Theorem eupth2eucrct 27077

Description: Append one path segment to an Eulerian path ⟨𝐹, 𝑃⟩ which may not be an (Eulerian) circuit to become an Eulerian circuit ⟨𝐻, 𝑄⟩ of the supergraph 𝑆 obtained by adding the new edge to the graph 𝐺. (Contributed by AV, 11-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)

Hypotheses

Ref	Expression
eupthp1.v	⊢ 𝑉 = (Vtx‘𝐺)
eupthp1.i	⊢ 𝐼 = (iEdg‘𝐺)
eupthp1.f	⊢ (𝜑 → Fun 𝐼)
eupthp1.a	⊢ (𝜑 → 𝐼 ∈ Fin)
eupthp1.b	⊢ (𝜑 → 𝐵 ∈ V)
eupthp1.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
eupthp1.d	⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)
eupthp1.p	⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃)
eupthp1.n	⊢ 𝑁 = (#‘𝐹)
eupthp1.e	⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))
eupthp1.x	⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸)
eupthp1.u	⊢ (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩})
eupthp1.h	⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})
eupthp1.q	⊢ 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})
eupthp1.s	⊢ (Vtx‘𝑆) = 𝑉
eupthp1.l	⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶})
eupth2eucrct.c	⊢ (𝜑 → 𝐶 = (𝑃‘0))

Assertion

Ref	Expression
eupth2eucrct	⊢ (𝜑 → (𝐻(EulerPaths‘𝑆)𝑄 ∧ 𝐻(Circuits‘𝑆)𝑄))

