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Mirrors > Home > MPE Home > Th. List > rusgrpropnb | Structured version Visualization version GIF version |
Description: The properties of a k-regular simple graph expressed with neighbors. (Contributed by Alexander van der Vekens, 26-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
Ref | Expression |
---|---|
rusgrpropnb.v | β’ π = (VtxβπΊ) |
Ref | Expression |
---|---|
rusgrpropnb | β’ (πΊ RegUSGraph πΎ β (πΊ β USGraph β§ πΎ β β0* β§ βπ£ β π (β―β(πΊ NeighbVtx π£)) = πΎ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rusgrpropnb.v | . . 3 β’ π = (VtxβπΊ) | |
2 | eqid 2736 | . . 3 β’ (VtxDegβπΊ) = (VtxDegβπΊ) | |
3 | 1, 2 | rusgrprop0 28515 | . 2 β’ (πΊ RegUSGraph πΎ β (πΊ β USGraph β§ πΎ β β0* β§ βπ£ β π ((VtxDegβπΊ)βπ£) = πΎ)) |
4 | simp1 1136 | . . 3 β’ ((πΊ β USGraph β§ πΎ β β0* β§ βπ£ β π ((VtxDegβπΊ)βπ£) = πΎ) β πΊ β USGraph) | |
5 | simp2 1137 | . . 3 β’ ((πΊ β USGraph β§ πΎ β β0* β§ βπ£ β π ((VtxDegβπΊ)βπ£) = πΎ) β πΎ β β0*) | |
6 | 1 | hashnbusgrvd 28476 | . . . . . . 7 β’ ((πΊ β USGraph β§ π£ β π) β (β―β(πΊ NeighbVtx π£)) = ((VtxDegβπΊ)βπ£)) |
7 | 6 | adantlr 713 | . . . . . 6 β’ (((πΊ β USGraph β§ πΎ β β0*) β§ π£ β π) β (β―β(πΊ NeighbVtx π£)) = ((VtxDegβπΊ)βπ£)) |
8 | eqeq2 2748 | . . . . . . 7 β’ (πΎ = ((VtxDegβπΊ)βπ£) β ((β―β(πΊ NeighbVtx π£)) = πΎ β (β―β(πΊ NeighbVtx π£)) = ((VtxDegβπΊ)βπ£))) | |
9 | 8 | eqcoms 2744 | . . . . . 6 β’ (((VtxDegβπΊ)βπ£) = πΎ β ((β―β(πΊ NeighbVtx π£)) = πΎ β (β―β(πΊ NeighbVtx π£)) = ((VtxDegβπΊ)βπ£))) |
10 | 7, 9 | syl5ibrcom 246 | . . . . 5 β’ (((πΊ β USGraph β§ πΎ β β0*) β§ π£ β π) β (((VtxDegβπΊ)βπ£) = πΎ β (β―β(πΊ NeighbVtx π£)) = πΎ)) |
11 | 10 | ralimdva 3164 | . . . 4 β’ ((πΊ β USGraph β§ πΎ β β0*) β (βπ£ β π ((VtxDegβπΊ)βπ£) = πΎ β βπ£ β π (β―β(πΊ NeighbVtx π£)) = πΎ)) |
12 | 11 | 3impia 1117 | . . 3 β’ ((πΊ β USGraph β§ πΎ β β0* β§ βπ£ β π ((VtxDegβπΊ)βπ£) = πΎ) β βπ£ β π (β―β(πΊ NeighbVtx π£)) = πΎ) |
13 | 4, 5, 12 | 3jca 1128 | . 2 β’ ((πΊ β USGraph β§ πΎ β β0* β§ βπ£ β π ((VtxDegβπΊ)βπ£) = πΎ) β (πΊ β USGraph β§ πΎ β β0* β§ βπ£ β π (β―β(πΊ NeighbVtx π£)) = πΎ)) |
14 | 3, 13 | syl 17 | 1 β’ (πΊ RegUSGraph πΎ β (πΊ β USGraph β§ πΎ β β0* β§ βπ£ β π (β―β(πΊ NeighbVtx π£)) = πΎ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βwral 3064 class class class wbr 5105 βcfv 6496 (class class class)co 7357 β0*cxnn0 12485 β―chash 14230 Vtxcvtx 27947 USGraphcusgr 28100 NeighbVtx cnbgr 28280 VtxDegcvtxdg 28413 RegUSGraph crusgr 28504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-n0 12414 df-xnn0 12486 df-z 12500 df-uz 12764 df-xadd 13034 df-fz 13425 df-hash 14231 df-edg 27999 df-uhgr 28009 df-ushgr 28010 df-upgr 28033 df-umgr 28034 df-uspgr 28101 df-usgr 28102 df-nbgr 28281 df-vtxdg 28414 df-rgr 28505 df-rusgr 28506 |
This theorem is referenced by: rusgrpropedg 28532 rusgrpropadjvtx 28533 rusgr1vtx 28536 numclwwlk1 29305 |
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