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Mirrors > Home > MPE Home > Th. List > vdegp1ai | Structured version Visualization version GIF version |
Description: The induction step for a vertex degree calculation. If the degree of π in the edge set πΈ is π, then adding {π, π} to the edge set, where π β π β π, yields degree π as well. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.) |
Ref | Expression |
---|---|
vdegp1ai.vg | β’ π = (VtxβπΊ) |
vdegp1ai.u | β’ π β π |
vdegp1ai.i | β’ πΌ = (iEdgβπΊ) |
vdegp1ai.w | β’ πΌ β Word {π₯ β (π« π β {β }) β£ (β―βπ₯) β€ 2} |
vdegp1ai.d | β’ ((VtxDegβπΊ)βπ) = π |
vdegp1ai.vf | β’ (VtxβπΉ) = π |
vdegp1ai.x | β’ π β π |
vdegp1ai.xu | β’ π β π |
vdegp1ai.y | β’ π β π |
vdegp1ai.yu | β’ π β π |
vdegp1ai.f | β’ (iEdgβπΉ) = (πΌ ++ β¨β{π, π}ββ©) |
Ref | Expression |
---|---|
vdegp1ai | β’ ((VtxDegβπΉ)βπ) = π |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prex 5394 | . . 3 β’ {π, π} β V | |
2 | vdegp1ai.vg | . . . 4 β’ π = (VtxβπΊ) | |
3 | vdegp1ai.i | . . . 4 β’ πΌ = (iEdgβπΊ) | |
4 | vdegp1ai.w | . . . . 5 β’ πΌ β Word {π₯ β (π« π β {β }) β£ (β―βπ₯) β€ 2} | |
5 | wrdf 14414 | . . . . . 6 β’ (πΌ β Word {π₯ β (π« π β {β }) β£ (β―βπ₯) β€ 2} β πΌ:(0..^(β―βπΌ))βΆ{π₯ β (π« π β {β }) β£ (β―βπ₯) β€ 2}) | |
6 | 5 | ffund 6677 | . . . . 5 β’ (πΌ β Word {π₯ β (π« π β {β }) β£ (β―βπ₯) β€ 2} β Fun πΌ) |
7 | 4, 6 | mp1i 13 | . . . 4 β’ ({π, π} β V β Fun πΌ) |
8 | vdegp1ai.vf | . . . . 5 β’ (VtxβπΉ) = π | |
9 | 8 | a1i 11 | . . . 4 β’ ({π, π} β V β (VtxβπΉ) = π) |
10 | vdegp1ai.f | . . . . 5 β’ (iEdgβπΉ) = (πΌ ++ β¨β{π, π}ββ©) | |
11 | wrdv 14424 | . . . . . . 7 β’ (πΌ β Word {π₯ β (π« π β {β }) β£ (β―βπ₯) β€ 2} β πΌ β Word V) | |
12 | 4, 11 | ax-mp 5 | . . . . . 6 β’ πΌ β Word V |
13 | cats1un 14616 | . . . . . 6 β’ ((πΌ β Word V β§ {π, π} β V) β (πΌ ++ β¨β{π, π}ββ©) = (πΌ βͺ {β¨(β―βπΌ), {π, π}β©})) | |
14 | 12, 13 | mpan 689 | . . . . 5 β’ ({π, π} β V β (πΌ ++ β¨β{π, π}ββ©) = (πΌ βͺ {β¨(β―βπΌ), {π, π}β©})) |
15 | 10, 14 | eqtrid 2789 | . . . 4 β’ ({π, π} β V β (iEdgβπΉ) = (πΌ βͺ {β¨(β―βπΌ), {π, π}β©})) |
16 | fvexd 6862 | . . . 4 β’ ({π, π} β V β (β―βπΌ) β V) | |
17 | wrdlndm 14425 | . . . . 5 β’ (πΌ β Word {π₯ β (π« π β {β }) β£ (β―βπ₯) β€ 2} β (β―βπΌ) β dom πΌ) | |
18 | 4, 17 | mp1i 13 | . . . 4 β’ ({π, π} β V β (β―βπΌ) β dom πΌ) |
19 | vdegp1ai.u | . . . . 5 β’ π β π | |
20 | 19 | a1i 11 | . . . 4 β’ ({π, π} β V β π β π) |
21 | id 22 | . . . 4 β’ ({π, π} β V β {π, π} β V) | |
22 | vdegp1ai.xu | . . . . . . 7 β’ π β π | |
23 | 22 | necomi 2999 | . . . . . 6 β’ π β π |
24 | vdegp1ai.yu | . . . . . . 7 β’ π β π | |
25 | 24 | necomi 2999 | . . . . . 6 β’ π β π |
26 | 23, 25 | prneli 4621 | . . . . 5 β’ π β {π, π} |
27 | 26 | a1i 11 | . . . 4 β’ ({π, π} β V β π β {π, π}) |
28 | 2, 3, 7, 9, 15, 16, 18, 20, 21, 27 | p1evtxdeq 28503 | . . 3 β’ ({π, π} β V β ((VtxDegβπΉ)βπ) = ((VtxDegβπΊ)βπ)) |
29 | 1, 28 | ax-mp 5 | . 2 β’ ((VtxDegβπΉ)βπ) = ((VtxDegβπΊ)βπ) |
30 | vdegp1ai.d | . 2 β’ ((VtxDegβπΊ)βπ) = π | |
31 | 29, 30 | eqtri 2765 | 1 β’ ((VtxDegβπΉ)βπ) = π |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 β wcel 2107 β wne 2944 β wnel 3050 {crab 3410 Vcvv 3448 β cdif 3912 βͺ cun 3913 β c0 4287 π« cpw 4565 {csn 4591 {cpr 4593 β¨cop 4597 class class class wbr 5110 dom cdm 5638 Fun wfun 6495 βcfv 6501 (class class class)co 7362 0cc0 11058 β€ cle 11197 2c2 12215 ..^cfzo 13574 β―chash 14237 Word cword 14409 ++ cconcat 14465 β¨βcs1 14490 Vtxcvtx 27989 iEdgciedg 27990 VtxDegcvtxdg 28455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-oadd 8421 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-xnn0 12493 df-z 12507 df-uz 12771 df-xadd 13041 df-fz 13432 df-fzo 13575 df-hash 14238 df-word 14410 df-concat 14466 df-s1 14491 df-vtx 27991 df-iedg 27992 df-vtxdg 28456 |
This theorem is referenced by: konigsberglem1 29238 konigsberglem2 29239 konigsberglem3 29240 |
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