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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 4670 | . . . . 5 ⊢ {𝑋, 𝑈} = {𝑈, 𝑋} | |
11 | s1eq 13956 | . . . . 5 ⊢ ({𝑋, 𝑈} = {𝑈, 𝑋} → 〈“{𝑋, 𝑈}”〉 = 〈“{𝑈, 𝑋}”〉) | |
12 | 10, 11 | ax-mp 5 | . . . 4 ⊢ 〈“{𝑋, 𝑈}”〉 = 〈“{𝑈, 𝑋}”〉 |
13 | 12 | oveq2i 7169 | . . 3 ⊢ (𝐼 ++ 〈“{𝑋, 𝑈}”〉) = (𝐼 ++ 〈“{𝑈, 𝑋}”〉) |
14 | 9, 13 | eqtri 2846 | . 2 ⊢ (iEdg‘𝐹) = (𝐼 ++ 〈“{𝑈, 𝑋}”〉) |
15 | 1, 2, 3, 4, 5, 6, 7, 8, 14 | vdegp1bi 27321 | 1 ⊢ ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ≠ wne 3018 {crab 3144 ∖ cdif 3935 ∅c0 4293 𝒫 cpw 4541 {csn 4569 {cpr 4571 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 1c1 10540 + caddc 10542 ≤ cle 10678 2c2 11695 ♯chash 13693 Word cword 13864 ++ cconcat 13924 〈“cs1 13951 Vtxcvtx 26783 iEdgciedg 26784 VtxDegcvtxdg 27249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-xadd 12511 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13925 df-s1 13952 df-vtx 26785 df-iedg 26786 df-vtxdg 27250 |
This theorem is referenced by: konigsberglem2 28034 konigsberglem3 28035 |
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