Theorem vtxdg0e 29455

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 29517, vdegp1bi 29518 and vdegp1ci 29519). (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 2738	. . . 4 ⊢ (𝐼 = ∅ ↔ (iEdg‘𝐺) = ∅)
3		dmeq 5847	. . . . . 6 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅)
4		dm0 5864	. . . . . 6 ⊢ dom ∅ = ∅
5	3, 4	eqtrdi 2784	. . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = ∅)
6		0fi 8971	. . . . 5 ⊢ ∅ ∈ Fin
7	5, 6	eqeltrdi 2841	. . . 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 2733	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
12		eqid 2733	. . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺)
13	10, 11, 12	vtxdgfival 29450	. . 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 3410	. . . . . . . 8 ⊢ (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)})
18		rab0 4335	. . . . . . . 8 ⊢ {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅
19	17, 18	eqtrdi 2784	. . . . . . 7 ⊢ (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅)
20	19	fveq2d 6832	. . . . . 6 ⊢ (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = (♯‘∅))
21		hash0 14276	. . . . . 6 ⊢ (♯‘∅) = 0
22	20, 21	eqtrdi 2784	. . . . 5 ⊢ (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = 0)
23		rabeq 3410	. . . . . . 7 ⊢ (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})
24	23	fveq2d 6832	. . . . . 6 ⊢ (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}))
25		rab0 4335	. . . . . . . 8 ⊢ {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = ∅
26	25	fveq2i 6831	. . . . . . 7 ⊢ (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (♯‘∅)
27	26, 21	eqtri 2756	. . . . . 6 ⊢ (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0
28	24, 27	eqtrdi 2784	. . . . 5 ⊢ (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0)
29	22, 28	oveq12d 7370	. . . 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 11295	. . 3 ⊢ (0 + 0) = 0
32	30, 31	eqtrdi 2784	. 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 2768	1 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3396  ∅c0 4282  {csn 4575  dom cdm 5619  ‘cfv 6486  (class class class)co 7352  Fincfn 8875  0cc0 11013   + caddc 11016  ♯chash 14239  Vtxcvtx 28976  iEdgciedg 28977  VtxDegcvtxdg 29446

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-n0 12389  df-z 12476  df-uz 12739  df-xadd 13014  df-fz 13410  df-hash 14240  df-vtxdg 29447

This theorem is referenced by:  vtxduhgr0e  29459  0edg0rgr  29553  eupth2lemb  30219  konigsberglem1  30234  konigsberglem2  30235  konigsberglem3  30236