Theorem trlsegvdegfi 16449

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 2232	. 2 |- (iEdg ` X ) = (iEdg ` X )
2		eqid 2232	. 2 |- (iEdg ` Y ) = (iEdg ` Y )
3		eqid 2232	. 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 2268	. 2 |- ( ph -> (Vtx ` Y ) = (Vtx ` X ) )
7		trlsegvdeg.vz	. . 3 |- ( ph -> (Vtx ` Z ) = V )
8	7, 5	eqtr4d 2268	. 2 |- ( ph -> (Vtx ` Z ) = (Vtx ` X ) )
9		trlsegvdegfi.v	. . 3 |- ( ph -> V e. Fin )
10	5, 9	eqeltrd 2309	. 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 2312	. . . 4 |- ( ph -> U e. (Vtx ` X ) )
15		df-vtx 15996	. . . . 5 |- Vtx = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) )
16	15	mptrcl 5759	. . . 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 16261	. 2 |- ( ph -> X e. UPGraph )
21	13, 4	eleqtrrd 2312	. . . 4 |- ( ph -> U e. (Vtx ` Y ) )
22	15	mptrcl 5759	. . . 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 5382	. . . . 5 |- ( ph -> I Fn dom I )
27		trlsegvdeg.w	. . . . . . 7 |- ( ph -> F (Trails ` G ) P )
28	12	trlf1 16370	. . . . . . 7 |- ( F (Trails ` G ) P -> F : ( 0..^ (♯ ` F ) ) -1-1-> dom I )
29		f1f 5572	. . . . . . 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 5812	. . . . 5 |- ( ph -> ( F ` N ) e. dom I )
33		fnressn 5869	. . . . 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 2268	. . 3 |- ( ph -> (iEdg ` Y ) = ( I |` { ( F ` N ) } ) )
36	11, 12, 23, 4, 35, 19	upgrspan 16261	. 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 16445	. . . 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 16446	. . . 4 |- ( ph -> dom (iEdg ` Y ) = { ( F ` N ) } )
40	38, 39	ineq12d 3422	. . 3 |- ( ph -> ( dom (iEdg ` X ) i^i dom (iEdg ` Y ) ) = ( ( ( F " ( 0..^ N ) ) i^i dom I ) i^i { ( F ` N ) } ) )
41		fzonel 10491	. . . . . . 7 |- -. N e. ( 0..^ N )
42	27, 28	syl 14	. . . . . . . 8 |- ( ph -> F : ( 0..^ (♯ ` F ) ) -1-1-> dom I )
43		elfzouz2 10492	. . . . . . . . 9 |- ( N e. ( 0..^ (♯ ` F ) ) -> (♯ ` F ) e. ( ZZ>= ` N ) )
44		fzoss2 10504	. . . . . . . . 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 5945	. . . . . . . 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 1274	. . . . . . 7 |- ( ph -> ( ( F ` N ) e. ( F " ( 0..^ N ) ) <-> N e. ( 0..^ N ) ) )
48	41, 47	mtbiri 682	. . . . . 6 |- ( ph -> -. ( F ` N ) e. ( F " ( 0..^ N ) ) )
49	48	intnanrd 940	. . . . 5 |- ( ph -> -. ( ( F ` N ) e. ( F " ( 0..^ N ) ) /\ ( F ` N ) e. dom I ) )
50		elin 3401	. . . . 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 684	. . . 4 |- ( ph -> -. ( F ` N ) e. ( ( F " ( 0..^ N ) ) i^i dom I ) )
52		disjsn 3750	. . . 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 2265	. 2 |- ( ph -> ( dom (iEdg ` X ) i^i dom (iEdg ` Y ) ) = (/) )
55	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem2 16443	. 2 |- ( ph -> Fun (iEdg ` X ) )
56	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem3 16444	. 2 |- ( ph -> Fun (iEdg ` Y ) )
57	25, 30, 31	resunimafz0 11191	. . 3 |- ( ph -> ( I |` ( F " ( 0 ... N ) ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) )
58	18, 24	uneq12d 3373	. . 3 |- ( ph -> ( (iEdg ` X ) u. (iEdg ` Y ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) u. { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } ) )
59	57, 37, 58	3eqtr4d 2275	. 2 |- ( ph -> (iEdg ` Z ) = ( (iEdg ` X ) u. (iEdg ` Y ) ) )
60	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem6 16447	. 2 |- ( ph -> dom (iEdg ` X ) e. Fin )
61	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem7 16448	. 2 |- ( ph -> dom (iEdg ` Y ) e. Fin )
62	1, 2, 3, 6, 8, 10, 20, 36, 54, 55, 56, 14, 59, 60, 61	vtxdfifiun 16279	1 |- ( ph -> ( (VtxDeg ` Z ) ` U ) = ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   _Vcvv 2812   u. cun 3208   i^i cin 3209   C_ wss 3210   (/)c0 3507   ifcif 3619   {csn 3688   <.cop 3691   class class class wbr 4108   X. cxp 4746   dom cdm 4748   |` cres 4750   "cima 4751   Fun wfun 5345   Fn wfn 5346   -->wf 5347   -1-1->wf1 5348   ` cfv 5351  (class class class)co 6049   1stc1st 6331   Fincfn 6974   0cc0 8123   + caddc 8126   ZZ>=cuz 9849   ...cfz 10338  ..^cfzo 10472  ♯chash 11133   Basecbs 13201  Vtxcvtx 15994  iEdgciedg 15995  UPGraphcupgr 16073  VtxDegcvtxdg 16268  Trailsctrls 16362

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239

This theorem depends on definitions:  df-bi 117  df-dc 843  df-ifp 987  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-2o 6647  df-oadd 6650  df-er 6766  df-map 6883  df-en 6975  df-dom 6976  df-fin 6977  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-z 9574  df-dec 9706  df-uz 9850  df-xadd 10102  df-fz 10339  df-fzo 10473  df-ihash 11134  df-word 11218  df-ndx 13204  df-slot 13205  df-base 13207  df-edgf 15987  df-vtx 15996  df-iedg 15997  df-edg 16040  df-uhgrm 16051  df-upgren 16075  df-subgr 16236  df-vtxdg 16269  df-wlks 16300  df-trls 16363

This theorem is referenced by:  eupth2lem3lem7fi  16456