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Mirrors > Home > MPE Home > Th. List > vdegp1ci | Structured version Visualization version GIF version |
Description: The induction step for a vertex degree calculation, for example in the Königsberg graph. If the degree of 𝑈 in the edge set 𝐸 is 𝑃, then adding {𝑋, 𝑈} to the edge set, where 𝑋 ≠ 𝑈, yields degree 𝑃 + 1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.) |
Ref | Expression |
---|---|
vdegp1ai.vg | ⊢ 𝑉 = (Vtx‘𝐺) |
vdegp1ai.u | ⊢ 𝑈 ∈ 𝑉 |
vdegp1ai.i | ⊢ 𝐼 = (iEdg‘𝐺) |
vdegp1ai.w | ⊢ 𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} |
vdegp1ai.d | ⊢ ((VtxDeg‘𝐺)‘𝑈) = 𝑃 |
vdegp1ai.vf | ⊢ (Vtx‘𝐹) = 𝑉 |
vdegp1bi.x | ⊢ 𝑋 ∈ 𝑉 |
vdegp1bi.xu | ⊢ 𝑋 ≠ 𝑈 |
vdegp1ci.f | ⊢ (iEdg‘𝐹) = (𝐼 ++ 〈“{𝑋, 𝑈}”〉) |
Ref | Expression |
---|---|
vdegp1ci | ⊢ ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vdegp1ai.vg | . 2 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | vdegp1ai.u | . 2 ⊢ 𝑈 ∈ 𝑉 | |
3 | vdegp1ai.i | . 2 ⊢ 𝐼 = (iEdg‘𝐺) | |
4 | vdegp1ai.w | . 2 ⊢ 𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} | |
5 | vdegp1ai.d | . 2 ⊢ ((VtxDeg‘𝐺)‘𝑈) = 𝑃 | |
6 | vdegp1ai.vf | . 2 ⊢ (Vtx‘𝐹) = 𝑉 | |
7 | vdegp1bi.x | . 2 ⊢ 𝑋 ∈ 𝑉 | |
8 | vdegp1bi.xu | . 2 ⊢ 𝑋 ≠ 𝑈 | |
9 | vdegp1ci.f | . . 3 ⊢ (iEdg‘𝐹) = (𝐼 ++ 〈“{𝑋, 𝑈}”〉) | |
10 | prcom 4661 | . . . . 5 ⊢ {𝑋, 𝑈} = {𝑈, 𝑋} | |
11 | s1eq 13948 | . . . . 5 ⊢ ({𝑋, 𝑈} = {𝑈, 𝑋} → 〈“{𝑋, 𝑈}”〉 = 〈“{𝑈, 𝑋}”〉) | |
12 | 10, 11 | ax-mp 5 | . . . 4 ⊢ 〈“{𝑋, 𝑈}”〉 = 〈“{𝑈, 𝑋}”〉 |
13 | 12 | oveq2i 7161 | . . 3 ⊢ (𝐼 ++ 〈“{𝑋, 𝑈}”〉) = (𝐼 ++ 〈“{𝑈, 𝑋}”〉) |
14 | 9, 13 | eqtri 2844 | . 2 ⊢ (iEdg‘𝐹) = (𝐼 ++ 〈“{𝑈, 𝑋}”〉) |
15 | 1, 2, 3, 4, 5, 6, 7, 8, 14 | vdegp1bi 27313 | 1 ⊢ ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 ≠ wne 3016 {crab 3142 ∖ cdif 3932 ∅c0 4290 𝒫 cpw 4538 {csn 4560 {cpr 4562 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 1c1 10532 + caddc 10534 ≤ cle 10670 2c2 11686 ♯chash 13684 Word cword 13855 ++ cconcat 13916 〈“cs1 13943 Vtxcvtx 26775 iEdgciedg 26776 VtxDegcvtxdg 27241 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-xadd 12502 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 df-vtx 26777 df-iedg 26778 df-vtxdg 27242 |
This theorem is referenced by: konigsberglem2 28026 konigsberglem3 28027 |
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