Theorem vtxdg0e 29451

Description: The degree of a vertex in an empty graph is zero, because there are no edges. This is the base case for the induction for calculating the degree of a vertex, for example in a Königsberg graph (see also the induction steps vdegp1ai 29513, vdegp1bi 29514 and vdegp1ci 29515). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.) (Revised by AV, 22-Mar-2021.)

Hypotheses

Ref	Expression
vtxdgf.v	⊢ 𝑉 = (Vtx‘𝐺)
vtxdg0e.i	⊢ 𝐼 = (iEdg‘𝐺)

Assertion

Ref	Expression
vtxdg0e	⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)

Proof of Theorem vtxdg0e

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vtxdg0e.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
2	1	eqeq1i 2736	. . . 4 ⊢ (𝐼 = ∅ ↔ (iEdg‘𝐺) = ∅)
3		dmeq 5843	. . . . . 6 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅)
4		dm0 5860	. . . . . 6 ⊢ dom ∅ = ∅
5	3, 4	eqtrdi 2782	. . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = ∅)
6		0fi 8964	. . . . 5 ⊢ ∅ ∈ Fin
7	5, 6	eqeltrdi 2839	. . . 4 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) ∈ Fin)
8	2, 7	sylbi 217	. . 3 ⊢ (𝐼 = ∅ → dom (iEdg‘𝐺) ∈ Fin)
9		simpl 482	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → 𝑈 ∈ 𝑉)
10		vtxdgf.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
11		eqid 2731	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
12		eqid 2731	. . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
13	10, 11, 12	vtxdgfival 29446	. . 3 ⊢ ((dom (iEdg‘𝐺) ∈ Fin ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) + (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})))
14	8, 9, 13	syl2an2 686	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) + (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})))
15	2, 5	sylbi 217	. . . . 5 ⊢ (𝐼 = ∅ → dom (iEdg‘𝐺) = ∅)
16	15	adantl 481	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → dom (iEdg‘𝐺) = ∅)
17		rabeq 3409	. . . . . . . 8 ⊢ (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)})
18		rab0 4336	. . . . . . . 8 ⊢ {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅
19	17, 18	eqtrdi 2782	. . . . . . 7 ⊢ (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅)
20	19	fveq2d 6826	. . . . . 6 ⊢ (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = (♯‘∅))
21		hash0 14271	. . . . . 6 ⊢ (♯‘∅) = 0
22	20, 21	eqtrdi 2782	. . . . 5 ⊢ (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = 0)
23		rabeq 3409	. . . . . . 7 ⊢ (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})
24	23	fveq2d 6826	. . . . . 6 ⊢ (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}))
25		rab0 4336	. . . . . . . 8 ⊢ {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = ∅
26	25	fveq2i 6825	. . . . . . 7 ⊢ (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (♯‘∅)
27	26, 21	eqtri 2754	. . . . . 6 ⊢ (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0
28	24, 27	eqtrdi 2782	. . . . 5 ⊢ (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0)
29	22, 28	oveq12d 7364	. . . 4 ⊢ (dom (iEdg‘𝐺) = ∅ → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) + (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})) = (0 + 0))
30	16, 29	syl 17	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) + (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})) = (0 + 0))
31		00id 11285	. . 3 ⊢ (0 + 0) = 0
32	30, 31	eqtrdi 2782	. 2 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → ((♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) + (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})) = 0)
33	14, 32	eqtrd 2766	1 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395  ∅c0 4283  {csn 4576  dom cdm 5616  ‘cfv 6481  (class class class)co 7346  Fincfn 8869  0cc0 11003   + caddc 11006  ♯chash 14234  Vtxcvtx 28972  iEdgciedg 28973  VtxDegcvtxdg 29442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-card 9829  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-n0 12379  df-z 12466  df-uz 12730  df-xadd 13009  df-fz 13405  df-hash 14235  df-vtxdg 29443

This theorem is referenced by:  vtxduhgr0e  29455  0edg0rgr  29549  eupth2lemb  30212  konigsberglem1  30227  konigsberglem2  30228  konigsberglem3  30229