Proof of Theorem eupth2eucrct

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eupthp1.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		eupthp1.i	. . 3 ⊢ 𝐼 = (iEdg‘𝐺)
3		eupthp1.f	. . 3 ⊢ (𝜑 → Fun 𝐼)
4		eupthp1.a	. . 3 ⊢ (𝜑 → 𝐼 ∈ Fin)
5		eupthp1.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
6		eupthp1.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
7		eupthp1.d	. . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)
8		eupthp1.p	. . 3 ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃)
9		eupthp1.n	. . 3 ⊢ 𝑁 = (#‘𝐹)
10		eupthp1.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺))
11		eupthp1.x	. . 3 ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸)
12		eupthp1.u	. . 3 ⊢ (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩})
13		eupthp1.h	. . 3 ⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})
14		eupthp1.q	. . 3 ⊢ 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})
15		eupthp1.s	. . 3 ⊢ (Vtx‘𝑆) = 𝑉
16		eupthp1.l	. . 3 ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶})
17	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16	eupthp1 27076	. 2 ⊢ (𝜑 → 𝐻(EulerPaths‘𝑆)𝑄)
18		simpr 477	. . 3 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → 𝐻(EulerPaths‘𝑆)𝑄)
19		eupthistrl 27071	. . . . 5 ⊢ (𝐻(EulerPaths‘𝑆)𝑄 → 𝐻(Trails‘𝑆)𝑄)
20	19	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → 𝐻(Trails‘𝑆)𝑄)
21		fveq2 6191	. . . . . . . 8 ⊢ (𝑘 = 0 → (𝑄‘𝑘) = (𝑄‘0))
22		fveq2 6191	. . . . . . . 8 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0))
23	21, 22	eqeq12d 2637	. . . . . . 7 ⊢ (𝑘 = 0 → ((𝑄‘𝑘) = (𝑃‘𝑘) ↔ (𝑄‘0) = (𝑃‘0)))
24		eupthiswlk 27072	. . . . . . . . 9 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
25	8, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)
26	12	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
27	15	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
28	1, 2, 3, 4, 5, 6, 7, 25, 9, 10, 11, 26, 13, 14, 27	wlkp1lem5 26574	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑄‘𝑘) = (𝑃‘𝑘))
29	2	wlkf 26510	. . . . . . . . 9 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
30	24, 29	syl 17	. . . . . . . 8 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
31		lencl 13324	. . . . . . . . 9 ⊢ (𝐹 ∈ Word dom 𝐼 → (#‘𝐹) ∈ ℕ0)
32	9	eleq1i 2692	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 ↔ (#‘𝐹) ∈ ℕ0)
33		0elfz 12436	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁))
34	32, 33	sylbir 225	. . . . . . . . 9 ⊢ ((#‘𝐹) ∈ ℕ0 → 0 ∈ (0...𝑁))
35	31, 34	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ Word dom 𝐼 → 0 ∈ (0...𝑁))
36	8, 30, 35	3syl 18	. . . . . . 7 ⊢ (𝜑 → 0 ∈ (0...𝑁))
37	23, 28, 36	rspcdva 3316	. . . . . 6 ⊢ (𝜑 → (𝑄‘0) = (𝑃‘0))
38	37	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → (𝑄‘0) = (𝑃‘0))
39		eupth2eucrct.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 = (𝑃‘0))
40	39	eqcomd 2628	. . . . . 6 ⊢ (𝜑 → (𝑃‘0) = 𝐶)
41	40	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → (𝑃‘0) = 𝐶)
42	14	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}))
43	13	fveq2i 6194	. . . . . . . . 9 ⊢ (#‘𝐻) = (#‘(𝐹 ∪ {⟨𝑁, 𝐵⟩}))
44	43	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → (#‘𝐻) = (#‘(𝐹 ∪ {⟨𝑁, 𝐵⟩})))
45		wrdfin 13323	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹 ∈ Fin)
46	29, 45	syl 17	. . . . . . . . . . 11 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Fin)
47	8, 24, 46	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ Fin)
48	47	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → 𝐹 ∈ Fin)
49		snfi 8038	. . . . . . . . . 10 ⊢ {⟨𝑁, 𝐵⟩} ∈ Fin
50	49	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → {⟨𝑁, 𝐵⟩} ∈ Fin)
51		wrddm 13312	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ Word dom 𝐼 → dom 𝐹 = (0..^(#‘𝐹)))
52	8, 30, 51	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐹 = (0..^(#‘𝐹)))
53		fzonel 12483	. . . . . . . . . . . . . . . 16 ⊢ ¬ (#‘𝐹) ∈ (0..^(#‘𝐹))
