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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 27807, using the definition RegUSGraph (df-rusgr 27027). (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 ∧ 𝐺RegUSGraph𝐾) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rusgrrgr 27032 | . . . . . . 7 ⊢ (𝐺RegUSGraph𝐾 → 𝐺RegGraph𝐾) | |
2 | frrusgrord0.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | eqid 2797 | . . . . . . . 8 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺) | |
4 | 2, 3 | rgrprop 27029 | . . . . . . 7 ⊢ (𝐺RegGraph𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾)) |
5 | 1, 4 | syl 17 | . . . . . 6 ⊢ (𝐺RegUSGraph𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾)) |
6 | 5 | simprd 496 | . . . . 5 ⊢ (𝐺RegUSGraph𝐾 → ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) |
7 | 2 | frrusgrord0 27807 | . . . . 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 ≠ ∅) → (𝐺RegUSGraph𝐾 → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) |
9 | 8 | 3expb 1113 | . . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → (𝐺RegUSGraph𝐾 → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) |
10 | 9 | expcom 414 | . 2 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝐺 ∈ FriendGraph → (𝐺RegUSGraph𝐾 → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1)))) |
11 | 10 | impd 411 | 1 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺RegUSGraph𝐾) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 ≠ wne 2986 ∀wral 3107 ∅c0 4217 class class class wbr 4968 ‘cfv 6232 (class class class)co 7023 Fincfn 8364 1c1 10391 + caddc 10393 · cmul 10395 − cmin 10723 ℕ0*cxnn0 11821 ♯chash 13544 Vtxcvtx 26468 VtxDegcvtxdg 26934 RegGraphcrgr 27024 RegUSGraphcrusgr 27025 FriendGraph cfrgr 27723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-inf2 8957 ax-ac2 9738 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-ifp 1056 df-3or 1081 df-3an 1082 df-tru 1528 df-fal 1538 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-disj 4937 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-2o 7961 df-oadd 7964 df-er 8146 df-map 8265 df-pm 8266 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-sup 8759 df-oi 8827 df-dju 9183 df-card 9221 df-ac 9395 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-n0 11752 df-xnn0 11822 df-z 11836 df-uz 12098 df-rp 12244 df-xadd 12362 df-fz 12747 df-fzo 12888 df-seq 13224 df-exp 13284 df-hash 13545 df-word 13712 df-concat 13773 df-s1 13798 df-s2 14050 df-s3 14051 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-clim 14683 df-sum 14881 df-vtx 26470 df-iedg 26471 df-edg 26520 df-uhgr 26530 df-ushgr 26531 df-upgr 26554 df-umgr 26555 df-uspgr 26622 df-usgr 26623 df-fusgr 26786 df-nbgr 26802 df-vtxdg 26935 df-rgr 27026 df-rusgr 27027 df-wlks 27068 df-wlkson 27069 df-trls 27160 df-trlson 27161 df-pths 27183 df-spths 27184 df-pthson 27185 df-spthson 27186 df-wwlks 27294 df-wwlksn 27295 df-wwlksnon 27296 df-wspthsn 27297 df-wspthsnon 27298 df-frgr 27724 |
This theorem is referenced by: numclwwlk7 27858 frgrregord013 27862 |
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