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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 4674 | . . . . 5 ⊢ {𝑋, 𝑈} = {𝑈, 𝑋} | |
11 | s1eq 14316 | . . . . 5 ⊢ ({𝑋, 𝑈} = {𝑈, 𝑋} → 〈“{𝑋, 𝑈}”〉 = 〈“{𝑈, 𝑋}”〉) | |
12 | 10, 11 | ax-mp 5 | . . . 4 ⊢ 〈“{𝑋, 𝑈}”〉 = 〈“{𝑈, 𝑋}”〉 |
13 | 12 | oveq2i 7283 | . . 3 ⊢ (𝐼 ++ 〈“{𝑋, 𝑈}”〉) = (𝐼 ++ 〈“{𝑈, 𝑋}”〉) |
14 | 9, 13 | eqtri 2768 | . 2 ⊢ (iEdg‘𝐹) = (𝐼 ++ 〈“{𝑈, 𝑋}”〉) |
15 | 1, 2, 3, 4, 5, 6, 7, 8, 14 | vdegp1bi 27915 | 1 ⊢ ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2110 ≠ wne 2945 {crab 3070 ∖ cdif 3889 ∅c0 4262 𝒫 cpw 4539 {csn 4567 {cpr 4569 class class class wbr 5079 ‘cfv 6432 (class class class)co 7272 1c1 10883 + caddc 10885 ≤ cle 11021 2c2 12039 ♯chash 14055 Word cword 14228 ++ cconcat 14284 〈“cs1 14311 Vtxcvtx 27377 iEdgciedg 27378 VtxDegcvtxdg 27843 |
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 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-1st 7825 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-oadd 8293 df-er 8490 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-dju 9670 df-card 9708 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-nn 11985 df-2 12047 df-n0 12245 df-xnn0 12317 df-z 12331 df-uz 12594 df-xadd 12860 df-fz 13251 df-fzo 13394 df-hash 14056 df-word 14229 df-concat 14285 df-s1 14312 df-vtx 27379 df-iedg 27380 df-vtxdg 27844 |
This theorem is referenced by: konigsberglem2 28626 konigsberglem3 28627 |
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