Theorem trlsegvdegfi 16691

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 2238	. 2 ⊢ (iEdg‘𝑋) = (iEdg‘𝑋)
2		eqid 2238	. 2 ⊢ (iEdg‘𝑌) = (iEdg‘𝑌)
3		eqid 2238	. 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 2274	. 2 ⊢ (𝜑 → (Vtx‘𝑌) = (Vtx‘𝑋))
7		trlsegvdeg.vz	. . 3 ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)
8	7, 5	eqtr4d 2274	. 2 ⊢ (𝜑 → (Vtx‘𝑍) = (Vtx‘𝑋))
9		trlsegvdegfi.v	. . 3 ⊢ (𝜑 → 𝑉 ∈ Fin)
10	5, 9	eqeltrd 2315	. 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 2318	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (Vtx‘𝑋))
15		df-vtx 16238	. . . . 5 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
16	15	mptrcl 5785	. . . 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 16503	. 2 ⊢ (𝜑 → 𝑋 ∈ UPGraph)
21	13, 4	eleqtrrd 2318	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (Vtx‘𝑌))
22	15	mptrcl 5785	. . . 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 5406	. . . . 5 ⊢ (𝜑 → 𝐼 Fn dom 𝐼)
27		trlsegvdeg.w	. . . . . . 7 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)
28	12	trlf1 16612	. . . . . . 7 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
29		f1f 5596	. . . . . . 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 5838	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑁) ∈ dom 𝐼)
33		fnressn 5895	. . . . 5 ⊢ ((𝐼 Fn dom 𝐼 ∧ (𝐹‘𝑁) ∈ dom 𝐼) → (𝐼 ↾ {(𝐹‘𝑁)}) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})
34	26, 32, 33	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝐼 ↾ {(𝐹‘𝑁)}) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})
35	24, 34	eqtr4d 2274	. . 3 ⊢ (𝜑 → (iEdg‘𝑌) = (𝐼 ↾ {(𝐹‘𝑁)}))
36	11, 12, 23, 4, 35, 19	upgrspan 16503	. 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 16687	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
39	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem5 16688	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)})
40	38, 39	ineq12d 3433	. . 3 ⊢ (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)}))
41		fzonel 10551	. . . . . . 7 ⊢ ¬ 𝑁 ∈ (0..^𝑁)
42	27, 28	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
43		elfzouz2 10552	. . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝑁))
44		fzoss2 10564	. . . . . . . . 9 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
45	31, 43, 44	3syl 17	. . . . . . . 8 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
46		f1elima 5973	. . . . . . . 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 1278	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁)))
48	41, 47	mtbiri 686	. . . . . 6 ⊢ (𝜑 → ¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)))
49	48	intnanrd 944	. . . . 5 ⊢ (𝜑 → ¬ ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹‘𝑁) ∈ dom 𝐼))
50		elin 3412	. . . . 5 ⊢ ((𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹‘𝑁) ∈ dom 𝐼))
51	49, 50	sylnibr 688	. . . 4 ⊢ (𝜑 → ¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
52		disjsn 3770	. . . 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 2271	. 2 ⊢ (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = ∅)
55	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem2 16685	. 2 ⊢ (𝜑 → Fun (iEdg‘𝑋))
56	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem3 16686	. 2 ⊢ (𝜑 → Fun (iEdg‘𝑌))
57	25, 30, 31	resunimafz0 11257	. . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}))
58	18, 24	uneq12d 3384	. . 3 ⊢ (𝜑 → ((iEdg‘𝑋) ∪ (iEdg‘𝑌)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}))
59	57, 37, 58	3eqtr4d 2281	. 2 ⊢ (𝜑 → (iEdg‘𝑍) = ((iEdg‘𝑋) ∪ (iEdg‘𝑌)))
60	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem6 16689	. 2 ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin)
61	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem7 16690	. 2 ⊢ (𝜑 → dom (iEdg‘𝑌) ∈ Fin)
62	1, 2, 3, 6, 8, 10, 20, 36, 54, 55, 56, 14, 59, 60, 61	vtxdfifiun 16521	1 ⊢ (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∪ cun 3218   ∩ cin 3219   ⊆ wss 3220  ∅c0 3520  ifcif 3638  {csn 3708  ⟨cop 3711   class class class wbr 4128   × cxp 4770  dom cdm 4772   ↾ cres 4774   “ cima 4775  Fun wfun 5369   Fn wfn 5370  ⟶wf 5371  –1-1→wf1 5372  ‘cfv 5375  (class class class)co 6079  1^st c1st 6366  Fincfn 7016  0cc0 8173   + caddc 8176  ℤ_≥cuz 9904  ...cfz 10394  ..^cfzo 10532  ♯chash 11197  Basecbs 13335  Vtxcvtx 16236  iEdgciedg 16237  UPGraphcupgr 16315  VtxDegcvtxdg 16510  Trailsctrls 16604

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-ifp 991  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-2o 6682  df-oadd 6685  df-er 6801  df-map 6918  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-z 9628  df-dec 9761  df-uz 9905  df-xadd 10158  df-fz 10395  df-fzo 10533  df-ihash 11198  df-word 11288  df-ndx 13338  df-slot 13339  df-base 13341  df-edgf 16229  df-vtx 16238  df-iedg 16239  df-edg 16282  df-uhgrm 16293  df-upgren 16317  df-subgr 16478  df-vtxdg 16511  df-wlks 16542  df-trls 16605

This theorem is referenced by:  eupth2lem3lem7fi  16698