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

Theorem trlsegvdeg 29989
Description: Formerly part of proof of eupth2lem3 29998: 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 π‘Œ with the vertices 𝑉 of 𝐺 and one edges being the last edge of the walk, then the vertex degree of any vertex π‘ˆ of 𝐺 within 𝑍 is the sum of the vertex degree of π‘ˆ within 𝑋 and the vertex degree of π‘ˆ within π‘Œ. 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 = (Vtxβ€˜πΊ)
trlsegvdeg.i 𝐼 = (iEdgβ€˜πΊ)
trlsegvdeg.f (πœ‘ β†’ Fun 𝐼)
trlsegvdeg.n (πœ‘ β†’ 𝑁 ∈ (0..^(β™―β€˜πΉ)))
trlsegvdeg.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
trlsegvdeg.w (πœ‘ β†’ 𝐹(Trailsβ€˜πΊ)𝑃)
trlsegvdeg.vx (πœ‘ β†’ (Vtxβ€˜π‘‹) = 𝑉)
trlsegvdeg.vy (πœ‘ β†’ (Vtxβ€˜π‘Œ) = 𝑉)
trlsegvdeg.vz (πœ‘ β†’ (Vtxβ€˜π‘) = 𝑉)
trlsegvdeg.ix (πœ‘ β†’ (iEdgβ€˜π‘‹) = (𝐼 β†Ύ (𝐹 β€œ (0..^𝑁))))
trlsegvdeg.iy (πœ‘ β†’ (iEdgβ€˜π‘Œ) = {⟨(πΉβ€˜π‘), (πΌβ€˜(πΉβ€˜π‘))⟩})
trlsegvdeg.iz (πœ‘ β†’ (iEdgβ€˜π‘) = (𝐼 β†Ύ (𝐹 β€œ (0...𝑁))))
Assertion
Ref Expression
trlsegvdeg (πœ‘ β†’ ((VtxDegβ€˜π‘)β€˜π‘ˆ) = (((VtxDegβ€˜π‘‹)β€˜π‘ˆ) + ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ)))

Proof of Theorem trlsegvdeg
StepHypRef Expression
1 eqid 2726 . 2 (iEdgβ€˜π‘‹) = (iEdgβ€˜π‘‹)
2 eqid 2726 . 2 (iEdgβ€˜π‘Œ) = (iEdgβ€˜π‘Œ)
3 eqid 2726 . 2 (Vtxβ€˜π‘‹) = (Vtxβ€˜π‘‹)
4 trlsegvdeg.vy . . 3 (πœ‘ β†’ (Vtxβ€˜π‘Œ) = 𝑉)
5 trlsegvdeg.vx . . 3 (πœ‘ β†’ (Vtxβ€˜π‘‹) = 𝑉)
64, 5eqtr4d 2769 . 2 (πœ‘ β†’ (Vtxβ€˜π‘Œ) = (Vtxβ€˜π‘‹))
7 trlsegvdeg.vz . . 3 (πœ‘ β†’ (Vtxβ€˜π‘) = 𝑉)
87, 5eqtr4d 2769 . 2 (πœ‘ β†’ (Vtxβ€˜π‘) = (Vtxβ€˜π‘‹))
9 trlsegvdeg.v . . . . 5 𝑉 = (Vtxβ€˜πΊ)
10 trlsegvdeg.i . . . . 5 𝐼 = (iEdgβ€˜πΊ)
11 trlsegvdeg.f . . . . 5 (πœ‘ β†’ Fun 𝐼)
12 trlsegvdeg.n . . . . 5 (πœ‘ β†’ 𝑁 ∈ (0..^(β™―β€˜πΉ)))
13 trlsegvdeg.u . . . . 5 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
14 trlsegvdeg.w . . . . 5 (πœ‘ β†’ 𝐹(Trailsβ€˜πΊ)𝑃)
15 trlsegvdeg.ix . . . . 5 (πœ‘ β†’ (iEdgβ€˜π‘‹) = (𝐼 β†Ύ (𝐹 β€œ (0..^𝑁))))
16 trlsegvdeg.iy . . . . 5 (πœ‘ β†’ (iEdgβ€˜π‘Œ) = {⟨(πΉβ€˜π‘), (πΌβ€˜(πΉβ€˜π‘))⟩})
