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Mirrors > Home > MPE Home > Th. List > finsumvtxdgeven | Structured version Visualization version GIF version |
Description: The sum of the degrees of all vertices of a finite pseudograph of finite size is even. See equation (2) in section I.1 in [Bollobas] p. 4, where it is also called the handshaking lemma. (Contributed by AV, 22-Dec-2021.) |
Ref | Expression |
---|---|
finsumvtxdgeven.v | β’ π = (VtxβπΊ) |
finsumvtxdgeven.i | β’ πΌ = (iEdgβπΊ) |
finsumvtxdgeven.d | β’ π· = (VtxDegβπΊ) |
Ref | Expression |
---|---|
finsumvtxdgeven | β’ ((πΊ β UPGraph β§ π β Fin β§ πΌ β Fin) β 2 β₯ Ξ£π£ β π (π·βπ£)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashcl 14339 | . . . . 5 β’ (πΌ β Fin β (β―βπΌ) β β0) | |
2 | 1 | 3ad2ant3 1133 | . . . 4 β’ ((πΊ β UPGraph β§ π β Fin β§ πΌ β Fin) β (β―βπΌ) β β0) |
3 | 2 | nn0zd 12606 | . . 3 β’ ((πΊ β UPGraph β§ π β Fin β§ πΌ β Fin) β (β―βπΌ) β β€) |
4 | eqidd 2728 | . . 3 β’ ((πΊ β UPGraph β§ π β Fin β§ πΌ β Fin) β (2 Β· (β―βπΌ)) = (2 Β· (β―βπΌ))) | |
5 | 2teven 16323 | . . 3 β’ (((β―βπΌ) β β€ β§ (2 Β· (β―βπΌ)) = (2 Β· (β―βπΌ))) β 2 β₯ (2 Β· (β―βπΌ))) | |
6 | 3, 4, 5 | syl2anc 583 | . 2 β’ ((πΊ β UPGraph β§ π β Fin β§ πΌ β Fin) β 2 β₯ (2 Β· (β―βπΌ))) |
7 | finsumvtxdgeven.v | . . 3 β’ π = (VtxβπΊ) | |
8 | finsumvtxdgeven.i | . . 3 β’ πΌ = (iEdgβπΊ) | |
9 | finsumvtxdgeven.d | . . 3 β’ π· = (VtxDegβπΊ) | |
10 | 7, 8, 9 | finsumvtxdg2size 29351 | . 2 β’ ((πΊ β UPGraph β§ π β Fin β§ πΌ β Fin) β Ξ£π£ β π (π·βπ£) = (2 Β· (β―βπΌ))) |
11 | 6, 10 | breqtrrd 5170 | 1 β’ ((πΊ β UPGraph β§ π β Fin β§ πΌ β Fin) β 2 β₯ Ξ£π£ β π (π·βπ£)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1085 = wceq 1534 β wcel 2099 class class class wbr 5142 βcfv 6542 (class class class)co 7414 Fincfn 8955 Β· cmul 11135 2c2 12289 β0cn0 12494 β€cz 12580 β―chash 14313 Ξ£csu 15656 β₯ cdvds 16222 Vtxcvtx 28796 iEdgciedg 28797 UPGraphcupgr 28880 VtxDegcvtxdg 29266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-disj 5108 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9457 df-oi 9525 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-xnn0 12567 df-z 12581 df-uz 12845 df-rp 12999 df-xadd 13117 df-fz 13509 df-fzo 13652 df-seq 13991 df-exp 14051 df-hash 14314 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-clim 15456 df-sum 15657 df-dvds 16223 df-vtx 28798 df-iedg 28799 df-edg 28848 df-uhgr 28858 df-upgr 28882 df-vtxdg 29267 |
This theorem is referenced by: vtxdgoddnumeven 29354 |
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