![]() |
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 2734 | . . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
3 | 1, 2 | 3cyclfrgrrn 30314 | . 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 30289 | . . . . . . . 8 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph) | |
5 | usgrumgr 29212 | . . . . . . . 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 484 | . . . . . . . . 9 ⊢ (((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) | |
9 | 8 | anim1i 615 | . . . . . . . 8 ⊢ ((((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) ∧ 𝑣 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑣 ∈ 𝑉 ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) |
10 | 3anass 1094 | . . . . . . . 8 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ↔ (𝑣 ∈ 𝑉 ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))) | |
11 | 9, 10 | sylibr 234 | . . . . . . 7 ⊢ ((((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) ∧ 𝑣 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑣 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) |
12 | 11 | adantr 480 | . . . . . 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 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, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑐} ∈ (Edg‘𝐺) ∧ {𝑐, 𝑣} ∈ (Edg‘𝐺))) | |
14 | 1, 2 | umgr3cyclex 30211 | . . . . . 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 1370 | . . . . 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 412 | . . . 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 3164 | . 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 395 ∧ w3a 1086 = wceq 1536 ∃wex 1775 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 {cpr 4632 class class class wbr 5147 ‘cfv 6562 0cc0 11152 1c1 11153 < clt 11292 3c3 12319 ♯chash 14365 Vtxcvtx 29027 Edgcedg 29078 UMGraphcumgr 29112 USGraphcusgr 29180 Cyclesccycls 29817 FriendGraph cfrgr 30286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-n0 12524 df-xnn0 12597 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-hash 14366 df-word 14549 df-concat 14605 df-s1 14630 df-s2 14883 df-s3 14884 df-s4 14885 df-edg 29079 df-uhgr 29089 df-upgr 29113 df-umgr 29114 df-usgr 29182 df-wlks 29631 df-trls 29724 df-pths 29748 df-cycls 29819 df-frgr 30287 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |