Theorem vtxdg0e 26779

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 26841, vdegp1bi 26842 and vdegp1ci 26843). (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 2830	. . . 4 ⊢ (𝐼 = ∅ ↔ (iEdg‘𝐺) = ∅)
3		dmeq 5560	. . . . . 6 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅)
4		dm0 5575	. . . . . 6 ⊢ dom ∅ = ∅
5	3, 4	syl6eq 2877	. . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = ∅)
6		0fin 8463	. . . . 5 ⊢ ∅ ∈ Fin
7	5, 6	syl6eqel 2914	. . . 4 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) ∈ Fin)
8	2, 7	sylbi 209	. . 3 ⊢ (𝐼 = ∅ → dom (iEdg‘𝐺) ∈ Fin)
9		simpl 476	. . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → 𝑈 ∈ 𝑉)
10		vtxdgf.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
11		eqid 2825	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
12		eqid 2825	. . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
13	10, 11, 12	vtxdgfival 26774	. . 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 677	. 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 209	. . . . 5 ⊢ (𝐼 = ∅ → dom (iEdg‘𝐺) = ∅)
16	15	adantl 475	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → dom (iEdg‘𝐺) = ∅)
17		rabeq 3405	. . . . . . . 8 ⊢ (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)})
18		rab0 4187	. . . . . . . 8 ⊢ {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅
19	17, 18	syl6eq 2877	. . . . . . 7 ⊢ (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅)
20	19	fveq2d 6441	. . . . . 6 ⊢ (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = (♯‘∅))
21		hash0 13455	. . . . . 6 ⊢ (♯‘∅) = 0
22	20, 21	syl6eq 2877	. . . . 5 ⊢ (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = 0)
23		rabeq 3405	. . . . . . 7 ⊢ (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})
24	23	fveq2d 6441	. . . . . 6 ⊢ (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}))
25		rab0 4187	. . . . . . . 8 ⊢ {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = ∅
26	25	fveq2i 6440	. . . . . . 7 ⊢ (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (♯‘∅)
27	26, 21	eqtri 2849	. . . . . 6 ⊢ (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0
28	24, 27	syl6eq 2877	. . . . 5 ⊢ (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0)
29	22, 28	oveq12d 6928	. . . 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 10537	. . 3 ⊢ (0 + 0) = 0
32	30, 31	syl6eq 2877	. 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 2861	1 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1656   ∈ wcel 2164  {crab 3121  ∅c0 4146  {csn 4399  dom cdm 5346  ‘cfv 6127  (class class class)co 6910  Fincfn 8228  0cc0 10259   + caddc 10262  ♯chash 13417  Vtxcvtx 26301  iEdgciedg 26302  VtxDegcvtxdg 26770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-card 9085  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-n0 11626  df-z 11712  df-uz 11976  df-xadd 12240  df-fz 12627  df-hash 13418  df-vtxdg 26771

This theorem is referenced by:  vtxduhgr0e  26783  0edg0rgr  26877  eupth2lemb  27610  konigsberglem1  27627  konigsberglem2  27628  konigsberglem3  27629