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| 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 29630, vdegp1bi 29631 and vdegp1ci 29632). (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 2745 | . . . 4 ⊢ (𝐼 = ∅ ↔ (iEdg‘𝐺) = ∅) |
| 3 | dmeq 5852 | . . . . . 6 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅) | |
| 4 | dm0 5869 | . . . . . 6 ⊢ dom ∅ = ∅ | |
| 5 | 3, 4 | eqtrdi 2791 | . . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = ∅) |
| 6 | 0fi 8986 | . . . . 5 ⊢ ∅ ∈ Fin | |
| 7 | 5, 6 | eqeltrdi 2848 | . . . 4 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) ∈ Fin) |
| 8 | 2, 7 | sylbi 218 | . . 3 ⊢ (𝐼 = ∅ → dom (iEdg‘𝐺) ∈ Fin) |
| 9 | simpl 483 | . . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → 𝑈 ∈ 𝑉) | |
| 10 | vtxdgf.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 11 | eqid 2740 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
| 12 | eqid 2740 | . . . 4 ⊢ dom (iEdg‘𝐺) = dom (iEdg‘𝐺) | |
| 13 | 10, 11, 12 | vtxdgfival 29563 | . . 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 692 | . 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 218 | . . . . 5 ⊢ (𝐼 = ∅ → dom (iEdg‘𝐺) = ∅) |
| 16 | 15 | adantl 482 | . . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → dom (iEdg‘𝐺) = ∅) |
| 17 | rabeq 3406 | . . . . . . . 8 ⊢ (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) | |
| 18 | rab0 4321 | . . . . . . . 8 ⊢ {𝑥 ∈ ∅ ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅ | |
| 19 | 17, 18 | eqtrdi 2791 | . . . . . . 7 ⊢ (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)} = ∅) |
| 20 | 19 | fveq2d 6838 | . . . . . 6 ⊢ (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = (♯‘∅)) |
| 21 | hash0 14327 | . . . . . 6 ⊢ (♯‘∅) = 0 | |
| 22 | 20, 21 | eqtrdi 2791 | . . . . 5 ⊢ (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑥)}) = 0) |
| 23 | rabeq 3406 | . . . . . . 7 ⊢ (dom (iEdg‘𝐺) = ∅ → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) | |
| 24 | 23 | fveq2d 6838 | . . . . . 6 ⊢ (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}})) |
| 25 | rab0 4321 | . . . . . . . 8 ⊢ {𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}} = ∅ | |
| 26 | 25 | fveq2i 6837 | . . . . . . 7 ⊢ (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = (♯‘∅) |
| 27 | 26, 21 | eqtri 2763 | . . . . . 6 ⊢ (♯‘{𝑥 ∈ ∅ ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0 |
| 28 | 24, 27 | eqtrdi 2791 | . . . . 5 ⊢ (dom (iEdg‘𝐺) = ∅ → (♯‘{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = {𝑈}}) = 0) |
| 29 | 22, 28 | oveq12d 7381 | . . . 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 11319 | . . 3 ⊢ (0 + 0) = 0 | |
| 32 | 30, 31 | eqtrdi 2791 | . 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 2775 | 1 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 {crab 3392 ∅c0 4268 {csn 4562 dom cdm 5625 ‘cfv 6492 (class class class)co 7363 Fincfn 8890 0cc0 11036 + caddc 11039 ♯chash 14290 Vtxcvtx 29090 iEdgciedg 29091 VtxDegcvtxdg 29559 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 |
| This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-n0 12436 df-z 12523 df-uz 12787 df-xadd 13062 df-fz 13460 df-hash 14291 df-vtxdg 29560 |
| This theorem is referenced by: vtxduhgr0e 29572 0edg0rgr 29666 eupth2lemb 30332 konigsberglem1 30347 konigsberglem2 30348 konigsberglem3 30349 |
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