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Mirrors > Home > MPE Home > Th. List > frgrregorufr | Structured version Visualization version GIF version |
Description: If there is a vertex having degree 𝐾 for each (nonnegative integer) 𝐾 in a friendship graph, then either all vertices have degree 𝐾 or there is a universal friend. This corresponds to claim 2 in [Huneke] p. 2: "Suppose there is a vertex of degree k > 1. ... all vertices have degree k, unless there is a universal friend. ... It follows that G is k-regular, i.e., the degree of every vertex is k". (Contributed by Alexander van der Vekens, 1-Jan-2018.) |
Ref | Expression |
---|---|
frgrregorufr0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
frgrregorufr0.e | ⊢ 𝐸 = (Edg‘𝐺) |
frgrregorufr0.d | ⊢ 𝐷 = (VtxDeg‘𝐺) |
Ref | Expression |
---|---|
frgrregorufr | ⊢ (𝐺 ∈ FriendGraph → (∃𝑎 ∈ 𝑉 (𝐷‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrregorufr0.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | frgrregorufr0.e | . . 3 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | frgrregorufr0.d | . . 3 ⊢ 𝐷 = (VtxDeg‘𝐺) | |
4 | 1, 2, 3 | frgrregorufr0 27188 | . 2 ⊢ (𝐺 ∈ FriendGraph → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) |
5 | orc 400 | . . . 4 ⊢ (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) | |
6 | 5 | a1d 25 | . . 3 ⊢ (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 → (∃𝑎 ∈ 𝑉 (𝐷‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) |
7 | fveq2 6191 | . . . . . . . 8 ⊢ (𝑣 = 𝑎 → (𝐷‘𝑣) = (𝐷‘𝑎)) | |
8 | 7 | neeq1d 2853 | . . . . . . 7 ⊢ (𝑣 = 𝑎 → ((𝐷‘𝑣) ≠ 𝐾 ↔ (𝐷‘𝑎) ≠ 𝐾)) |
9 | 8 | rspcva 3307 | . . . . . 6 ⊢ ((𝑎 ∈ 𝑉 ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾) → (𝐷‘𝑎) ≠ 𝐾) |
10 | df-ne 2795 | . . . . . . 7 ⊢ ((𝐷‘𝑎) ≠ 𝐾 ↔ ¬ (𝐷‘𝑎) = 𝐾) | |
11 | pm2.21 120 | . . . . . . 7 ⊢ (¬ (𝐷‘𝑎) = 𝐾 → ((𝐷‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) | |
12 | 10, 11 | sylbi 207 | . . . . . 6 ⊢ ((𝐷‘𝑎) ≠ 𝐾 → ((𝐷‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) |
13 | 9, 12 | syl 17 | . . . . 5 ⊢ ((𝑎 ∈ 𝑉 ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾) → ((𝐷‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) |
14 | 13 | ancoms 469 | . . . 4 ⊢ ((∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∧ 𝑎 ∈ 𝑉) → ((𝐷‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) |
15 | 14 | rexlimdva 3031 | . . 3 ⊢ (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 → (∃𝑎 ∈ 𝑉 (𝐷‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) |
16 | olc 399 | . . . 4 ⊢ (∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) | |
17 | 16 | a1d 25 | . . 3 ⊢ (∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸 → (∃𝑎 ∈ 𝑉 (𝐷‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) |
18 | 6, 15, 17 | 3jaoi 1391 | . 2 ⊢ ((∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸) → (∃𝑎 ∈ 𝑉 (𝐷‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) |
19 | 4, 18 | syl 17 | 1 ⊢ (𝐺 ∈ FriendGraph → (∃𝑎 ∈ 𝑉 (𝐷‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 383 ∧ wa 384 ∨ w3o 1036 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ∖ cdif 3571 {csn 4177 {cpr 4179 ‘cfv 5888 Vtxcvtx 25874 Edgcedg 25939 VtxDegcvtxdg 26361 FriendGraph cfrgr 27120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-xadd 11947 df-fz 12327 df-hash 13118 df-edg 25940 df-uhgr 25953 df-ushgr 25954 df-upgr 25977 df-umgr 25978 df-uspgr 26045 df-usgr 26046 df-nbgr 26228 df-vtxdg 26362 df-frgr 27121 |
This theorem is referenced by: frgrregorufrg 27190 |
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