Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 3cyclfrgr | Structured version Visualization version GIF version |
Description: Every vertex in a friendship graph (with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 19-Nov-2017.) (Revised by AV, 2-Apr-2021.) |
Ref | Expression |
---|---|
3cyclfrgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
3cyclfrgr | ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → ∀𝑣 ∈ 𝑉 ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝑣)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3cyclfrgr.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | eqid 2737 | . . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | 3cyclfrgrrn 28366 | . 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → ∀𝑣 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑣, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑣} ∈ (Edg‘𝐺))) |
4 | frgrusgr 28341 | . . . . . . . 8 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph) | |
5 | usgrumgr 27267 | . . . . . . . 8 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ UMGraph) |
7 | 6 | ad4antr 732 | . . . . . 6 ⊢ (((((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) ∧ 𝑣 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ ({𝑣, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑣} ∈ (Edg‘𝐺))) → 𝐺 ∈ UMGraph) |
8 | simpr 488 | . . . . . . . . 9 ⊢ (((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) | |
9 | 8 | anim1i 618 | . . . . . . . 8 ⊢ ((((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) ∧ 𝑣 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑣 ∈ 𝑉 ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) |
10 | 3anass 1097 | . . . . . . . 8 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ↔ (𝑣 ∈ 𝑉 ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) | |
11 | 9, 10 | sylibr 237 | . . . . . . 7 ⊢ ((((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) ∧ 𝑣 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑣 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) |
12 | 11 | adantr 484 | . . . . . 6 ⊢ (((((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) ∧ 𝑣 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ ({𝑣, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑣} ∈ (Edg‘𝐺))) → (𝑣 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) |
13 | simpr 488 | . . . . . 6 ⊢ (((((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) ∧ 𝑣 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ ({𝑣, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑣} ∈ (Edg‘𝐺))) → ({𝑣, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑣} ∈ (Edg‘𝐺))) | |
14 | 1, 2 | umgr3cyclex 28263 | . . . . . 6 ⊢ ((𝐺 ∈ UMGraph ∧ (𝑣 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ ({𝑣, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑣} ∈ (Edg‘𝐺))) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝑣)) |
15 | 7, 12, 13, 14 | syl3anc 1373 | . . . . 5 ⊢ (((((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) ∧ 𝑣 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ ({𝑣, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑣} ∈ (Edg‘𝐺))) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝑣)) |
16 | 15 | ex 416 | . . . 4 ⊢ ((((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) ∧ 𝑣 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (({𝑣, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑣} ∈ (Edg‘𝐺)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝑣))) |
17 | 16 | rexlimdvva 3210 | . . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) ∧ 𝑣 ∈ 𝑉) → (∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑣, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑣} ∈ (Edg‘𝐺)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝑣))) |
18 | 17 | ralimdva 3097 | . 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → (∀𝑣 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑣, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑣} ∈ (Edg‘𝐺)) → ∀𝑣 ∈ 𝑉 ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝑣))) |
19 | 3, 18 | mpd 15 | 1 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → ∀𝑣 ∈ 𝑉 ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝑣)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∃wex 1787 ∈ wcel 2110 ∀wral 3058 ∃wrex 3059 {cpr 4540 class class class wbr 5050 ‘cfv 6377 0cc0 10726 1c1 10727 < clt 10864 3c3 11883 ♯chash 13893 Vtxcvtx 27084 Edgcedg 27135 UMGraphcumgr 27169 USGraphcusgr 27237 Cyclesccycls 27869 FriendGraph cfrgr 28338 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ifp 1064 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-int 4857 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-1st 7758 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-1o 8199 df-oadd 8203 df-er 8388 df-map 8507 df-en 8624 df-dom 8625 df-sdom 8626 df-fin 8627 df-dju 9514 df-card 9552 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-nn 11828 df-2 11890 df-3 11891 df-4 11892 df-n0 12088 df-xnn0 12160 df-z 12174 df-uz 12436 df-fz 13093 df-fzo 13236 df-hash 13894 df-word 14067 df-concat 14123 df-s1 14150 df-s2 14410 df-s3 14411 df-s4 14412 df-edg 27136 df-uhgr 27146 df-upgr 27170 df-umgr 27171 df-usgr 27239 df-wlks 27684 df-trls 27777 df-pths 27800 df-cycls 27871 df-frgr 28339 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |