Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > frgrregorufrg | 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 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". Variant of frgrregorufr 41490 with generalization. (Contributed by Alexander van der Vekens, 6-Sep-2018.) (Revised by AV, 26-May-2021.) |
Ref | Expression |
---|---|
frrusgrord0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
frgrregorufrg.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
frgrregorufrg | ⊢ (𝐺 ∈ FriendGraph → ∀𝑘 ∈ ℕ0 (∃𝑎 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frrusgrord0.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | frgrregorufrg.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
3 | eqid 2610 | . . . . 5 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺) | |
4 | 1, 2, 3 | frgrregorufr 41490 | . . . 4 ⊢ (𝐺 ∈ FriendGraph → (∃𝑎 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) |
5 | 4 | adantr 480 | . . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (∃𝑎 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) |
6 | frgrusgr 41432 | . . . . . . . 8 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph ) | |
7 | 6 | adantr 480 | . . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → 𝐺 ∈ USGraph ) |
8 | 7 | adantr 480 | . . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → 𝐺 ∈ USGraph ) |
9 | nn0xnn0 11244 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℕ0*) | |
10 | 9 | adantl 481 | . . . . . . . 8 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0*) |
11 | 10 | anim1i 590 | . . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘)) |
12 | 1, 3 | isrgr 40759 | . . . . . . . 8 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (𝐺 RegGraph 𝑘 ↔ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘))) |
13 | 12 | adantr 480 | . . . . . . 7 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → (𝐺 RegGraph 𝑘 ↔ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘))) |
14 | 11, 13 | mpbird 246 | . . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → 𝐺 RegGraph 𝑘) |
15 | isrusgr 40761 | . . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (𝐺 RegUSGraph 𝑘 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝑘))) | |
16 | 15 | adantr 480 | . . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → (𝐺 RegUSGraph 𝑘 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝑘))) |
17 | 8, 14, 16 | mpbir2and 959 | . . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → 𝐺 RegUSGraph 𝑘) |
18 | 17 | ex 449 | . . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘 → 𝐺 RegUSGraph 𝑘)) |
19 | 18 | orim1d 880 | . . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → ((∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸) → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) |
20 | 5, 19 | syld 46 | . 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (∃𝑎 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) |
21 | 20 | ralrimiva 2949 | 1 ⊢ (𝐺 ∈ FriendGraph → ∀𝑘 ∈ ℕ0 (∃𝑎 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ∖ cdif 3537 {csn 4125 {cpr 4127 class class class wbr 4583 ‘cfv 5804 ℕ0cn0 11169 ℕ0*cxnn0 11240 Vtxcvtx 25673 Edgcedga 25792 USGraph cusgr 40379 VtxDegcvtxdg 40681 RegGraph crgr 40755 RegUSGraph crusgr 40756 FriendGraph cfrgr 41428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-xadd 11823 df-fz 12198 df-hash 12980 df-uhgr 25724 df-ushgr 25725 df-upgr 25749 df-umgr 25750 df-edga 25793 df-uspgr 40380 df-usgr 40381 df-nbgr 40554 df-vtxdg 40682 df-rgr 40757 df-rusgr 40758 df-frgr 41429 |
This theorem is referenced by: av-friendshipgt3 41552 |
Copyright terms: Public domain | W3C validator |