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Theorem trlsegvdeg 27505
Description: Formerly part of proof of eupth2lem3 27514: 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 2765 . 2 (iEdg‘𝑋) = (iEdg‘𝑋)
2 eqid 2765 . 2 (iEdg‘𝑌) = (iEdg‘𝑌)
3 eqid 2765 . 2 (Vtx‘𝑋) = (Vtx‘𝑋)
4 trlsegvdeg.vy . . 3 (𝜑 → (Vtx‘𝑌) = 𝑉)
5 trlsegvdeg.vx . . 3 (𝜑 → (Vtx‘𝑋) = 𝑉)
64, 5eqtr4d 2802 . 2 (𝜑 → (Vtx‘𝑌) = (Vtx‘𝑋))
7 trlsegvdeg.vz . . 3 (𝜑 → (Vtx‘𝑍) = 𝑉)
87, 5eqtr4d 2802 . 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 27501 . . . 4 (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
199, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem5 27502 . . . 4 (𝜑 → dom (iEdg‘𝑌) = {(𝐹𝑁)})
2018, 19ineq12d 3977 . . 3 (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹𝑁)}))
21 fzonel 12691 . . . . . . 7 ¬ 𝑁 ∈ (0..^𝑁)
2210trlf1 26887 . . . . . . . . 9 (𝐹(Trails‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
2314, 22syl 17 . . . . . . . 8 (𝜑𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
24 elfzouz2 12692 . . . . . . . . 9 (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ𝑁))
25 fzoss2 12704 . . . . . . . . 9 ((♯‘𝐹) ∈ (ℤ𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
2612, 24, 253syl 18 . . . . . . . 8 (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
27 f1elima 6712 . . . . . . . 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 1490 . . . . . . 7 (𝜑 → ((𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁)))
2921, 28mtbiri 318 . . . . . 6 (𝜑 → ¬ (𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)))
3029orcd 899 . . . . 5 (𝜑 → (¬ (𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹𝑁) ∈ dom 𝐼))
31 ianor 1004 . . . . . 6 (¬ ((𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹𝑁) ∈ dom 𝐼) ↔ (¬ (𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹𝑁) ∈ dom 𝐼))
32 elin 3958 . . . . . 6 ((𝐹𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ ((𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹𝑁) ∈ dom 𝐼))
3331, 32xchnxbir 324 . . . . 5 (¬ (𝐹𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ (¬ (𝐹𝑁) ∈ (𝐹 “ (0..^𝑁)) ∨ ¬ (𝐹𝑁) ∈ dom 𝐼))
3430, 33sylibr 225 . . . 4 (𝜑 → ¬ (𝐹𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
35 disjsn 4402 . . . 4 ((((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹𝑁)}) = ∅ ↔ ¬ (𝐹𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
3634, 35sylibr 225 . . 3 (𝜑 → (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹𝑁)}) = ∅)
3720, 36eqtrd 2799 . 2 (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = ∅)
389, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem2 27499 . 2 (𝜑 → Fun (iEdg‘𝑋))
399, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem3 27500 . 2 (𝜑 → Fun (iEdg‘𝑌))
4013, 5eleqtrrd 2847 . 2 (𝜑𝑈 ∈ (Vtx‘𝑋))
41 f1f 6283 . . . . 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 13430 . . 3 (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))
4415, 16uneq12d 3930 . . 3 (𝜑 → ((iEdg‘𝑋) ∪ (iEdg‘𝑌)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))
4543, 17, 443eqtr4d 2809 . 2 (𝜑 → (iEdg‘𝑍) = ((iEdg‘𝑋) ∪ (iEdg‘𝑌)))
469, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem6 27503 . 2 (𝜑 → dom (iEdg‘𝑋) ∈ Fin)
479, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem7 27504 . 2 (𝜑 → dom (iEdg‘𝑌) ∈ Fin)
481, 2, 3, 6, 8, 37, 38, 39, 40, 45, 46, 47vtxdfiun 26669 1 (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  cun 3730  cin 3731  wss 3732  c0 4079  {csn 4334  cop 4340   class class class wbr 4809  dom cdm 5277  cres 5279  cima 5280  Fun wfun 6062  wf 6064  1-1wf1 6065  cfv 6068  (class class class)co 6842  0cc0 10189   + caddc 10192  cuz 11886  ...cfz 12533  ..^cfzo 12673  chash 13321  Vtxcvtx 26165  iEdgciedg 26166  VtxDegcvtxdg 26652  Trailsctrls 26879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-ifp 1086  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-xnn0 11611  df-z 11625  df-uz 11887  df-xadd 12147  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13487  df-vtxdg 26653  df-wlks 26786  df-trls 26881
This theorem is referenced by:  eupth2lem3lem7  27512
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