Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 (see also vdegp1ai-av 40752, vdegp1ai-av 40752 and vdegp1ai-av 40752). (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 2615 | . . . 4 ⊢ (𝐼 = ∅ ↔ (iEdg‘𝐺) = ∅) |
3 | dmeq 5246 | . . . . . 6 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅) | |
4 | dm0 5260 | . . . . . 6 ⊢ dom ∅ = ∅ | |
5 | 3, 4 | syl6eq 2660 | . . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = ∅) |
6 | 0fin 8073 | . . . . 5 ⊢ ∅ ∈ Fin | |
7 | 5, 6 | syl6eqel 2696 | . . . 4 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) ∈ Fin) |
8 | 2, 7 | sylbi 206 | . . 3 ⊢ (𝐼 = ∅ → dom (iEdg‘𝐺) ∈ Fin) |
9 | simpl 472 | . . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → 𝑈 ∈ 𝑉) | |
10 | vtxdgf.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
11 | eqid 2610 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
12 | eqid 2610 | . . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺) | |
13 | 10, 11, 12 | vtxdgfival 40685 | . . 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 871 | . 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 206 | . . . . 5 ⊢ (𝐼 = ∅ → dom (iEdg‘𝐺) = ∅) |
16 | 15 | adantl 481 | . . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → dom (iEdg‘𝐺) = ∅) |
17 | rabeq 3166 | . . . . . . . 8 ⊢ (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) | |
18 | rab0 3909 | . . . . . . . 8 ⊢ {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅ | |
19 | 17, 18 | syl6eq 2660 | . . . . . . 7 ⊢ (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅) |
20 | 19 | fveq2d 6107 | . . . . . 6 ⊢ (dom (iEdg‘𝐺) = ∅ → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = (#‘∅)) |
21 | hash0 13019 | . . . . . 6 ⊢ (#‘∅) = 0 | |
22 | 20, 21 | syl6eq 2660 | . . . . 5 ⊢ (dom (iEdg‘𝐺) = ∅ → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = 0) |
23 | rabeq 3166 | . . . . . . 7 ⊢ (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) | |
24 | 23 | fveq2d 6107 | . . . . . 6 ⊢ (dom (iEdg‘𝐺) = ∅ → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (#‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})) |
25 | rab0 3909 | . . . . . . . 8 ⊢ {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = ∅ | |
26 | 25 | fveq2i 6106 | . . . . . . 7 ⊢ (#‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (#‘∅) |
27 | 26, 21 | eqtri 2632 | . . . . . 6 ⊢ (#‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0 |
28 | 24, 27 | syl6eq 2660 | . . . . 5 ⊢ (dom (iEdg‘𝐺) = ∅ → (#‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0) |
29 | 22, 28 | oveq12d 6567 | . . . 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 10090 | . . 3 ⊢ (0 + 0) = 0 | |
32 | 30, 31 | syl6eq 2660 | . 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 2644 | 1 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 ∅c0 3874 {csn 4125 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 0cc0 9815 + caddc 9818 #chash 12979 Vtxcvtx 25673 iEdgciedg 25674 VtxDegcvtxdg 40681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-xadd 11823 df-fz 12198 df-hash 12980 df-vtxdg 40682 |
This theorem is referenced by: vtxduhgr0e 40693 0edg0rgr 40772 eupth2lemb 41405 konigsberglem1 41422 konigsberglem2 41423 konigsberglem3 41424 |
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