Theorem trlsegvdegfi 16588

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	⊢ (𝜑 → Fun 𝐼)
trlsegvdeg.n	⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))
trlsegvdeg.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
trlsegvdeg.w	⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)
trlsegvdeg.vx	⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)
trlsegvdeg.vy	⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)
trlsegvdeg.vz	⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)
trlsegvdeg.ix	⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
trlsegvdeg.iy	⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})
trlsegvdeg.iz	⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))
trlsegvdegfi.g	⊢ (𝜑 → 𝐺 ∈ UPGraph)
trlsegvdegfi.v	⊢ (𝜑 → 𝑉 ∈ Fin)

Assertion

Ref	Expression
trlsegvdegfi	⊢ (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))

Proof of Theorem trlsegvdegfi

Step	Hyp	Ref	Expression
1		eqid 2234	. 2 ⊢ (iEdg‘𝑋) = (iEdg‘𝑋)
2		eqid 2234	. 2 ⊢ (iEdg‘𝑌) = (iEdg‘𝑌)
3		eqid 2234	. 2 ⊢ (Vtx‘𝑋) = (Vtx‘𝑋)
4		trlsegvdeg.vy	. . 3 ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)
5		trlsegvdeg.vx	. . 3 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)
6	4, 5	eqtr4d 2270	. 2 ⊢ (𝜑 → (Vtx‘𝑌) = (Vtx‘𝑋))
7		trlsegvdeg.vz	. . 3 ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)
8	7, 5	eqtr4d 2270	. 2 ⊢ (𝜑 → (Vtx‘𝑍) = (Vtx‘𝑋))
9		trlsegvdegfi.v	. . 3 ⊢ (𝜑 → 𝑉 ∈ Fin)
10	5, 9	eqeltrd 2311	. 2 ⊢ (𝜑 → (Vtx‘𝑋) ∈ Fin)
11		trlsegvdeg.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
12		trlsegvdeg.i	. . 3 ⊢ 𝐼 = (iEdg‘𝐺)
13		trlsegvdeg.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
14	13, 5	eleqtrrd 2314	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (Vtx‘𝑋))
15		df-vtx 16135	. . . . 5 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
16	15	mptrcl 5765	. . . 4 ⊢ (𝑈 ∈ (Vtx‘𝑋) → 𝑋 ∈ V)
17	14, 16	syl 14	. . 3 ⊢ (𝜑 → 𝑋 ∈ V)
18		trlsegvdeg.ix	. . 3 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
19		trlsegvdegfi.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
20	11, 12, 17, 5, 18, 19	upgrspan 16400	. 2 ⊢ (𝜑 → 𝑋 ∈ UPGraph)
21	13, 4	eleqtrrd 2314	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (Vtx‘𝑌))
22	15	mptrcl 5765	. . . 4 ⊢ (𝑈 ∈ (Vtx‘𝑌) → 𝑌 ∈ V)
23	21, 22	syl 14	. . 3 ⊢ (𝜑 → 𝑌 ∈ V)
24		trlsegvdeg.iy	. . . 4 ⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})
25		trlsegvdeg.f	. . . . . 6 ⊢ (𝜑 → Fun 𝐼)
26	25	funfnd 5388	. . . . 5 ⊢ (𝜑 → 𝐼 Fn dom 𝐼)
27		trlsegvdeg.w	. . . . . . 7 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)
28	12	trlf1 16509	. . . . . . 7 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
29		f1f 5578	. . . . . . 7 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
30	27, 28, 29	3syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼)
31		trlsegvdeg.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))
32	30, 31	ffvelcdmd 5818	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑁) ∈ dom 𝐼)
33		fnressn 5875	. . . . 5 ⊢ ((𝐼 Fn dom 𝐼 ∧ (𝐹‘𝑁) ∈ dom 𝐼) → (𝐼 ↾ {(𝐹‘𝑁)}) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})
34	26, 32, 33	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐼 ↾ {(𝐹‘𝑁)}) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})
35	24, 34	eqtr4d 2270	. . 3 ⊢ (𝜑 → (iEdg‘𝑌) = (𝐼 ↾ {(𝐹‘𝑁)}))
36	11, 12, 23, 4, 35, 19	upgrspan 16400	. 2 ⊢ (𝜑 → 𝑌 ∈ UPGraph)
37		trlsegvdeg.iz	. . . . 5 ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))
