Theorem wlkp1 29765

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 30303. (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 29700	. . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
4		wrdf 14453	. . . . . . 7 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
5		wlkp1.n	. . . . . . . . . 10 ⊢ 𝑁 = (♯‘𝐹)
6	5	eqcomi 2746	. . . . . . . . 9 ⊢ (♯‘𝐹) = 𝑁
7	6	oveq2i 7379	. . . . . . . 8 ⊢ (0..^(♯‘𝐹)) = (0..^𝑁)
8	7	feq2i 6662	. . . . . . 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 6856	. . . . . . 7 ⊢ 𝑁 ∈ V
12	11	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ V)
13		wlkp1.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
14		snidg 4619	. . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵})
15	13, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
16		wlkp1.e	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))
17		dmsnopg 6179	. . . . . . . 8 ⊢ (𝐸 ∈ (Edg‘𝐺) → dom {⟨𝐵, 𝐸⟩} = {𝐵})
18	16, 17	syl 17	. . . . . . 7 ⊢ (𝜑 → dom {⟨𝐵, 𝐸⟩} = {𝐵})
19	15, 18	eleqtrrd 2840	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ dom {⟨𝐵, 𝐸⟩})
20	12, 19	fsnd 6826	. . . . 5 ⊢ (𝜑 → {⟨𝑁, 𝐵⟩}:{𝑁}⟶dom {⟨𝐵, 𝐸⟩})
21		fzodisjsn 13625	. . . . . 6 ⊢ ((0..^𝑁) ∩ {𝑁}) = ∅
22	21	a1i 11	. . . . 5 ⊢ (𝜑 → ((0..^𝑁) ∩ {𝑁}) = ∅)
23		fun 6704	. . . . 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 838	. . . 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 29758	. . . . . . 7 ⊢ (𝜑 → (♯‘𝐻) = (𝑁 + 1))
35	34	oveq2d 7384	. . . . . 6 ⊢ (𝜑 → (0..^(♯‘𝐻)) = (0..^(𝑁 + 1)))
36		wlkcl 29701	. . . . . . . 8 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0)
37		eleq1 2825	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) = 𝑁 → ((♯‘𝐹) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
38	37	eqcoms 2745	. . . . . . . . . 10 ⊢ (𝑁 = (♯‘𝐹) → ((♯‘𝐹) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
39		elnn0uz 12804	. . . . . . . . . . 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 13704	. . . . . . 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 2772	. . . . 5 ⊢ (𝜑 → (0..^(♯‘𝐻)) = ((0..^𝑁) ∪ {𝑁}))
47	33	dmeqd 5862	. . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝑆) = dom (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
48		dmun 5867	. . . . . 6 ⊢ dom (𝐼 ∪ {⟨𝐵, 𝐸⟩}) = (dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩})
49	47, 48	eqtrdi 2788	. . . . 5 ⊢ (𝜑 → dom (iEdg‘𝑆) = (dom 𝐼 ∪ dom {⟨𝐵, 𝐸⟩}))
50	26, 46, 49	feq123d 6659	. . . 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 14455	. . 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 29702	. . . . . . 7 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
55	1, 54	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
56	5	oveq2i 7379	. . . . . . 7 ⊢ (0...𝑁) = (0...(♯‘𝐹))
57	56	feq2i 6662	. . . . . 6 ⊢ (𝑃:(0...𝑁)⟶𝑉 ↔ 𝑃:(0...(♯‘𝐹))⟶𝑉)
58	55, 57	sylibr 234	. . . . 5 ⊢ (𝜑 → 𝑃:(0...𝑁)⟶𝑉)
59		ovexd 7403	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ V)
60	59, 30	fsnd 6826	. . . . 5 ⊢ (𝜑 → {⟨(𝑁 + 1), 𝐶⟩}:{(𝑁 + 1)}⟶𝑉)
61		fzp1disj 13511	. . . . . 6 ⊢ ((0...𝑁) ∩ {(𝑁 + 1)}) = ∅
62	61	a1i 11	. . . . 5 ⊢ (𝜑 → ((0...𝑁) ∩ {(𝑁 + 1)}) = ∅)
63		fun 6704	. . . . 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 838	. . . 4 ⊢ (𝜑 → (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}):((0...𝑁) ∪ {(𝑁 + 1)})⟶(𝑉 ∪ 𝑉))
65		fzsuc 13499	. . . . . 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 4111	. . . . . . 7 ⊢ (𝑉 ∪ 𝑉) = 𝑉
68	67	eqcomi 2746	. . . . . 6 ⊢ 𝑉 = (𝑉 ∪ 𝑉)
69	68	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑉 = (𝑉 ∪ 𝑉))
70	66, 69	feq23d 6665	. . . 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 7384	. . . 4 ⊢ (𝜑 → (0...(♯‘𝐻)) = (0...(𝑁 + 1)))
75		wlkp1.s	. . . 4 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
76	73, 74, 75	feq123d 6659	. . 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 29764	. 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 29760	. . 3 ⊢ (𝜑 → (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
81		eqid 2737	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
82		eqid 2737	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
83	81, 82	iswlk 29696	. . 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 1344	1 ⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  if-wif 1063   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  {csn 4582  {cpr 4584  ⟨cop 4588   class class class wbr 5100  dom cdm 5632  Fun wfun 6494  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  Fincfn 8895  0cc0 11038  1c1 11039   + caddc 11041  ℕ₀cn0 12413  ℤ_≥cuz 12763  ...cfz 13435  ..^cfzo 13582  ♯chash 14265  Word cword 14448  Vtxcvtx 29081  iEdgciedg 29082  Edgcedg 29132  Walkscwlks 29682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-er 8645  df-map 8777  df-pm 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-dju 9825  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-n0 12414  df-z 12501  df-uz 12764  df-fz 13436  df-fzo 13583  df-hash 14266  df-word 14449  df-wlks 29685

This theorem is referenced by:  eupthp1  30303