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Theorem trlsegvdeg 27300
Description: Formerly part of proof of eupth2lem3 27309: 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 2724 . 2 (iEdg‘𝑋) = (iEdg‘𝑋)
2 eqid 2724 . 2 (iEdg‘𝑌) = (iEdg‘𝑌)
3 eqid 2724 . 2 (Vtx‘𝑋) = (Vtx‘𝑋)
4 trlsegvdeg.vy . . 3 (𝜑 → (Vtx‘𝑌) = 𝑉)
5 trlsegvdeg.vx . . 3 (𝜑 → (Vtx‘𝑋) = 𝑉)
64, 5eqtr4d 2761 . 2 (𝜑 → (Vtx‘𝑌) = (Vtx‘𝑋))
7 trlsegvdeg.vz . . 3 (𝜑 → (Vtx‘𝑍) = 𝑉)
87, 5eqtr4d 2761 . 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 27296 . . . 4 (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
199, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem5 27297 . . . 4 (𝜑 → dom (iEdg‘𝑌) = {(𝐹𝑁)})
2018, 19ineq12d 3923 . . 3 (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹𝑁)}))
21 fzonel 12598 . . . . . . 7 ¬ 𝑁 ∈ (0..^𝑁)
2210trlf1 26726 . . . . . . . . 9 (𝐹(Trails‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
2314, 22syl 17 . . . . . . . 8 (𝜑𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
24 elfzouz2 12599 . . . . . . . . 9 (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ𝑁))
25 fzoss2 12611 . . . . . . . . 9 ((♯‘𝐹) ∈ (ℤ𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
2612, 24, 253syl 18 . . . . . . . 8 (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
27 f1elima 6635 . . . . . . . 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 1439 . . . . . . 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 3904 . . . . . 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 4353 . . . 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 2758 . 2 (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = ∅)
389, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem2 27294 . 2 (𝜑 → Fun (iEdg‘𝑋))
399, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem3 27295 . 2 (𝜑 → Fun (iEdg‘𝑌))
4013, 5eleqtrrd 2806 . 2 (𝜑𝑈 ∈ (Vtx‘𝑋))
41 f1f 6214 . . . . 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 13342 . . 3 (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))
4415, 16uneq12d 3876 . . 3 (𝜑 → ((iEdg‘𝑋) ∪ (iEdg‘𝑌)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))
4543, 17, 443eqtr4d 2768 . 2 (𝜑 → (iEdg‘𝑍) = ((iEdg‘𝑋) ∪ (iEdg‘𝑌)))
469, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem6 27298 . 2 (𝜑 → dom (iEdg‘𝑋) ∈ Fin)
479, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem7 27299 . 2 (𝜑 → dom (iEdg‘𝑌) ∈ Fin)
481, 2, 3, 6, 8, 37, 38, 39, 40, 45, 46, 47vtxdfiun 26509 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 1596  wcel 2103  cun 3678  cin 3679  wss 3680  c0 4023  {csn 4285  cop 4291   class class class wbr 4760  dom cdm 5218  cres 5220  cima 5221  Fun wfun 5995  wf 5997  1-1wf1 5998  cfv 6001  (class class class)co 6765  0cc0 10049   + caddc 10052  cuz 11800  ...cfz 12440  ..^cfzo 12580  chash 13232  Vtxcvtx 25994  iEdgciedg 25995  VtxDegcvtxdg 26492  Trailsctrls 26718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
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 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-map 7976  df-pm 7977  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-card 8878  df-cda 9103  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-n0 11406  df-xnn0 11477  df-z 11491  df-uz 11801  df-xadd 12061  df-fz 12441  df-fzo 12581  df-hash 13233  df-word 13406  df-vtxdg 26493  df-wlks 26626  df-trls 26720
This theorem is referenced by:  eupth2lem3lem7  27307
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