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Theorem trlsegvdeg 28176
Description: Formerly part of proof of eupth2lem3 28185: 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 2739 . 2 (iEdg‘𝑋) = (iEdg‘𝑋)
2 eqid 2739 . 2 (iEdg‘𝑌) = (iEdg‘𝑌)
3 eqid 2739 . 2 (Vtx‘𝑋) = (Vtx‘𝑋)
4 trlsegvdeg.vy . . 3 (𝜑 → (Vtx‘𝑌) = 𝑉)
5 trlsegvdeg.vx . . 3 (𝜑 → (Vtx‘𝑋) = 𝑉)
64, 5eqtr4d 2777 . 2 (𝜑 → (Vtx‘𝑌) = (Vtx‘𝑋))
7 trlsegvdeg.vz . . 3 (𝜑 → (Vtx‘𝑍) = 𝑉)
87, 5eqtr4d 2777 . 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 28172 . . . 4 (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
199, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem5 28173 . . . 4 (𝜑 → dom (iEdg‘𝑌) = {(𝐹𝑁)})
2018, 19ineq12d 4114 . . 3 (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹𝑁)}))
21 fzonel 13154 . . . . . . 7 ¬ 𝑁 ∈ (0..^𝑁)
2210trlf1 27652 . . . . . . . . 9 (𝐹(Trails‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
2314, 22syl 17 . . . . . . . 8 (𝜑𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
24 elfzouz2 13155 . . . . . . . . 9 (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ𝑁))
25 fzoss2 13168 . . . . . . . . 9 ((♯‘𝐹) ∈ (ℤ𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
2612, 24, 253syl 18 . . . . . . . 8 (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
27 f1elima 7044 . . . . . . . 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 1372 . . . . . . 7 (𝜑 → ((𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁)))
2921, 28mtbiri 330 . . . . . 6 (𝜑 → ¬ (𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)))
3029orcd 872 . . . . 5 (𝜑 → (¬ (𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹𝑁) ∈ dom 𝐼))
31 ianor 981 . . . . . 6 (¬ ((𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹𝑁) ∈ dom 𝐼) ↔ (¬ (𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹𝑁) ∈ dom 𝐼))
32 elin 3869 . . . . . 6 ((𝐹𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ ((𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹𝑁) ∈ dom 𝐼))
3331, 32xchnxbir 336 . . . . 5 (¬ (𝐹𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ (¬ (𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹𝑁) ∈ dom 𝐼))
3430, 33sylibr 237 . . . 4 (𝜑 → ¬ (𝐹𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
35 disjsn 4612 . . . 4 ((((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹𝑁)}) = ∅ ↔ ¬ (𝐹𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
3634, 35sylibr 237 . . 3 (𝜑 → (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹𝑁)}) = ∅)
3720, 36eqtrd 2774 . 2 (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = ∅)
389, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem2 28170 . 2 (𝜑 → Fun (iEdg‘𝑋))
399, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem3 28171 . 2 (𝜑 → Fun (iEdg‘𝑌))
4013, 5eleqtrrd 2837 . 2 (𝜑𝑈 ∈ (Vtx‘𝑋))
41 f1f 6584 . . . . 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 13907 . . 3 (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))
4415, 16uneq12d 4064 . . 3 (𝜑 → ((iEdg‘𝑋) ∪ (iEdg‘𝑌)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))
4543, 17, 443eqtr4d 2784 . 2 (𝜑 → (iEdg‘𝑍) = ((iEdg‘𝑋) ∪ (iEdg‘𝑌)))
469, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem6 28174 . 2 (𝜑 → dom (iEdg‘𝑋) ∈ Fin)
479, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem7 28175 . 2 (𝜑 → dom (iEdg‘𝑌) ∈ Fin)
481, 2, 3, 6, 8, 37, 38, 39, 40, 45, 46, 47vtxdfiun 27436 1 (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 846   = wceq 1542  wcel 2114  cun 3851  cin 3852  wss 3853  c0 4221  {csn 4526  cop 4532   class class class wbr 5040  dom cdm 5535  cres 5537  cima 5538  Fun wfun 6343  wf 6345  1-1wf1 6346  cfv 6349  (class class class)co 7182  0cc0 10627   + caddc 10630  cuz 12336  ...cfz 12993  ..^cfzo 13136  chash 13794  Vtxcvtx 26953  iEdgciedg 26954  VtxDegcvtxdg 27419  Trailsctrls 27644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491  ax-cnex 10683  ax-resscn 10684  ax-1cn 10685  ax-icn 10686  ax-addcl 10687  ax-addrcl 10688  ax-mulcl 10689  ax-mulrcl 10690  ax-mulcom 10691  ax-addass 10692  ax-mulass 10693  ax-distr 10694  ax-i2m1 10695  ax-1ne0 10696  ax-1rid 10697  ax-rnegex 10698  ax-rrecex 10699  ax-cnre 10700  ax-pre-lttri 10701  ax-pre-lttrn 10702  ax-pre-ltadd 10703  ax-pre-mulgt0 10704
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-int 4847  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-pred 6139  df-ord 6185  df-on 6186  df-lim 6187  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7139  df-ov 7185  df-oprab 7186  df-mpo 7187  df-om 7612  df-1st 7726  df-2nd 7727  df-wrecs 7988  df-recs 8049  df-rdg 8087  df-1o 8143  df-oadd 8147  df-er 8332  df-map 8451  df-en 8568  df-dom 8569  df-sdom 8570  df-fin 8571  df-dju 9415  df-card 9453  df-pnf 10767  df-mnf 10768  df-xr 10769  df-ltxr 10770  df-le 10771  df-sub 10962  df-neg 10963  df-nn 11729  df-n0 11989  df-xnn0 12061  df-z 12075  df-uz 12337  df-xadd 12603  df-fz 12994  df-fzo 13137  df-hash 13795  df-word 13968  df-vtxdg 27420  df-wlks 27553  df-trls 27646
This theorem is referenced by:  eupth2lem3lem7  28183
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