54	53	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ (#‘𝐹) ∈ (0..^(#‘𝐹)))
55	9	eleq1i 2692	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (0..^(#‘𝐹)) ↔ (#‘𝐹) ∈ (0..^(#‘𝐹)))
56	54, 55	sylnibr 319	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ 𝑁 ∈ (0..^(#‘𝐹)))
57		eleq2 2690	. . . . . . . . . . . . . . 15 ⊢ (dom 𝐹 = (0..^(#‘𝐹)) → (𝑁 ∈ dom 𝐹 ↔ 𝑁 ∈ (0..^(#‘𝐹))))
58	57	notbid 308	. . . . . . . . . . . . . 14 ⊢ (dom 𝐹 = (0..^(#‘𝐹)) → (¬ 𝑁 ∈ dom 𝐹 ↔ ¬ 𝑁 ∈ (0..^(#‘𝐹))))
59	56, 58	syl5ibrcom 237	. . . . . . . . . . . . 13 ⊢ (𝜑 → (dom 𝐹 = (0..^(#‘𝐹)) → ¬ 𝑁 ∈ dom 𝐹))
60		fvex 6201	. . . . . . . . . . . . . . . 16 ⊢ (#‘𝐹) ∈ V
61	9, 60	eqeltri 2697	. . . . . . . . . . . . . . 15 ⊢ 𝑁 ∈ V
62	61	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ V)
63	62, 5	opeldmd 5327	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⟨𝑁, 𝐵⟩ ∈ 𝐹 → 𝑁 ∈ dom 𝐹))
64	59, 63	nsyld 154	. . . . . . . . . . . 12 ⊢ (𝜑 → (dom 𝐹 = (0..^(#‘𝐹)) → ¬ ⟨𝑁, 𝐵⟩ ∈ 𝐹))
65	52, 64	mpd 15	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ ⟨𝑁, 𝐵⟩ ∈ 𝐹)
66	65	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → ¬ ⟨𝑁, 𝐵⟩ ∈ 𝐹)
67		disjsn 4246	. . . . . . . . . 10 ⊢ ((𝐹 ∩ {⟨𝑁, 𝐵⟩}) = ∅ ↔ ¬ ⟨𝑁, 𝐵⟩ ∈ 𝐹)
68	66, 67	sylibr 224	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → (𝐹 ∩ {⟨𝑁, 𝐵⟩}) = ∅)
69		hashun 13171	. . . . . . . . 9 ⊢ ((𝐹 ∈ Fin ∧ {⟨𝑁, 𝐵⟩} ∈ Fin ∧ (𝐹 ∩ {⟨𝑁, 𝐵⟩}) = ∅) → (#‘(𝐹 ∪ {⟨𝑁, 𝐵⟩})) = ((#‘𝐹) + (#‘{⟨𝑁, 𝐵⟩})))
70	48, 50, 68, 69	syl3anc 1326	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → (#‘(𝐹 ∪ {⟨𝑁, 𝐵⟩})) = ((#‘𝐹) + (#‘{⟨𝑁, 𝐵⟩})))
71	9	eqcomi 2631	. . . . . . . . . 10 ⊢ (#‘𝐹) = 𝑁
72		opex 4932	. . . . . . . . . . 11 ⊢ ⟨𝑁, 𝐵⟩ ∈ V
73		hashsng 13159	. . . . . . . . . . 11 ⊢ (⟨𝑁, 𝐵⟩ ∈ V → (#‘{⟨𝑁, 𝐵⟩}) = 1)
74	72, 73	ax-mp 5	. . . . . . . . . 10 ⊢ (#‘{⟨𝑁, 𝐵⟩}) = 1
75	71, 74	oveq12i 6662	. . . . . . . . 9 ⊢ ((#‘𝐹) + (#‘{⟨𝑁, 𝐵⟩})) = (𝑁 + 1)
76	75	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → ((#‘𝐹) + (#‘{⟨𝑁, 𝐵⟩})) = (𝑁 + 1))
77	44, 70, 76	3eqtrd 2660	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → (#‘𝐻) = (𝑁 + 1))
78	42, 77	fveq12d 6197	. . . . . 6 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → (𝑄‘(#‘𝐻)) = ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1)))
79		ovexd 6680	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ V)
80	1, 2, 3, 4, 5, 6, 7, 25, 9	wlkp1lem1 26570	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃)
81	79, 6, 80	3jca 1242	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 + 1) ∈ V ∧ 𝐶 ∈ 𝑉 ∧ ¬ (𝑁 + 1) ∈ dom 𝑃))
82	81	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → ((𝑁 + 1) ∈ V ∧ 𝐶 ∈ 𝑉 ∧ ¬ (𝑁 + 1) ∈ dom 𝑃))
83		fsnunfv 6453	. . . . . . 7 ⊢ (((𝑁 + 1) ∈ V ∧ 𝐶 ∈ 𝑉 ∧ ¬ (𝑁 + 1) ∈ dom 𝑃) → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1)) = 𝐶)
84	82, 83	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → ((𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})‘(𝑁 + 1)) = 𝐶)
85	78, 84	eqtr2d 2657	. . . . 5 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → 𝐶 = (𝑄‘(#‘𝐻)))
86	38, 41, 85	3eqtrd 2660	. . . 4 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → (𝑄‘0) = (𝑄‘(#‘𝐻)))
87		iscrct 26685	. . . 4 ⊢ (𝐻(Circuits‘𝑆)𝑄 ↔ (𝐻(Trails‘𝑆)𝑄 ∧ (𝑄‘0) = (𝑄‘(#‘𝐻))))
88	20, 86, 87	sylanbrc 698	. . 3 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → 𝐻(Circuits‘𝑆)𝑄)
89	18, 88	jca 554	. 2 ⊢ ((𝜑 ∧ 𝐻(EulerPaths‘𝑆)𝑄) → (𝐻(EulerPaths‘𝑆)𝑄 ∧ 𝐻(Circuits‘𝑆)𝑄))
90	17, 89	mpdan 702	1 ⊢ (𝜑 → (𝐻(EulerPaths‘𝑆)𝑄 ∧ 𝐻(Circuits‘𝑆)𝑄))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  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  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  0cc0 9936  1c1 9937   + caddc 9939  ℕ₀cn0 11292  ...cfz 12326  ..^cfzo 12465  #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939  Walkscwlks 26492  Trailsctrls 26587  Circuitsccrcts 26679  EulerPathsceupth 27057

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  df-trls 26589  df-crcts 26681  df-eupth 27058

This theorem is referenced by: (None)