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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 28109 | . 2 ⊢ (𝐺 ∈ FriendGraph → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) |
5 | orc 864 | . . . 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 6645 | . . . . . . . 8 ⊢ (𝑣 = 𝑎 → (𝐷‘𝑣) = (𝐷‘𝑎)) | |
8 | 7 | neeq1d 3046 | . . . . . . 7 ⊢ (𝑣 = 𝑎 → ((𝐷‘𝑣) ≠ 𝐾 ↔ (𝐷‘𝑎) ≠ 𝐾)) |
9 | 8 | rspcva 3569 | . . . . . 6 ⊢ ((𝑎 ∈ 𝑉 ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾) → (𝐷‘𝑎) ≠ 𝐾) |
10 | df-ne 2988 | . . . . . . 7 ⊢ ((𝐷‘𝑎) ≠ 𝐾 ↔ ¬ (𝐷‘𝑎) = 𝐾) | |
11 | pm2.21 123 | . . . . . . 7 ⊢ (¬ (𝐷‘𝑎) = 𝐾 → ((𝐷‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) | |
12 | 10, 11 | sylbi 220 | . . . . . 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 462 | . . . 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 3243 | . . 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 865 | . . . 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 1424 | . 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 ∧ wa 399 ∨ wo 844 ∨ w3o 1083 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 ∖ cdif 3878 {csn 4525 {cpr 4527 ‘cfv 6324 Vtxcvtx 26789 Edgcedg 26840 VtxDegcvtxdg 27255 FriendGraph cfrgr 28043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-xadd 12496 df-fz 12886 df-hash 13687 df-edg 26841 df-uhgr 26851 df-ushgr 26852 df-upgr 26875 df-umgr 26876 df-uspgr 26943 df-usgr 26944 df-nbgr 27123 df-vtxdg 27256 df-frgr 28044 |
This theorem is referenced by: frgrregorufrg 28111 |
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