17 trlsegvdeg.iz . . . . 5 (πœ‘ β†’ (iEdgβ€˜π‘) = (𝐼 β†Ύ (𝐹 β€œ (0...𝑁))))
189, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem4 29985 . . . 4 (πœ‘ β†’ dom (iEdgβ€˜π‘‹) = ((𝐹 β€œ (0..^𝑁)) ∩ dom 𝐼))
199, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem5 29986 . . . 4 (πœ‘ β†’ dom (iEdgβ€˜π‘Œ) = {(πΉβ€˜π‘)})
2018, 19ineq12d 4208 . . 3 (πœ‘ β†’ (dom (iEdgβ€˜π‘‹) ∩ dom (iEdgβ€˜π‘Œ)) = (((𝐹 β€œ (0..^𝑁)) ∩ dom 𝐼) ∩ {(πΉβ€˜π‘)}))
21 fzonel 13652 . . . . . . 7 Β¬ 𝑁 ∈ (0..^𝑁)
2210trlf1 29464 . . . . . . . . 9 (𝐹(Trailsβ€˜πΊ)𝑃 β†’ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼)
2314, 22syl 17 . . . . . . . 8 (πœ‘ β†’ 𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼)
24 elfzouz2 13653 . . . . . . . . 9 (𝑁 ∈ (0..^(β™―β€˜πΉ)) β†’ (β™―β€˜πΉ) ∈ (β„€β‰₯β€˜π‘))
25 fzoss2 13666 . . . . . . . . 9 ((β™―β€˜πΉ) ∈ (β„€β‰₯β€˜π‘) β†’ (0..^𝑁) βŠ† (0..^(β™―β€˜πΉ)))
2612, 24, 253syl 18 . . . . . . . 8 (πœ‘ β†’ (0..^𝑁) βŠ† (0..^(β™―β€˜πΉ)))
27 f1elima 7258 . . . . . . . 8 ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ)) ∧ (0..^𝑁) βŠ† (0..^(β™―β€˜πΉ))) β†’ ((πΉβ€˜π‘) ∈ (𝐹 β€œ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁)))
2823, 12, 26, 27syl3anc 1368 . . . . . . 7 (πœ‘ β†’ ((πΉβ€˜π‘) ∈ (𝐹 β€œ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁)))
2921, 28mtbiri 327 . . . . . 6 (πœ‘ β†’ Β¬ (πΉβ€˜π‘) ∈ (𝐹 β€œ (0..^𝑁)))
3029orcd 870 . . . . 5 (πœ‘ β†’ (Β¬ (πΉβ€˜π‘) ∈ (𝐹 β€œ (0..^𝑁)) ∨ Β¬ (πΉβ€˜π‘) ∈ dom 𝐼))
31 ianor 978 . . . . . 6 (Β¬ ((πΉβ€˜π‘) ∈ (𝐹 β€œ (0..^𝑁)) ∧ (πΉβ€˜π‘) ∈ dom 𝐼) ↔ (Β¬ (πΉβ€˜π‘) ∈ (𝐹 β€œ (0..^𝑁)) ∨ Β¬ (πΉβ€˜π‘) ∈ dom 𝐼))
32 elin 3959 . . . . . 6 ((πΉβ€˜π‘) ∈ ((𝐹 β€œ (0..^𝑁)) ∩ dom 𝐼) ↔ ((πΉβ€˜π‘) ∈ (𝐹 β€œ (0..^𝑁)) ∧ (πΉβ€˜π‘) ∈ dom 𝐼))
3331, 32xchnxbir 333 . . . . 5 (Β¬ (πΉβ€˜π‘) ∈ ((𝐹 β€œ (0..^𝑁)) ∩ dom 𝐼) ↔ (Β¬ (πΉβ€˜π‘) ∈ (𝐹 β€œ (0..^𝑁)) ∨ Β¬ (πΉβ€˜π‘) ∈ dom 𝐼))
3430, 33sylibr 233 . . . 4 (πœ‘ β†’ Β¬ (πΉβ€˜π‘) ∈ ((𝐹 β€œ (0..^𝑁)) ∩ dom 𝐼))
35 disjsn 4710 . . . 4 ((((𝐹 β€œ (0..^𝑁)) ∩ dom 𝐼) ∩ {(πΉβ€˜π‘)}) = βˆ… ↔ Β¬ (πΉβ€˜π‘) ∈ ((𝐹 β€œ (0..^𝑁)) ∩ dom 𝐼))
3634, 35sylibr 233 . . 3 (πœ‘ β†’ (((𝐹 β€œ (0..^𝑁)) ∩ dom 𝐼) ∩ {(πΉβ€˜π‘)}) = βˆ…)
