Theorem trlsegvdeg 27371

Description: Formerly part of proof of eupth2lem3 27380: If a trail in a graph 𝐺 induces a subgraph 𝑍 with the vertices 𝑉 of 𝐺 and the edges being the edges of the walk, and a subgraph 𝑋 with the vertices 𝑉 of 𝐺 and the edges being the edges of the walk except the last one, and a subgraph 𝑌 with the vertices 𝑉 of 𝐺 and one edges being the last edge of the walk, then the vertex degree of any vertex 𝑈 of 𝐺 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.) (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 2752	. 2 ⊢ (iEdg‘𝑋) = (iEdg‘𝑋)
2		eqid 2752	. 2 ⊢ (iEdg‘𝑌) = (iEdg‘𝑌)
3		eqid 2752	. 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 2789	. 2 ⊢ (𝜑 → (Vtx‘𝑌) = (Vtx‘𝑋))
7		trlsegvdeg.vz	. . 3 ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)
8	7, 5	eqtr4d 2789	. 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 27367	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
19	9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17	trlsegvdeglem5 27368	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)})
20	18, 19	ineq12d 3950	. . 3 ⊢ (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)}))
21		fzonel 12669	. . . . . . 7 ⊢ ¬ 𝑁 ∈ (0..^𝑁)
22	10	trlf1 26797	. . . . . . . . 9 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
23	14, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
24		elfzouz2 12670	. . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝑁))
25		fzoss2 12682	. . . . . . . . 9 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
26	12, 24, 25	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
27		f1elima 6675	. . . . . . . 8 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑁 ∈ (0..^(♯‘𝐹)) ∧ (0..^𝑁) ⊆ (0..^(♯‘𝐹))) → ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁)))
28	23, 12, 26, 27	syl3anc 1473	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁)))
29	21, 28	mtbiri 316	. . . . . 6 ⊢ (𝜑 → ¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)))
30	29	orcd 406	. . . . 5 ⊢ (𝜑 → (¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹‘𝑁) ∈ dom 𝐼))
31		ianor 510	. . . . . 6 ⊢ (¬ ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹‘𝑁) ∈ dom 𝐼) ↔ (¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹‘𝑁) ∈ dom 𝐼))
32		elin 3931	. . . . . 6 ⊢ ((𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹‘𝑁) ∈ dom 𝐼))
33	31, 32	xchnxbir 322	. . . . 5 ⊢ (¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ (¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹‘𝑁) ∈ dom 𝐼))
34	30, 33	sylibr 224	. . . 4 ⊢ (𝜑 → ¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
35		disjsn 4382	. . . 4 ⊢ ((((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)}) = ∅ ↔ ¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
36	34, 35	sylibr 224	. . 3 ⊢ (𝜑 → (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)}) = ∅)
37	20, 36	eqtrd 2786	. 2 ⊢ (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = ∅)
38	9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17	trlsegvdeglem2 27365	. 2 ⊢ (𝜑 → Fun (iEdg‘𝑋))
39	9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17	trlsegvdeglem3 27366	. 2 ⊢ (𝜑 → Fun (iEdg‘𝑌))
40	13, 5	eleqtrrd 2834	. 2 ⊢ (𝜑 → 𝑈 ∈ (Vtx‘𝑋))
41		f1f 6254	. . . . 5 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
42	14, 22, 41	3syl 18	. . . 4 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
43	11, 42, 12	resunimafz0 13413	. . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}))
44	15, 16	uneq12d 3903	. . 3 ⊢ (𝜑 → ((iEdg‘𝑋) ∪ (iEdg‘𝑌)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}))
45	43, 17, 44	3eqtr4d 2796	. 2 ⊢ (𝜑 → (iEdg‘𝑍) = ((iEdg‘𝑋) ∪ (iEdg‘𝑌)))
46	9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17	trlsegvdeglem6 27369	. 2 ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin)
47	9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17	trlsegvdeglem7 27370	. 2 ⊢ (𝜑 → dom (iEdg‘𝑌) ∈ Fin)
48	1, 2, 3, 6, 8, 37, 38, 39, 40, 45, 46, 47	vtxdfiun 26580	1 ⊢ (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   = wceq 1624   ∈ wcel 2131   ∪ cun 3705   ∩ cin 3706   ⊆ wss 3707  ∅c0 4050  {csn 4313  ⟨cop 4319   class class class wbr 4796  dom cdm 5258   ↾ cres 5260   “ cima 5261  Fun wfun 6035  ⟶wf 6037  –1-1→wf1 6038  ‘cfv 6041  (class class class)co 6805  0cc0 10120   + caddc 10123  ℤ_≥cuz 11871  ...cfz 12511  ..^cfzo 12651  ♯chash 13303  Vtxcvtx 26065  iEdgciedg 26066  VtxDegcvtxdg 26563  Trailsctrls 26789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1051  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-map 8017  df-pm 8018  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8947  df-cda 9174  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-n0 11477  df-xnn0 11548  df-z 11562  df-uz 11872  df-xadd 12132  df-fz 12512  df-fzo 12652  df-hash 13304  df-word 13477  df-vtxdg 26564  df-wlks 26697  df-trls 26791

This theorem is referenced by:  eupth2lem3lem7  27378