Theorem trlsegvdegfi 16345

Description: The effect on vertex degree of adding one edge to a trail. In the following, a subgraph induced by a segment of a trail is called a "subtrail": For any subtrail Z of a trail <. F , P >. in a pseudograph G which is composed of subtrails X and Y, where Y consists of a single edge, the vertex degree of any vertex U 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 ) ) ) )
trlsegvdegfi.g	|- ( ph -> G e. UPGraph )
trlsegvdegfi.v	|- ( ph -> V e. Fin )

Assertion

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

Proof of Theorem trlsegvdegfi

Step	Hyp	Ref	Expression
1		eqid 2231	. 2 |- (iEdg ` X ) = (iEdg ` X )
2		eqid 2231	. 2 |- (iEdg ` Y ) = (iEdg ` Y )
3		eqid 2231	. 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 2267	. 2 |- ( ph -> (Vtx ` Y ) = (Vtx ` X ) )
7		trlsegvdeg.vz	. . 3 |- ( ph -> (Vtx ` Z ) = V )
8	7, 5	eqtr4d 2267	. 2 |- ( ph -> (Vtx ` Z ) = (Vtx ` X ) )
9		trlsegvdegfi.v	. . 3 |- ( ph -> V e. Fin )
10	5, 9	eqeltrd 2308	. 2 |- ( ph -> (Vtx ` X ) e. Fin )
11		trlsegvdeg.v	. . 3 |- V = (Vtx ` G )
12		trlsegvdeg.i	. . 3 |- I = (iEdg ` G )
13		trlsegvdeg.u	. . . . 5 |- ( ph -> U e. V )
14	13, 5	eleqtrrd 2311	. . . 4 |- ( ph -> U e. (Vtx ` X ) )
15		df-vtx 15892	. . . . 5 |- Vtx = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) )
16	15	mptrcl 5730	. . . 4 |- ( U e. (Vtx ` X ) -> X e. _V )
17	14, 16	syl 14	. . 3 |- ( ph -> X e. _V )
18		trlsegvdeg.ix	. . 3 |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )
19		trlsegvdegfi.g	. . 3 |- ( ph -> G e. UPGraph )
20	11, 12, 17, 5, 18, 19	upgrspan 16157	. 2 |- ( ph -> X e. UPGraph )
21	13, 4	eleqtrrd 2311	. . . 4 |- ( ph -> U e. (Vtx ` Y ) )
22	15	mptrcl 5730	. . . 4 |- ( U e. (Vtx ` Y ) -> Y e. _V )
23	21, 22	syl 14	. . 3 |- ( ph -> Y e. _V )
24		trlsegvdeg.iy	. . . 4 |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
25		trlsegvdeg.f	. . . . . 6 |- ( ph -> Fun I )
26	25	funfnd 5357	. . . . 5 |- ( ph -> I Fn dom I )
27		trlsegvdeg.w	. . . . . . 7 |- ( ph -> F (Trails ` G ) P )
28	12	trlf1 16266	. . . . . . 7 |- ( F (Trails ` G ) P -> F : ( 0..^ (♯ ` F ) ) -1-1-> dom I )
29		f1f 5543	. . . . . . 7 |- ( F : ( 0..^ (♯ ` F ) ) -1-1-> dom I -> F : ( 0..^ (♯ ` F ) ) --> dom I )
30	27, 28, 29	3syl 17	. . . . . 6 |- ( ph -> F : ( 0..^ (♯ ` F ) ) --> dom I )
31		trlsegvdeg.n	. . . . . 6 |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )
32	30, 31	ffvelcdmd 5784	. . . . 5 |- ( ph -> ( F ` N ) e. dom I )
33		fnressn 5841	. . . . 5 |- ( ( I Fn dom I /\ ( F ` N ) e. dom I ) -> ( I |` { ( F ` N ) } ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
34	26, 32, 33	syl2anc 411	. . . 4 |- ( ph -> ( I |` { ( F ` N ) } ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
35	24, 34	eqtr4d 2267	. . 3 |- ( ph -> (iEdg ` Y ) = ( I |` { ( F ` N ) } ) )
36	11, 12, 23, 4, 35, 19	upgrspan 16157	. 2 |- ( ph -> Y e. UPGraph )
37		trlsegvdeg.iz	. . . . 5 |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )
38	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem4 16341	. . . 4 |- ( ph -> dom (iEdg ` X ) = ( ( F " ( 0..^ N ) ) i^i dom I ) )
39	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem5 16342	. . . 4 |- ( ph -> dom (iEdg ` Y ) = { ( F ` N ) } )
40	38, 39	ineq12d 3409	. . 3 |- ( ph -> ( dom (iEdg ` X ) i^i dom (iEdg ` Y ) ) = ( ( ( F " ( 0..^ N ) ) i^i dom I ) i^i { ( F ` N ) } ) )
41		fzonel 10399	. . . . . . 7 |- -. N e. ( 0..^ N )
42	27, 28	syl 14	. . . . . . . 8 |- ( ph -> F : ( 0..^ (♯ ` F ) ) -1-1-> dom I )
43		elfzouz2 10400	. . . . . . . . 9 |- ( N e. ( 0..^ (♯ ` F ) ) -> (♯ ` F ) e. ( ZZ>= ` N ) )
44		fzoss2 10412	. . . . . . . . 9 |- ( (♯ ` F ) e. ( ZZ>= ` N ) -> ( 0..^ N ) C_ ( 0..^ (♯ ` F ) ) )
45	31, 43, 44	3syl 17	. . . . . . . 8 |- ( ph -> ( 0..^ N ) C_ ( 0..^ (♯ ` F ) ) )
46		f1elima 5917	. . . . . . . 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 ) ) )
47	42, 31, 45, 46	syl3anc 1273	. . . . . . 7 |- ( ph -> ( ( F ` N ) e. ( F " ( 0..^ N ) ) <-> N e. ( 0..^ N ) ) )
48	41, 47	mtbiri 681	. . . . . 6 |- ( ph -> -. ( F ` N ) e. ( F " ( 0..^ N ) ) )
49	48	intnanrd 939	. . . . 5 |- ( ph -> -. ( ( F ` N ) e. ( F " ( 0..^ N ) ) /\ ( F ` N ) e. dom I ) )
50		elin 3390	. . . . 5 |- ( ( F ` N ) e. ( ( F " ( 0..^ N ) ) i^i dom I ) <-> ( ( F ` N ) e. ( F " ( 0..^ N ) ) /\ ( F ` N ) e. dom I ) )
51	49, 50	sylnibr 683	. . . 4 |- ( ph -> -. ( F ` N ) e. ( ( F " ( 0..^ N ) ) i^i dom I ) )
52		disjsn 3731	. . . 4 |- ( ( ( ( F " ( 0..^ N ) ) i^i dom I ) i^i { ( F ` N ) } ) = (/) <-> -. ( F ` N ) e. ( ( F " ( 0..^ N ) ) i^i dom I ) )
53	51, 52	sylibr 134	. . 3 |- ( ph -> ( ( ( F " ( 0..^ N ) ) i^i dom I ) i^i { ( F ` N ) } ) = (/) )
54	40, 53	eqtrd 2264	. 2 |- ( ph -> ( dom (iEdg ` X ) i^i dom (iEdg ` Y ) ) = (/) )
55	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem2 16339	. 2 |- ( ph -> Fun (iEdg ` X ) )
56	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem3 16340	. 2 |- ( ph -> Fun (iEdg ` Y ) )
57	25, 30, 31	resunimafz0 11099	. . 3 |- ( ph -> ( I |` ( F " ( 0 ... N ) ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) )
58	18, 24	uneq12d 3362	. . 3 |- ( ph -> ( (iEdg ` X ) u. (iEdg ` Y ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) )
59	57, 37, 58	3eqtr4d 2274	. 2 |- ( ph -> (iEdg ` Z ) = ( (iEdg ` X ) u. (iEdg ` Y ) ) )
60	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem6 16343	. 2 |- ( ph -> dom (iEdg ` X ) e. Fin )
61	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem7 16344	. 2 |- ( ph -> dom (iEdg ` Y ) e. Fin )
62	1, 2, 3, 6, 8, 10, 20, 36, 54, 55, 56, 14, 59, 60, 61	vtxdfifiun 16175	1 |- ( ph -> ( (VtxDeg ` Z ) ` U ) = ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   _Vcvv 2802   u. cun 3198   i^i cin 3199   C_ wss 3200   (/)c0 3494   ifcif 3605   {csn 3669   <.cop 3672   class class class wbr 4088   X. cxp 4723   dom cdm 4725   |` cres 4727   "cima 4728   Fun wfun 5320   Fn wfn 5321   -->wf 5322   -1-1->wf1 5323   ` cfv 5326  (class class class)co 6021   1stc1st 6304   Fincfn 6912   0cc0 8035   + caddc 8038   ZZ>=cuz 9758   ...cfz 10246  ..^cfzo 10380  ♯chash 11041   Basecbs 13103  Vtxcvtx 15890  iEdgciedg 15891  UPGraphcupgr 15969  VtxDegcvtxdg 16164  Trailsctrls 16258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8126  ax-resscn 8127  ax-1cn 8128  ax-1re 8129  ax-icn 8130  ax-addcl 8131  ax-addrcl 8132  ax-mulcl 8133  ax-addcom 8135  ax-mulcom 8136  ax-addass 8137  ax-mulass 8138  ax-distr 8139  ax-i2m1 8140  ax-0lt1 8141  ax-1rid 8142  ax-0id 8143  ax-rnegex 8144  ax-cnre 8146  ax-pre-ltirr 8147  ax-pre-ltwlin 8148  ax-pre-lttrn 8149  ax-pre-apti 8150  ax-pre-ltadd 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-ifp 986  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5974  df-ov 6024  df-oprab 6025  df-mpo 6026  df-1st 6306  df-2nd 6307  df-recs 6474  df-irdg 6539  df-frec 6560  df-1o 6585  df-2o 6586  df-oadd 6589  df-er 6705  df-map 6822  df-en 6913  df-dom 6914  df-fin 6915  df-pnf 8219  df-mnf 8220  df-xr 8221  df-ltxr 8222  df-le 8223  df-sub 8355  df-neg 8356  df-inn 9147  df-2 9205  df-3 9206  df-4 9207  df-5 9208  df-6 9209  df-7 9210  df-8 9211  df-9 9212  df-n0 9406  df-z 9483  df-dec 9615  df-uz 9759  df-xadd 10011  df-fz 10247  df-fzo 10381  df-ihash 11042  df-word 11121  df-ndx 13106  df-slot 13107  df-base 13109  df-edgf 15883  df-vtx 15892  df-iedg 15893  df-edg 15936  df-uhgrm 15947  df-upgren 15971  df-subgr 16132  df-vtxdg 16165  df-wlks 16196  df-trls 16259

This theorem is referenced by:  eupth2lem3lem7fi  16352