Theorem trlsegvdeg 27087

Description: Formerly part of proof of eupth2lem3 27096: 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	|- ( ph -> Fun I )
trlsegvdeg.n	|- ( ph -> N e. ( 0..^ ( # ` F ) ) )
trlsegvdeg.u	|- ( ph -> U e. V )
trlsegvdeg.w	|- ( ph -> F (Trails ` G ) P )
trlsegvdeg.vx	|- ( ph -> (Vtx ` X ) = V )
trlsegvdeg.vy	|- ( ph -> (Vtx ` Y ) = V )
trlsegvdeg.vz	|- ( ph -> (Vtx ` Z ) = V )
trlsegvdeg.ix	|- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )
trlsegvdeg.iy	|- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
trlsegvdeg.iz	|- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )

Assertion

Ref	Expression
trlsegvdeg	|- ( ph -> ( (VtxDeg ` Z ) ` U ) = ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) )

Proof of Theorem trlsegvdeg

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- (iEdg ` X ) = (iEdg ` X )
2		eqid 2622	. 2 |- (iEdg ` Y ) = (iEdg ` Y )
3		eqid 2622	. 2 |- (Vtx ` X ) = (Vtx ` X )
4		trlsegvdeg.vy	. . 3 |- ( ph -> (Vtx ` Y ) = V )
5		trlsegvdeg.vx	. . 3 |- ( ph -> (Vtx ` X ) = V )
6	4, 5	eqtr4d 2659	. 2 |- ( ph -> (Vtx ` Y ) = (Vtx ` X ) )
7		trlsegvdeg.vz	. . 3 |- ( ph -> (Vtx ` Z ) = V )
8	7, 5	eqtr4d 2659	. 2 |- ( ph -> (Vtx ` Z ) = (Vtx ` X ) )
9		trlsegvdeg.v	. . . . 5 |- V = (Vtx ` G )
10		trlsegvdeg.i	. . . . 5 |- I = (iEdg ` G )
11		trlsegvdeg.f	. . . . 5 |- ( ph -> Fun I )
12		trlsegvdeg.n	. . . . 5 |- ( ph -> N e. ( 0..^ ( # ` F ) ) )
13		trlsegvdeg.u	. . . . 5 |- ( ph -> U e. V )
14		trlsegvdeg.w	. . . . 5 |- ( ph -> F (Trails ` G ) P )
15		trlsegvdeg.ix	. . . . 5 |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )
16		trlsegvdeg.iy	. . . . 5 |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
17		trlsegvdeg.iz	. . . . 5 |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )
18	9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17	trlsegvdeglem4 27083	. . . 4 |- ( ph -> dom (iEdg ` X ) = ( ( F " ( 0..^ N ) ) i^i dom I ) )
19	9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17	trlsegvdeglem5 27084	. . . 4 |- ( ph -> dom (iEdg ` Y ) = { ( F ` N ) } )
20	18, 19	ineq12d 3815	. . 3 |- ( ph -> ( dom (iEdg ` X ) i^i dom (iEdg ` Y ) ) = ( ( ( F " ( 0..^ N ) ) i^i dom I ) i^i { ( F ` N ) } ) )
21		fzonel 12483	. . . . . . 7 |- -. N e. ( 0..^ N )
22	10	trlf1 26595	. . . . . . . . 9 |- ( F (Trails ` G ) P -> F : ( 0..^ ( # ` F ) ) -1-1-> dom I )
23	14, 22	syl 17	. . . . . . . 8 |- ( ph -> F : ( 0..^ ( # ` F ) ) -1-1-> dom I )
24		elfzouz2 12484	. . . . . . . . 9 |- ( N e. ( 0..^ ( # ` F ) ) -> ( # ` F ) e. ( ZZ>= ` N ) )
25		fzoss2 12496	. . . . . . . . 9 |- ( ( # ` F ) e. ( ZZ>= ` N ) -> ( 0..^ N ) C_ ( 0..^ ( # ` F ) ) )
26	12, 24, 25	3syl 18	. . . . . . . 8 |- ( ph -> ( 0..^ N ) C_ ( 0..^ ( # ` F ) ) )
27		f1elima 6520	. . . . . . . 8 |- ( ( F : ( 0..^ ( # ` F ) ) -1-1-> dom I /\ N e. ( 0..^ ( # ` F ) ) /\ ( 0..^ N ) C_ ( 0..^ ( # ` F ) ) ) -> ( ( F ` N ) e. ( F " ( 0..^ N ) ) <-> N e. ( 0..^ N ) ) )
