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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 (𝜑𝑁 ∈ (0..^(♯‘𝐹)))
trlsegvdeg.u (𝜑𝑈𝑉)
trlsegvdeg.w (𝜑𝐹(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) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
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
StepHypRef 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)
64, 5eqtr4d 2789 . 2 (𝜑 → (Vtx‘𝑌) = (Vtx‘𝑋))
7 trlsegvdeg.vz . . 3 (𝜑 → (Vtx‘𝑍) = 𝑉)
87, 5eqtr4d 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 (𝜑𝑁 ∈ (0..^(♯‘𝐹)))
13 trlsegvdeg.u . . . . 5 (𝜑𝑈𝑉)
14 trlsegvdeg.w . . . . 5 (𝜑𝐹(Trails‘𝐺)𝑃)
15 trlsegvdeg.ix . . . . 5 (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
16 trlsegvdeg.iy . . . . 5 (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})
17 trlsegvdeg.iz . . . . 5 (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))
189, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem4 27367 . . . 4 (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
199, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem5 27368 . . . 4 (𝜑 → dom (iEdg‘𝑌) = {(𝐹𝑁)})
2018, 19ineq12d 3950 . . 3 (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹𝑁)}))
21 fzonel 12669 . . . . . . 7 ¬ 𝑁 ∈ (0..^𝑁)
2210trlf1 26797 . . . . . . . . 9 (𝐹(Trails‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
2314, 22syl 17 . . . . . . . 8 (𝜑𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
24 elfzouz2 12670 . . . . . . . . 9 (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ𝑁))
25 fzoss2 12682 . . . . . . . . 9 ((♯‘𝐹) ∈ (ℤ𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
2612, 24, 253syl 18 . . . . . . . 8 (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
27 f1elima 6675 . . . . . . . 8 ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝑁 ∈ (0..^(♯‘𝐹)) ∧ (0..^𝑁) ⊆ (0..^(♯‘𝐹))) → ((𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁)))
2823, 12, 26, 27syl3anc 1473 . . . . . . 7 (𝜑 → ((𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁)))
2921, 28mtbiri 316 . . . . . 6 (𝜑 → ¬ (𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)))
3029orcd 406 . . . . 5 (𝜑 → (¬ (𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹𝑁) ∈ dom 𝐼))
31 ianor 510 . . . . . 6 (¬ ((𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹𝑁) ∈ dom 𝐼) ↔ (¬ (𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹𝑁) ∈ dom 𝐼))
32 elin 3931 . . . . . 6 ((𝐹𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ ((𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹𝑁) ∈ dom 𝐼))
3331, 32xchnxbir 322 . . . . 5 (¬ (𝐹𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ (¬ (𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹𝑁) ∈ dom 𝐼))
3430, 33sylibr 224 . . . 4 (𝜑 → ¬ (𝐹𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
35 disjsn 4382 . . . 4 ((((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹𝑁)}) = ∅ ↔ ¬ (𝐹𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
3634, 35sylibr 224 . . 3 (𝜑 → (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹𝑁)}) = ∅)
3720, 36eqtrd 2786 . 2 (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = ∅)
389, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem2 27365 . 2 (𝜑 → Fun (iEdg‘𝑋))
399, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem3 27366 . 2 (𝜑 → Fun (iEdg‘𝑌))
4013, 5eleqtrrd 2834 . 2 (𝜑𝑈 ∈ (Vtx‘𝑋))
41 f1f 6254 . . . . 5 (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
4214, 22, 413syl 18 . . . 4 (𝜑𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
4311, 42, 12resunimafz0 13413 . . 3 (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))
4415, 16uneq12d 3903 . . 3 (𝜑 → ((iEdg‘𝑋) ∪ (iEdg‘𝑌)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))
4543, 17, 443eqtr4d 2796 . 2 (𝜑 → (iEdg‘𝑍) = ((iEdg‘𝑋) ∪ (iEdg‘𝑌)))
469, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem6 27369 . 2 (𝜑 → dom (iEdg‘𝑋) ∈ Fin)
479, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem7 27370 . 2 (𝜑 → dom (iEdg‘𝑌) ∈ Fin)
481, 2, 3, 6, 8, 37, 38, 39, 40, 45, 46, 47vtxdfiun 26580 1 (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))
 Colors of variables: wff setvar class 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
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