38	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem4 16584	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
39	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem5 16585	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)})
40	38, 39	ineq12d 3427	. . 3 ⊢ (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)}))
41		fzonel 10517	. . . . . . 7 ⊢ ¬ 𝑁 ∈ (0..^𝑁)
42	27, 28	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
43		elfzouz2 10518	. . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝑁))
44		fzoss2 10530	. . . . . . . . 9 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
45	31, 43, 44	3syl 17	. . . . . . . 8 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
46		f1elima 5952	. . . . . . . 8 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑁 ∈ (0..^(♯‘𝐹)) ∧ (0..^𝑁) ⊆ (0..^(♯‘𝐹))) → ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁)))
47	42, 31, 45, 46	syl3anc 1274	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁)))
48	41, 47	mtbiri 682	. . . . . 6 ⊢ (𝜑 → ¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)))
49	48	intnanrd 940	. . . . 5 ⊢ (𝜑 → ¬ ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹‘𝑁) ∈ dom 𝐼))
50		elin 3406	. . . . 5 ⊢ ((𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹‘𝑁) ∈ dom 𝐼))
51	49, 50	sylnibr 684	. . . 4 ⊢ (𝜑 → ¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
52		disjsn 3756	. . . 4 ⊢ ((((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)}) = ∅ ↔ ¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
53	51, 52	sylibr 134	. . 3 ⊢ (𝜑 → (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)}) = ∅)
54	40, 53	eqtrd 2267	. 2 ⊢ (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = ∅)
55	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem2 16582	. 2 ⊢ (𝜑 → Fun (iEdg‘𝑋))
56	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem3 16583	. 2 ⊢ (𝜑 → Fun (iEdg‘𝑌))
57	25, 30, 31	resunimafz0 11223	. . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}))
58	18, 24	uneq12d 3378	. . 3 ⊢ (𝜑 → ((iEdg‘𝑋) ∪ (iEdg‘𝑌)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}))
59	57, 37, 58	3eqtr4d 2277	. 2 ⊢ (𝜑 → (iEdg‘𝑍) = ((iEdg‘𝑋) ∪ (iEdg‘𝑌)))
60	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem6 16586	. 2 ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin)
61	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem7 16587	. 2 ⊢ (𝜑 → dom (iEdg‘𝑌) ∈ Fin)
62	1, 2, 3, 6, 8, 10, 20, 36, 54, 55, 56, 14, 59, 60, 61	vtxdfifiun 16418	1 ⊢ (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ∪ cun 3212   ∩ cin 3213   ⊆ wss 3214  ∅c0 3512  ifcif 3624  {csn 3694  ⟨cop 3697   class class class wbr 4114   × cxp 4752  dom cdm 4754   ↾ cres 4756   “ cima 4757  Fun wfun 5351   Fn wfn 5352  ⟶wf 5353  –1-1→wf1 5354  ‘cfv 5357  (class class class)co 6058  1^st c1st 6345  Fincfn 6988  0cc0 8143   + caddc 8146  ℤ_≥cuz 9871  ...cfz 10361  ..^cfzo 10498  ♯chash 11163  Basecbs 13296  Vtxcvtx 16133  iEdgciedg 16134  UPGraphcupgr 16212  VtxDegcvtxdg 16407  Trailsctrls 16501

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-map 6897  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-dec 9728  df-uz 9872  df-xadd 10125  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-edg 16179  df-uhgrm 16190  df-upgren 16214  df-subgr 16375  df-vtxdg 16408  df-wlks 16439  df-trls 16502

This theorem is referenced by:  eupth2lem3lem7fi  16595