28	23, 12, 26, 27	syl3anc 1326	. . . . . . 7 |- ( ph -> ( ( F ` N ) e. ( F " ( 0..^ N ) ) <-> N e. ( 0..^ N ) ) )
29	21, 28	mtbiri 317	. . . . . 6 |- ( ph -> -. ( F ` N ) e. ( F " ( 0..^ N ) ) )
30	29	orcd 407	. . . . 5 |- ( ph -> ( -. ( F ` N ) e. ( F " ( 0..^ N ) ) \/ -. ( F ` N ) e. dom I ) )
31		ianor 509	. . . . . 6 |- ( -. ( ( F ` N ) e. ( F " ( 0..^ N ) ) /\ ( F ` N ) e. dom I ) <-> ( -. ( F ` N ) e. ( F " ( 0..^ N ) ) \/ -. ( F ` N ) e. dom I ) )
32		elin 3796	. . . . . 6 |- ( ( F ` N ) e. ( ( F " ( 0..^ N ) ) i^i dom I ) <-> ( ( F ` N ) e. ( F " ( 0..^ N ) ) /\ ( F ` N ) e. dom I ) )
33	31, 32	xchnxbir 323	. . . . 5 |- ( -. ( F ` N ) e. ( ( F " ( 0..^ N ) ) i^i dom I ) <-> ( -. ( F ` N ) e. ( F " ( 0..^ N ) ) \/ -. ( F ` N ) e. dom I ) )
34	30, 33	sylibr 224	. . . 4 |- ( ph -> -. ( F ` N ) e. ( ( F " ( 0..^ N ) ) i^i dom I ) )
35		disjsn 4246	. . . 4 |- ( ( ( ( F " ( 0..^ N ) ) i^i dom I ) i^i { ( F ` N ) } ) = (/) <-> -. ( F ` N ) e. ( ( F " ( 0..^ N ) ) i^i dom I ) )
36	34, 35	sylibr 224	. . 3 |- ( ph -> ( ( ( F " ( 0..^ N ) ) i^i dom I ) i^i { ( F ` N ) } ) = (/) )
37	20, 36	eqtrd 2656	. 2 |- ( ph -> ( dom (iEdg ` X ) i^i dom (iEdg ` Y ) ) = (/) )
38	9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17	trlsegvdeglem2 27081	. 2 |- ( ph -> Fun (iEdg ` X ) )
39	9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17	trlsegvdeglem3 27082	. 2 |- ( ph -> Fun (iEdg ` Y ) )
40	13, 5	eleqtrrd 2704	. 2 |- ( ph -> U e. (Vtx ` X ) )
41		f1f 6101	. . . . 5 |- ( F : ( 0..^ ( # ` F ) ) -1-1-> dom I -> F : ( 0..^ ( # ` F ) ) --> dom I )
42	14, 22, 41	3syl 18	. . . 4 |- ( ph -> F : ( 0..^ ( # ` F ) ) --> dom I )
43	11, 42, 12	resunimafz0 13229	. . 3 |- ( ph -> ( I |` ( F " ( 0 ... N ) ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) )
44	15, 16	uneq12d 3768	. . 3 |- ( ph -> ( (iEdg ` X ) u. (iEdg ` Y ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) )
45	43, 17, 44	3eqtr4d 2666	. 2 |- ( ph -> (iEdg ` Z ) = ( (iEdg ` X ) u. (iEdg ` Y ) ) )
46	9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17	trlsegvdeglem6 27085	. 2 |- ( ph -> dom (iEdg ` X ) e. Fin )
47	9, 10, 11, 12, 13, 14, 5, 4, 7, 15, 16, 17	trlsegvdeglem7 27086	. 2 |- ( ph -> dom (iEdg ` Y ) e. Fin )
48	1, 2, 3, 6, 8, 37, 38, 39, 40, 45, 46, 47	vtxdfiun 26378	1 |- ( ph -> ( (VtxDeg ` Z ) ` U ) = ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   class class class wbr 4653   dom cdm 5114   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   0cc0 9936   + caddc 9939   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  VtxDegcvtxdg 26361  Trailsctrls 26587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-xadd 11947  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-vtxdg 26362  df-wlks 26495  df-trls 26589

This theorem is referenced by:  eupth2lem3lem7  27094