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Theorem trlsegvdeg 30079
Description: Formerly part of proof of eupth2lem3 30088: 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 2725 . 2 (iEdgβ€˜π‘‹) = (iEdgβ€˜π‘‹)
2 eqid 2725 . 2 (iEdgβ€˜π‘Œ) = (iEdgβ€˜π‘Œ)
3 eqid 2725 . 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 2768 . 2 (πœ‘ β†’ (Vtxβ€˜π‘Œ) = (Vtxβ€˜π‘‹))
7 trlsegvdeg.vz . . 3 (πœ‘ β†’ (Vtxβ€˜π‘) = 𝑉)
87, 5eqtr4d 2768 . 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 30075 . . . 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 30076 . . . 4 (πœ‘ β†’ dom (iEdgβ€˜π‘Œ) = {(πΉβ€˜π‘)})
2018, 19ineq12d 4207 . . 3 (πœ‘ β†’ (dom (iEdgβ€˜π‘‹) ∩ dom (iEdgβ€˜π‘Œ)) = (((𝐹 β€œ (0..^𝑁)) ∩ dom 𝐼) ∩ {(πΉβ€˜π‘)}))
21 fzonel 13676 . . . . . . 7 Β¬ 𝑁 ∈ (0..^𝑁)
2210trlf1 29554 . . . . . . . . 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 13677 . . . . . . . . 9 (𝑁 ∈ (0..^(β™―β€˜πΉ)) β†’ (β™―β€˜πΉ) ∈ (β„€β‰₯β€˜π‘))
25 fzoss2 13690 . . . . . . . . 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 7268 . . . . . . . 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 326 . . . . . 6 (πœ‘ β†’ Β¬ (πΉβ€˜π‘) ∈ (𝐹 β€œ (0..^𝑁)))
3029orcd 871 . . . . 5 (πœ‘ β†’ (Β¬ (πΉβ€˜π‘) ∈ (𝐹 β€œ (0..^𝑁)) ∨ Β¬ (πΉβ€˜π‘) ∈ dom 𝐼))
31 ianor 979 . . . . . 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 3956 . . . . . 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 332 . . . . 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 4711 . . . 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 2765 . 2 (πœ‘ β†’ (dom (iEdgβ€˜π‘‹) ∩ dom (iEdgβ€˜π‘Œ)) = βˆ…)
389, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem2 30073 . 2 (πœ‘ β†’ Fun (iEdgβ€˜π‘‹))
399, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem3 30074 . 2 (πœ‘ β†’ Fun (iEdgβ€˜π‘Œ))
4013, 5eleqtrrd 2828 . 2 (πœ‘ β†’ π‘ˆ ∈ (Vtxβ€˜π‘‹))
41 f1f 6787 . . . . 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 14434 . . 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 4157 . . 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 2775 . 2 (πœ‘ β†’ (iEdgβ€˜π‘) = ((iEdgβ€˜π‘‹) βˆͺ (iEdgβ€˜π‘Œ)))
469, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem6 30077 . 2 (πœ‘ β†’ dom (iEdgβ€˜π‘‹) ∈ Fin)
479, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17trlsegvdeglem7 30078 . 2 (πœ‘ β†’ dom (iEdgβ€˜π‘Œ) ∈ Fin)
481, 2, 3, 6, 8, 37, 38, 39, 40, 45, 46, 47vtxdfiun 29338 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 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098   βˆͺ cun 3938   ∩ cin 3939   βŠ† wss 3940  βˆ…c0 4318  {csn 4624  βŸ¨cop 4630   class class class wbr 5143  dom cdm 5672   β†Ύ cres 5674   β€œ cima 5675  Fun wfun 6536  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€˜cfv 6542  (class class class)co 7415  0cc0 11136   + caddc 11139  β„€β‰₯cuz 12850  ...cfz 13514  ..^cfzo 13657  β™―chash 14319  Vtxcvtx 28851  iEdgciedg 28852  VtxDegcvtxdg 29321  Trailsctrls 29546
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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-oadd 8487  df-er 8721  df-map 8843  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-dju 9922  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-n0 12501  df-xnn0 12573  df-z 12587  df-uz 12851  df-xadd 13123  df-fz 13515  df-fzo 13658  df-hash 14320  df-word 14495  df-vtxdg 29322  df-wlks 29455  df-trls 29548
This theorem is referenced by:  eupth2lem3lem7  30086
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