Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > vtxdg0e | Structured version Visualization version GIF version |
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 26432, vdegp1bi 26433 and vdegp1ci 26434). (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.) |
Ref | Expression |
---|---|
vtxdgf.v | ⊢ 𝑉 = (Vtx‘𝐺) |
vtxdg0e.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
vtxdg0e | ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtxdg0e.i | . . . . 5 ⊢ 𝐼 = (iEdg‘𝐺) | |
2 | 1 | eqeq1i 2627 | . . . 4 ⊢ (𝐼 = ∅ ↔ (iEdg‘𝐺) = ∅) |
3 | dmeq 5324 | . . . . . 6 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅) | |
4 | dm0 5339 | . . . . . 6 ⊢ dom ∅ = ∅ | |
5 | 3, 4 | syl6eq 2672 | . . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = ∅) |
6 | 0fin 8188 | . . . . 5 ⊢ ∅ ∈ Fin | |
7 | 5, 6 | syl6eqel 2709 | . . . 4 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) ∈ Fin) |
8 | 2, 7 | sylbi 207 | . . 3 ⊢ (𝐼 = ∅ → dom (iEdg‘𝐺) ∈ Fin) |
9 | simpl 473 | . . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → 𝑈 ∈ 𝑉) | |
10 | vtxdgf.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
11 | eqid 2622 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
12 | eqid 2622 | . . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺) | |
13 | 10, 11, 12 | vtxdgfival 26365 | . . 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 875 | . 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 207 | . . . . 5 ⊢ (𝐼 = ∅ → dom (iEdg‘𝐺) = ∅) |
16 | 15 | adantl 482 | . . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → dom (iEdg‘𝐺) = ∅) |
17 | rabeq 3192 | . . . . . . . 8 ⊢ (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) | |
18 | rab0 3955 | . . . . . . . 8 ⊢ {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅ | |
19 | 17, 18 | syl6eq 2672 | . . . . . . 7 ⊢ (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅) |
20 | 19 | fveq2d 6195 | . . . . . 6 ⊢ (dom (iEdg‘𝐺) = ∅ → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = (#‘∅)) |
21 | hash0 13158 | . . . . . 6 ⊢ (#‘∅) = 0 | |
22 | 20, 21 | syl6eq 2672 | . . . . 5 ⊢ (dom (iEdg‘𝐺) = ∅ → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = 0) |
23 | rabeq 3192 | . . . . . . 7 ⊢ (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) | |
24 | 23 | fveq2d 6195 | . . . . . 6 ⊢ (dom (iEdg‘𝐺) = ∅ → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (#‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})) |
25 | rab0 3955 | . . . . . . . 8 ⊢ {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = ∅ | |
26 | 25 | fveq2i 6194 | . . . . . . 7 ⊢ (#‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (#‘∅) |
27 | 26, 21 | eqtri 2644 | . . . . . 6 ⊢ (#‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0 |
28 | 24, 27 | syl6eq 2672 | . . . . 5 ⊢ (dom (iEdg‘𝐺) = ∅ → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0) |
29 | 22, 28 | oveq12d 6668 | . . . 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 10211 | . . 3 ⊢ (0 + 0) = 0 | |
32 | 30, 31 | syl6eq 2672 | . 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 2656 | 1 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 ∅c0 3915 {csn 4177 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 0cc0 9936 + caddc 9939 #chash 13117 Vtxcvtx 25874 iEdgciedg 25875 VtxDegcvtxdg 26361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-xadd 11947 df-fz 12327 df-hash 13118 df-vtxdg 26362 |
This theorem is referenced by: vtxduhgr0e 26374 0edg0rgr 26468 eupth2lemb 27097 konigsberglem1 27114 konigsberglem2 27115 konigsberglem3 27116 |
Copyright terms: Public domain | W3C validator |