3720, 36eqtrd 2766 . 2 (πœ‘ β†’ (dom (iEdgβ€˜π‘‹) ∩ dom (iEdgβ€˜π‘Œ)) = βˆ…)
389, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem2 29983 . 2 (πœ‘ β†’ Fun (iEdgβ€˜π‘‹))
399, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem3 29984 . 2 (πœ‘ β†’ Fun (iEdgβ€˜π‘Œ))
4013, 5eleqtrrd 2830 . 2 (πœ‘ β†’ π‘ˆ ∈ (Vtxβ€˜π‘‹))
41 f1f 6781 . . . . 5 (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 β†’ 𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐼)
4214, 22, 413syl 18 . . . 4 (πœ‘ β†’ 𝐹:(0..^(β™―β€˜πΉ))⟢dom 𝐼)
4311, 42, 12resunimafz0 14410 . . 3 (πœ‘ β†’ (𝐼 β†Ύ (𝐹 β€œ (0...𝑁))) = ((𝐼 β†Ύ (𝐹 β€œ (0..^𝑁))) βˆͺ {⟨(πΉβ€˜π‘), (πΌβ€˜(πΉβ€˜π‘))⟩}))
4415, 16uneq12d 4159 . . 3 (πœ‘ β†’ ((iEdgβ€˜π‘‹) βˆͺ (iEdgβ€˜π‘Œ)) = ((𝐼 β†Ύ (𝐹 β€œ (0..^𝑁))) βˆͺ {⟨(πΉβ€˜π‘), (πΌβ€˜(πΉβ€˜π‘))⟩}))
4543, 17, 443eqtr4d 2776 . 2 (πœ‘ β†’ (iEdgβ€˜π‘) = ((iEdgβ€˜π‘‹) βˆͺ (iEdgβ€˜π‘Œ)))
469, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem6 29987 . 2 (πœ‘ β†’ dom (iEdgβ€˜π‘‹) ∈ Fin)
479, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem7 29988 . 2 (πœ‘ β†’ dom (iEdgβ€˜π‘Œ) ∈ Fin)
481, 2, 3, 6, 8, 37, 38, 39, 40, 45, 46, 47vtxdfiun 29248 1 (πœ‘ β†’ ((VtxDegβ€˜π‘)β€˜π‘ˆ) = (((VtxDegβ€˜π‘‹)β€˜π‘ˆ) + ((VtxDegβ€˜π‘Œ)β€˜π‘ˆ)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   = wceq 1533   ∈ wcel 2098   βˆͺ cun 3941   ∩ cin 3942   βŠ† wss 3943  βˆ…c0 4317  {csn 4623  βŸ¨cop 4629   class class class wbr 5141  dom cdm 5669   β†Ύ cres 5671   β€œ cima 5672  Fun wfun 6531  βŸΆwf 6533  β€“1-1β†’wf1 6534  β€˜cfv 6537  (class class class)co 7405  0cc0 11112   + caddc 11115  β„€β‰₯cuz 12826  ...cfz 13490  ..^cfzo 13633  β™―chash 14295  Vtxcvtx 28764  iEdgciedg 28765  VtxDegcvtxdg 29231  Trailsctrls 29456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1060  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-oadd 8471  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-xadd 13099  df-fz 13491  df-fzo 13634  df-hash 14296  df-word 14471  df-vtxdg 29232  df-wlks 29365  df-trls 29458
This theorem is referenced by:  eupth2lem3lem7  29996
  Copyright terms: Public domain W3C validator