![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fusgrn0eqdrusgr | Structured version Visualization version GIF version |
Description: If all vertices in a nonempty finite simple graph have the same (finite) degree, the graph is k-regular. (Contributed by AV, 26-Dec-2020.) |
Ref | Expression |
---|---|
isrusgr0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isrusgr0.d | ⊢ 𝐷 = (VtxDeg‘𝐺) |
Ref | Expression |
---|---|
fusgrn0eqdrusgr | ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 → 𝐺 RegUSGraph 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fusgrusgr 28115 | . . . 4 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph) | |
2 | 1 | ad2antrr 724 | . . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) → 𝐺 ∈ USGraph) |
3 | isrusgr0.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | isrusgr0.d | . . . . . 6 ⊢ 𝐷 = (VtxDeg‘𝐺) | |
5 | 3, 4 | fusgrregdegfi 28362 | . . . . 5 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 → 𝐾 ∈ ℕ0)) |
6 | 5 | imp 407 | . . . 4 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) → 𝐾 ∈ ℕ0) |
7 | 6 | nn0xnn0d 12452 | . . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) → 𝐾 ∈ ℕ0*) |
8 | simpr 485 | . . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) → ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) | |
9 | 3, 4 | usgreqdrusgr 28361 | . . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) → 𝐺 RegUSGraph 𝐾) |
10 | 2, 7, 8, 9 | syl3anc 1371 | . 2 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) → 𝐺 RegUSGraph 𝐾) |
11 | 10 | ex 413 | 1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 → 𝐺 RegUSGraph 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∀wral 3062 ∅c0 4280 class class class wbr 5103 ‘cfv 6493 ℕ0cn0 12371 ℕ0*cxnn0 12443 Vtxcvtx 27792 USGraphcusgr 27945 FinUSGraphcfusgr 28109 VtxDegcvtxdg 28258 RegUSGraph crusgr 28349 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-oadd 8408 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-dju 9795 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-n0 12372 df-xnn0 12444 df-z 12458 df-uz 12722 df-xadd 12988 df-fz 13379 df-hash 14185 df-vtx 27794 df-iedg 27795 df-edg 27844 df-uhgr 27854 df-upgr 27878 df-umgr 27879 df-uspgr 27946 df-usgr 27947 df-fusgr 28110 df-vtxdg 28259 df-rgr 28350 df-rusgr 28351 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |