 223	. . . . 5 ⊢ (𝜑 → 𝑃:(0...𝑁)⟶𝑉)
60		ovex 6577	. . . . . . 7 ⊢ (𝑁 + 1) ∈ V
61	60	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ V)
62	61, 31	fsnd 6091	. . . . 5 ⊢ (𝜑 → {⟨(𝑁 + 1), 𝐶⟩}:{(𝑁 + 1)}⟶𝑉)
63		fzp1disj 12269	. . . . . 6 ⊢ ((0...𝑁) ∩ {(𝑁 + 1)}) = ∅
64	63	a1i 11	. . . . 5 ⊢ (𝜑 → ((0...𝑁) ∩ {(𝑁 + 1)}) = ∅)
65		fun 5979	. . . . 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))
66	59, 62, 64, 65	syl21anc 1317	. . . 4 ⊢ (𝜑 → (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):((0...𝑁) ∪ {(𝑁 + 1)})⟶(𝑉 ∪ 𝑉))
67		fzsuc 12258	. . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘0) → (0...(𝑁 + 1)) = ((0...𝑁) ∪ {(𝑁 + 1)}))
68	44, 67	syl 17	. . . . 5 ⊢ (𝜑 → (0...(𝑁 + 1)) = ((0...𝑁) ∪ {(𝑁 + 1)}))
69		unidm 3718	. . . . . . 7 ⊢ (𝑉 ∪ 𝑉) = 𝑉
70	69	eqcomi 2619	. . . . . 6 ⊢ 𝑉 = (𝑉 ∪ 𝑉)
71	70	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑉 = (𝑉 ∪ 𝑉))
72	68, 71	feq23d 5953	. . . 4 ⊢ (𝜑 → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):(0...(𝑁 + 1))⟶𝑉 ↔ (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):((0...𝑁) ∪ {(𝑁 + 1)})⟶(𝑉 ∪ 𝑉)))
73	66, 72	mpbird 246	. . 3 ⊢ (𝜑 → (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):(0...(𝑁 + 1))⟶𝑉)
74		1wlkp1.q	. . . . 5 ⊢ 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})
75	74	a1i 11	. . . 4 ⊢ (𝜑 → 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}))
76	35	oveq2d 6565	. . . 4 ⊢ (𝜑 → (0...(#‘𝐻)) = (0...(𝑁 + 1)))
77		1wlkp1.s	. . . 4 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
78	75, 76, 77	feq123d 5947	. . 3 ⊢ (𝜑 → (𝑄:(0...(#‘𝐻))⟶(Vtx‘𝑆) ↔ (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):(0...(𝑁 + 1))⟶𝑉))
79	73, 78	mpbird 246	. 2 ⊢ (𝜑 → 𝑄:(0...(#‘𝐻))⟶(Vtx‘𝑆))
80		1wlkp1.l	. . 3 ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶})
81	28, 2, 29, 30, 14, 31, 32, 1, 5, 17, 33, 34, 26, 74, 77, 80	1wlkp1lem8 40889	. 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))))
82	28, 2, 29, 30, 14, 31, 32, 1, 5, 17, 33, 34, 26, 74, 77	1wlkp1lem4 40885	. . 3 ⊢ (𝜑 → (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
83		eqid 2610	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
84		eqid 2610	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
85	83, 84	is1wlk 40813	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V) → (𝐻(1Walks‘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	82, 85	syl 17	. 2 ⊢ (𝜑 → (𝐻(1Walks‘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))))))
87	54, 79, 81, 86	mpbir3and 1238	1 ⊢ (𝜑 → 𝐻(1Walks‘𝑆)𝑄)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  if-wif 1006   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583  dom cdm 5038  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  0cc0 9815  1c1 9816   + caddc 9818  ℕ₀cn0 11169  ℤ_≥cuz 11563  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146  Vtxcvtx 25673  iEdgciedg 25674  Edgcedga 25792  1Walksc1wlks 40796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-1wlks 40800

This theorem is referenced by:  eupthp1  41384