MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  trlsegvdeg Structured version   Visualization version   GIF version

Theorem trlsegvdeg 30246
Description: Formerly part of proof of eupth2lem3 30255: 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 2737 . 2 (iEdg‘𝑋) = (iEdg‘𝑋)
2 eqid 2737 . 2 (iEdg‘𝑌) = (iEdg‘𝑌)
3 eqid 2737 . 2 (Vtx‘𝑋) = (Vtx‘𝑋)
4 trlsegvdeg.vy . . 3 (𝜑 → (Vtx‘𝑌) = 𝑉)
5 trlsegvdeg.vx . . 3 (𝜑 → (Vtx‘𝑋) = 𝑉)
64, 5eqtr4d 2780 . 2 (𝜑 → (Vtx‘𝑌) = (Vtx‘𝑋))
7 trlsegvdeg.vz . . 3 (𝜑 → (Vtx‘𝑍) = 𝑉)
87, 5eqtr4d 2780 . 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 30242 . . . 4 (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
199, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem5 30243 . . . 4 (𝜑 → dom (iEdg‘𝑌) = {(𝐹𝑁)})
2018, 19ineq12d 4221 . . 3 (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹𝑁)}))
21 fzonel 13713 . . . . . . 7 ¬ 𝑁 ∈ (0..^𝑁)
2210trlf1 29716 . . . . . . . . 9 (𝐹(Trails‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
2314, 22syl 17 . . . . . . . 8 (𝜑𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
24 elfzouz2 13714 . . . . . . . . 9 (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ𝑁))
25 fzoss2 13727 . . . . . . . . 9 ((♯‘𝐹) ∈ (ℤ𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
2612, 24, 253syl 18 . . . . . . . 8 (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
27 f1elima 7283 . . . . . . . 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 1373 . . . . . . 7 (𝜑 → ((𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁)))
2921, 28mtbiri 327 . . . . . 6 (𝜑 → ¬ (𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)))
3029orcd 874 . . . . 5 (𝜑 → (¬ (𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹𝑁) ∈ dom 𝐼))
31 ianor 984 . . . . . 6 (¬ ((𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹𝑁) ∈ dom 𝐼) ↔ (¬ (𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹𝑁) ∈ dom 𝐼))
32 elin 3967 . . . . . 6 ((𝐹𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ ((𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹𝑁) ∈ dom 𝐼))
3331, 32xchnxbir 333 . . . . 5 (¬ (𝐹𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ (¬ (𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹𝑁) ∈ dom 𝐼))
3430, 33sylibr 234 . . . 4 (𝜑 → ¬ (𝐹𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
35 disjsn 4711 . . . 4 ((((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹𝑁)}) = ∅ ↔ ¬ (𝐹𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
3634, 35sylibr 234 . . 3 (𝜑 → (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹𝑁)}) = ∅)
3720, 36eqtrd 2777 . 2 (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = ∅)
389, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem2 30240 . 2 (𝜑 → Fun (iEdg‘𝑋))
399, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem3 30241 . 2 (𝜑 → Fun (iEdg‘𝑌))
4013, 5eleqtrrd 2844 . 2 (𝜑𝑈 ∈ (Vtx‘𝑋))
41 f1f 6804 . . . . 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 14484 . . 3 (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))
4415, 16uneq12d 4169 . . 3 (𝜑 → ((iEdg‘𝑋) ∪ (iEdg‘𝑌)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))
4543, 17, 443eqtr4d 2787 . 2 (𝜑 → (iEdg‘𝑍) = ((iEdg‘𝑋) ∪ (iEdg‘𝑌)))
469, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem6 30244 . 2 (𝜑 → dom (iEdg‘𝑋) ∈ Fin)
479, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem7 30245 . 2 (𝜑 → dom (iEdg‘𝑌) ∈ Fin)
481, 2, 3, 6, 8, 37, 38, 39, 40, 45, 46, 47vtxdfiun 29500 1 (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  cun 3949  cin 3950  wss 3951  c0 4333  {csn 4626  cop 4632   class class class wbr 5143  dom cdm 5685  cres 5687  cima 5688  Fun wfun 6555  wf 6557  1-1wf1 6558  cfv 6561  (class class class)co 7431  0cc0 11155   + caddc 11158  cuz 12878  ...cfz 13547  ..^cfzo 13694  chash 14369  Vtxcvtx 29013  iEdgciedg 29014  VtxDegcvtxdg 29483  Trailsctrls 29708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-xadd 13155  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-vtxdg 29484  df-wlks 29617  df-trls 29710
This theorem is referenced by:  eupth2lem3lem7  30253
  Copyright terms: Public domain W3C validator