Theorem trlsegvdeg 30885

Description: The effect on vertex degree of adding one edge to a trail.. In the following, a subgraph induced by a segment of a trail is called a "subtrail": For any subtrail 𝑍 of a trail ⟨𝐹, 𝑃⟩ in a pseudograph 𝐺 which is composed of subtrails 𝑋 and 𝑌, where 𝑌 consists of a single edge, the vertex degree of any vertex 𝑈 within 𝑍 is the sum of the vertex degree of 𝑈 within 𝑋 and the vertex degree of 𝑈 within 𝑌. Note that this theorem would not hold for arbitrary walks (if the last edge was identical with a previous edge, the degree of the vertices incident with this edge would not be increased because of this edge). (Contributed by Mario Carneiro, 8-Apr-2015.) Formerly part of proof of eupth2lem3 30894. (Revised by AV, 20-Feb-2021.)

Hypotheses

Ref	Expression
trlsegvdeg.v	⊢ 𝑉 = (Vtx‘𝐺)
trlsegvdeg.i	⊢ 𝐼 = (iEdg‘𝐺)
trlsegvdeg.f	⊢ (𝜑 → Fun 𝐼)
trlsegvdeg.n	⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))
trlsegvdeg.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
trlsegvdeg.w	⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)
trlsegvdeg.vx	⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)
trlsegvdeg.vy	⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)
trlsegvdeg.vz	⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)
trlsegvdeg.ix	⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
trlsegvdeg.iy	⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})
trlsegvdeg.iz	⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))

Assertion

Ref	Expression
trlsegvdeg	⊢ (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))

Proof of Theorem trlsegvdeg

Step	Hyp	Ref	Expression
1		eqid 2763	. 2 ⊢ (iEdg‘𝑋) = (iEdg‘𝑋)
2		eqid 2763	. 2 ⊢ (iEdg‘𝑌) = (iEdg‘𝑌)
3		eqid 2763	. 2 ⊢ (Vtx‘𝑋) = (Vtx‘𝑋)
4		trlsegvdeg.vy	. . 3 ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)
5		trlsegvdeg.vx	. . 3 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)
6	4, 5	eqtr4d 2801	. 2 ⊢ (𝜑 → (Vtx‘𝑌) = (Vtx‘𝑋))
7		trlsegvdeg.vz	. . 3 ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)
8	7, 5	eqtr4d 2801	. 2 ⊢ (𝜑 → (Vtx‘𝑍) = (Vtx‘𝑋))
9		trlsegvdeg.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
10		trlsegvdeg.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
11		trlsegvdeg.f	. . . . 5 ⊢ (𝜑 → Fun 𝐼)
12		trlsegvdeg.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))
13		trlsegvdeg.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
14		trlsegvdeg.w	. . . . 5 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)
15		trlsegvdeg.ix	. . . . 5 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
16		trlsegvdeg.iy	. . . . 5 ⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})
17		trlsegvdeg.iz	. . . . 5 ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))
18	9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17	trlsegvdeglem4 30881	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
19	9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17	trlsegvdeglem5 30882	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)})
20	18, 19	ineq12d 4121	. . 3 ⊢ (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)}))
21		fzonel 13848	. . . . . . 7 ⊢ ¬ 𝑁 ∈ (0..^𝑁)
22	10	trlf1 30338	. . . . . . . . 9 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))⟶1-1dom 𝐼)
23	14, 22	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))⟶1-1dom 𝐼)
24		elfzouz2 13849	. . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝑁))
25		fzoss2 13862	. . . . . . . . 9 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
26	12, 24, 25	3syl 19	. . . . . . . 8 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
27		f1elima 7298	. . . . . . . 8 ⊢ ((𝐹:(0..^(♯‘𝐹))⟶1-1dom 𝐼 ∧ 𝑁 ∈ (0..^(♯‘𝐹)) ∧ (0..^𝑁) ⊆ (0..^(♯‘𝐹))) → ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁)))
28	23, 12, 26, 27	syl3anc 1399	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁)))
29	21, 28	mtbiri 330	. . . . . 6 ⊢ (𝜑 → ¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)))
30	29	orcd 887	. . . . 5 ⊢ (𝜑 → (¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹‘𝑁) ∈ dom 𝐼))
31		ianor 997	. . . . . 6 ⊢ (¬ ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹‘𝑁) ∈ dom 𝐼) ↔ (¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹‘𝑁) ∈ dom 𝐼))
32		elin 4096	. . . . . 6 ⊢ ((𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹‘𝑁) ∈ dom 𝐼))
33	31, 32	xchnxbir 336	. . . . 5 ⊢ (¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ (¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹‘𝑁) ∈ dom 𝐼))
34	30, 33	sylibr 237	. . . 4 ⊢ (𝜑 → ¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
35		disjsn 4673	. . . 4 ⊢ ((((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)}) = ∅ ↔ ¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
36	34, 35	sylibr 237	. . 3 ⊢ (𝜑 → (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)}) = ∅)
37	20, 36	eqtrd 2798	. 2 ⊢ (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = ∅)
38	9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17	trlsegvdeglem2 30879	. 2 ⊢ (𝜑 → Fun (iEdg‘𝑋))
39	9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17	trlsegvdeglem3 30880	. 2 ⊢ (𝜑 → Fun (iEdg‘𝑌))
40	13, 5	eleqtrrd 2867	. 2 ⊢ (𝜑 → 𝑈 ∈ (Vtx‘𝑋))
41		f1f 6808	. . . . 5 ⊢ (𝐹:(0..^(♯‘𝐹))⟶1-1dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
42	14, 22, 41	3syl 19	. . . 4 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
43	11, 42, 12	resunimafz0 14631	. . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}))
44	15, 16	uneq12d 4171	. . 3 ⊢ (𝜑 → ((iEdg‘𝑋) ∪ (iEdg‘𝑌)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}))
45	43, 17, 44	3eqtr4d 2808	. 2 ⊢ (𝜑 → (iEdg‘𝑍) = ((iEdg‘𝑋) ∪ (iEdg‘𝑌)))
46	9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17	trlsegvdeglem6 30883	. 2 ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin)
47	9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17	trlsegvdeglem7 30884	. 2 ⊢ (𝜑 → dom (iEdg‘𝑌) ∈ Fin)
48	1, 2, 3, 6, 8, 37, 38, 39, 40, 45, 46, 47	vtxdfiun 30120	1 ⊢ (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588  ⟨cop 4840   class class class wbr 5108  dom cdm 5658   ↾ cres 5660   “ cima 5661  Fun wfun 6560  ⟶wf 6562  ⟶1-1wf1 6563  ‘cfv 6566  (class class class)co 7451  0cc0 11239   + caddc 11242  ℤ_≥cuz 13005  ...cfz 13679  ..^cfzo 13828  ♯chash 14515  Vtxcvtx 29631  iEdgciedg 29632  VtxDegcvtxdg 30103  Trailsctrls 30330

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-oadd 8518  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-dju 10019  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-n0 12647  df-xnn0 12720  df-z 12734  df-uz 13006  df-xadd 13282  df-fz 13680  df-fzo 13829  df-hash 14516  df-word 14700  df-vtxdg 30104  df-wlks 30237  df-trls 30332

This theorem is used by:  eupth2lem3lem7  30892