Theorem trlsegvdegfi 16321

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 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 ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)
5		trlsegvdeg.vx	. . 3 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)
6	4, 5	eqtr4d 2267	. 2 ⊢ (𝜑 → (Vtx‘𝑌) = (Vtx‘𝑋))
7		trlsegvdeg.vz	. . 3 ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)
8	7, 5	eqtr4d 2267	. 2 ⊢ (𝜑 → (Vtx‘𝑍) = (Vtx‘𝑋))
9		trlsegvdegfi.v	. . 3 ⊢ (𝜑 → 𝑉 ∈ Fin)
10	5, 9	eqeltrd 2308	. 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 2311	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (Vtx‘𝑋))
15		df-vtx 15868	. . . . 5 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st ‘𝑔), (Base‘𝑔)))
16	15	mptrcl 5729	. . . 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 16133	. 2 ⊢ (𝜑 → 𝑋 ∈ UPGraph)
21	13, 4	eleqtrrd 2311	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (Vtx‘𝑌))
22	15	mptrcl 5729	. . . 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 5357	. . . . 5 ⊢ (𝜑 → 𝐼 Fn dom 𝐼)
27		trlsegvdeg.w	. . . . . . 7 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)
28	12	trlf1 16242	. . . . . . 7 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
29		f1f 5542	. . . . . . 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 5783	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑁) ∈ dom 𝐼)
33		fnressn 5840	. . . . 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 2267	. . 3 ⊢ (𝜑 → (iEdg‘𝑌) = (𝐼 ↾ {(𝐹‘𝑁)}))
36	11, 12, 23, 4, 35, 19	upgrspan 16133	. 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 16317	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
39	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem5 16318	. . . 4 ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)})
40	38, 39	ineq12d 3409	. . 3 ⊢ (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = (((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∩ {(𝐹‘𝑁)}))
41		fzonel 10396	. . . . . . 7 ⊢ ¬ 𝑁 ∈ (0..^𝑁)
42	27, 28	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)
43		elfzouz2 10397	. . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝑁))
44		fzoss2 10409	. . . . . . . . 9 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
45	31, 43, 44	3syl 17	. . . . . . . 8 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
46		f1elima 5914	. . . . . . . 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 1273	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ↔ 𝑁 ∈ (0..^𝑁)))
48	41, 47	mtbiri 681	. . . . . 6 ⊢ (𝜑 → ¬ (𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)))
49	48	intnanrd 939	. . . . 5 ⊢ (𝜑 → ¬ ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹‘𝑁) ∈ dom 𝐼))
50		elin 3390	. . . . 5 ⊢ ((𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ↔ ((𝐹‘𝑁) ∈ (𝐹 “ (0..^𝑁)) ∧ (𝐹‘𝑁) ∈ dom 𝐼))
51	49, 50	sylnibr 683	. . . 4 ⊢ (𝜑 → ¬ (𝐹‘𝑁) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
52		disjsn 3731	. . . 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 2264	. 2 ⊢ (𝜑 → (dom (iEdg‘𝑋) ∩ dom (iEdg‘𝑌)) = ∅)
55	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem2 16315	. 2 ⊢ (𝜑 → Fun (iEdg‘𝑋))
56	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem3 16316	. 2 ⊢ (𝜑 → Fun (iEdg‘𝑌))
57	25, 30, 31	resunimafz0 11096	. . 3 ⊢ (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}))
58	18, 24	uneq12d 3362	. . 3 ⊢ (𝜑 → ((iEdg‘𝑋) ∪ (iEdg‘𝑌)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩}))
59	57, 37, 58	3eqtr4d 2274	. 2 ⊢ (𝜑 → (iEdg‘𝑍) = ((iEdg‘𝑋) ∪ (iEdg‘𝑌)))
60	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem6 16319	. 2 ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin)
61	11, 12, 25, 31, 13, 27, 5, 4, 7, 18, 24, 37	trlsegvdeglem7 16320	. 2 ⊢ (𝜑 → dom (iEdg‘𝑌) ∈ Fin)
62	1, 2, 3, 6, 8, 10, 20, 36, 54, 55, 56, 14, 59, 60, 61	vtxdfifiun 16151	1 ⊢ (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  Vcvv 2802   ∪ cun 3198   ∩ cin 3199   ⊆ wss 3200  ∅c0 3494  ifcif 3605  {csn 3669  ⟨cop 3672   class class class wbr 4088   × 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 6018  1^st c1st 6301  Fincfn 6909  0cc0 8032   + caddc 8035  ℤ_≥cuz 9755  ...cfz 10243  ..^cfzo 10377  ♯chash 11038  Basecbs 13084  Vtxcvtx 15866  iEdgciedg 15867  UPGraphcupgr 15945  VtxDegcvtxdg 16140  Trailsctrls 16234

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 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

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 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-2o 6583  df-oadd 6586  df-er 6702  df-map 6819  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-z 9480  df-dec 9612  df-uz 9756  df-xadd 10008  df-fz 10244  df-fzo 10378  df-ihash 11039  df-word 11115  df-ndx 13087  df-slot 13088  df-base 13090  df-edgf 15859  df-vtx 15868  df-iedg 15869  df-edg 15912  df-uhgrm 15923  df-upgren 15947  df-subgr 16108  df-vtxdg 16141  df-wlks 16172  df-trls 16235

This theorem is referenced by:  eupth2lem3lem7fi  16328