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Mirrors > Home > MPE Home > Th. List > frgrwopreglem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for frgrwopreg 30120. If the set π΄ of vertices of degree πΎ is not empty in a friendship graph with at least two vertices, then πΎ must be greater than 1 . This is only an observation, which is not required for the proof the friendship theorem. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 2-Jan-2022.) |
Ref | Expression |
---|---|
frgrwopreg.v | β’ π = (VtxβπΊ) |
frgrwopreg.d | β’ π· = (VtxDegβπΊ) |
frgrwopreg.a | β’ π΄ = {π₯ β π β£ (π·βπ₯) = πΎ} |
frgrwopreg.b | β’ π΅ = (π β π΄) |
Ref | Expression |
---|---|
frgrwopreglem2 | β’ ((πΊ β FriendGraph β§ 1 < (β―βπ) β§ π΄ β β ) β 2 β€ πΎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4342 | . . 3 β’ (π΄ β β β βπ₯ π₯ β π΄) | |
2 | frgrwopreg.a | . . . . . 6 β’ π΄ = {π₯ β π β£ (π·βπ₯) = πΎ} | |
3 | 2 | reqabi 3449 | . . . . 5 β’ (π₯ β π΄ β (π₯ β π β§ (π·βπ₯) = πΎ)) |
4 | frgrwopreg.v | . . . . . . . . . . 11 β’ π = (VtxβπΊ) | |
5 | 4 | vdgfrgrgt2 30095 | . . . . . . . . . 10 β’ ((πΊ β FriendGraph β§ π₯ β π) β (1 < (β―βπ) β 2 β€ ((VtxDegβπΊ)βπ₯))) |
6 | 5 | imp 406 | . . . . . . . . 9 β’ (((πΊ β FriendGraph β§ π₯ β π) β§ 1 < (β―βπ)) β 2 β€ ((VtxDegβπΊ)βπ₯)) |
7 | breq2 5146 | . . . . . . . . . . 11 β’ (πΎ = (π·βπ₯) β (2 β€ πΎ β 2 β€ (π·βπ₯))) | |
8 | frgrwopreg.d | . . . . . . . . . . . . 13 β’ π· = (VtxDegβπΊ) | |
9 | 8 | fveq1i 6892 | . . . . . . . . . . . 12 β’ (π·βπ₯) = ((VtxDegβπΊ)βπ₯) |
10 | 9 | breq2i 5150 | . . . . . . . . . . 11 β’ (2 β€ (π·βπ₯) β 2 β€ ((VtxDegβπΊ)βπ₯)) |
11 | 7, 10 | bitrdi 287 | . . . . . . . . . 10 β’ (πΎ = (π·βπ₯) β (2 β€ πΎ β 2 β€ ((VtxDegβπΊ)βπ₯))) |
12 | 11 | eqcoms 2735 | . . . . . . . . 9 β’ ((π·βπ₯) = πΎ β (2 β€ πΎ β 2 β€ ((VtxDegβπΊ)βπ₯))) |
13 | 6, 12 | syl5ibrcom 246 | . . . . . . . 8 β’ (((πΊ β FriendGraph β§ π₯ β π) β§ 1 < (β―βπ)) β ((π·βπ₯) = πΎ β 2 β€ πΎ)) |
14 | 13 | exp31 419 | . . . . . . 7 β’ (πΊ β FriendGraph β (π₯ β π β (1 < (β―βπ) β ((π·βπ₯) = πΎ β 2 β€ πΎ)))) |
15 | 14 | com14 96 | . . . . . 6 β’ ((π·βπ₯) = πΎ β (π₯ β π β (1 < (β―βπ) β (πΊ β FriendGraph β 2 β€ πΎ)))) |
16 | 15 | impcom 407 | . . . . 5 β’ ((π₯ β π β§ (π·βπ₯) = πΎ) β (1 < (β―βπ) β (πΊ β FriendGraph β 2 β€ πΎ))) |
17 | 3, 16 | sylbi 216 | . . . 4 β’ (π₯ β π΄ β (1 < (β―βπ) β (πΊ β FriendGraph β 2 β€ πΎ))) |
18 | 17 | exlimiv 1926 | . . 3 β’ (βπ₯ π₯ β π΄ β (1 < (β―βπ) β (πΊ β FriendGraph β 2 β€ πΎ))) |
19 | 1, 18 | sylbi 216 | . 2 β’ (π΄ β β β (1 < (β―βπ) β (πΊ β FriendGraph β 2 β€ πΎ))) |
20 | 19 | 3imp31 1110 | 1 β’ ((πΊ β FriendGraph β§ 1 < (β―βπ) β§ π΄ β β ) β 2 β€ πΎ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1085 = wceq 1534 βwex 1774 β wcel 2099 β wne 2935 {crab 3427 β cdif 3941 β c0 4318 class class class wbr 5142 βcfv 6542 1c1 11131 < clt 11270 β€ cle 11271 2c2 12289 β―chash 14313 Vtxcvtx 28796 VtxDegcvtxdg 29266 FriendGraph cfrgr 30055 |
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-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 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ifp 1062 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-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 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-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-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-oadd 8484 df-er 8718 df-map 8838 df-pm 8839 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 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-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-xnn0 12567 df-z 12581 df-uz 12845 df-xadd 13117 df-fz 13509 df-fzo 13652 df-hash 14314 df-word 14489 df-concat 14545 df-s1 14570 df-s2 14823 df-s3 14824 df-edg 28848 df-uhgr 28858 df-upgr 28882 df-umgr 28883 df-uspgr 28950 df-usgr 28951 df-vtxdg 29267 df-wlks 29400 df-wlkson 29401 df-trls 29493 df-trlson 29494 df-pths 29517 df-spths 29518 df-pthson 29519 df-spthson 29520 df-conngr 29984 df-frgr 30056 |
This theorem is referenced by: (None) |
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