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Mirrors > Home > MPE Home > Th. List > frrusgrord | Structured version Visualization version GIF version |
Description: If a nonempty finite friendship graph is k-regular, its order is k(k-1)+1. This corresponds to claim 3 in [Huneke] p. 2: "Next we claim that the number n of vertices in G is exactly k(k-1)+1.". Variant of frrusgrord0 30270, using the definition RegUSGraph (df-rusgr 29492). (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 26-May-2021.) (Proof shortened by AV, 12-Jan-2022.) |
Ref | Expression |
---|---|
frrusgrord0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
frrusgrord | ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rusgrrgr 29497 | . . . . . . 7 ⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 RegGraph 𝐾) | |
2 | frrusgrord0.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | eqid 2726 | . . . . . . . 8 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺) | |
4 | 2, 3 | rgrprop 29494 | . . . . . . 7 ⊢ (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾)) |
5 | 1, 4 | syl 17 | . . . . . 6 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾)) |
6 | 5 | simprd 494 | . . . . 5 ⊢ (𝐺 RegUSGraph 𝐾 → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) |
7 | 2 | frrusgrord0 30270 | . . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) |
8 | 6, 7 | syl5 34 | . . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐺 RegUSGraph 𝐾 → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) |
9 | 8 | 3expb 1117 | . . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → (𝐺 RegUSGraph 𝐾 → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) |
10 | 9 | expcom 412 | . 2 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐺 ∈ FriendGraph → (𝐺 RegUSGraph 𝐾 → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1)))) |
11 | 10 | impd 409 | 1 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ∅c0 4322 class class class wbr 5145 ‘cfv 6546 (class class class)co 7416 Fincfn 8966 1c1 11150 + caddc 11152 · cmul 11154 − cmin 11485 ℕ0*cxnn0 12590 ♯chash 14342 Vtxcvtx 28929 VtxDegcvtxdg 29399 RegGraph crgr 29489 RegUSGraph crusgr 29490 FriendGraph cfrgr 30188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-inf2 9677 ax-ac2 10497 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-disj 5111 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-oadd 8492 df-er 8726 df-map 8849 df-pm 8850 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9478 df-oi 9546 df-dju 9937 df-card 9975 df-ac 10152 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-n0 12519 df-xnn0 12591 df-z 12605 df-uz 12869 df-rp 13023 df-xadd 13141 df-fz 13533 df-fzo 13676 df-seq 14016 df-exp 14076 df-hash 14343 df-word 14518 df-concat 14574 df-s1 14599 df-s2 14852 df-s3 14853 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-clim 15485 df-sum 15686 df-vtx 28931 df-iedg 28932 df-edg 28981 df-uhgr 28991 df-ushgr 28992 df-upgr 29015 df-umgr 29016 df-uspgr 29083 df-usgr 29084 df-fusgr 29250 df-nbgr 29266 df-vtxdg 29400 df-rgr 29491 df-rusgr 29492 df-wlks 29533 df-wlkson 29534 df-trls 29626 df-trlson 29627 df-pths 29650 df-spths 29651 df-pthson 29652 df-spthson 29653 df-wwlks 29761 df-wwlksn 29762 df-wwlksnon 29763 df-wspthsn 29764 df-wspthsnon 29765 df-frgr 30189 |
This theorem is referenced by: numclwwlk7 30321 frgrregord013 30325 |
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