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Mirrors > Home > MPE Home > Th. List > frrusgrord0 | 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.". (Contributed by Alexander van der Vekens, 11-Mar-2018.) (Revised by AV, 26-May-2021.) (Proof shortened by AV, 12-Jan-2022.) |
Ref | Expression |
---|---|
frrusgrord0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
frrusgrord0 | ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrusgr 30290 | . . . . . . 7 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph) | |
2 | 1 | anim1i 615 | . . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
3 | frrusgrord0.v | . . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | 3 | isfusgr 29350 | . . . . . 6 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
5 | 2, 4 | sylibr 234 | . . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph) |
6 | 3 | fusgreghash2wsp 30367 | . . . . 5 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(2 WSPathsN 𝐺)) = ((♯‘𝑉) · (𝐾 · (𝐾 − 1))))) |
7 | 5, 6 | stoic3 1773 | . . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(2 WSPathsN 𝐺)) = ((♯‘𝑉) · (𝐾 · (𝐾 − 1))))) |
8 | 7 | imp 406 | . . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (♯‘(2 WSPathsN 𝐺)) = ((♯‘𝑉) · (𝐾 · (𝐾 − 1)))) |
9 | 3 | frgrhash2wsp 30361 | . . . . . . . 8 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (♯‘(2 WSPathsN 𝐺)) = ((♯‘𝑉) · ((♯‘𝑉) − 1))) |
10 | 9 | eqcomd 2741 | . . . . . . 7 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → ((♯‘𝑉) · ((♯‘𝑉) − 1)) = (♯‘(2 WSPathsN 𝐺))) |
11 | 10 | eqeq1d 2737 | . . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (((♯‘𝑉) · ((♯‘𝑉) − 1)) = ((♯‘𝑉) · (𝐾 · (𝐾 − 1))) ↔ (♯‘(2 WSPathsN 𝐺)) = ((♯‘𝑉) · (𝐾 · (𝐾 − 1))))) |
12 | 11 | 3adant3 1131 | . . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (((♯‘𝑉) · ((♯‘𝑉) − 1)) = ((♯‘𝑉) · (𝐾 · (𝐾 − 1))) ↔ (♯‘(2 WSPathsN 𝐺)) = ((♯‘𝑉) · (𝐾 · (𝐾 − 1))))) |
13 | 12 | adantr 480 | . . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (((♯‘𝑉) · ((♯‘𝑉) − 1)) = ((♯‘𝑉) · (𝐾 · (𝐾 − 1))) ↔ (♯‘(2 WSPathsN 𝐺)) = ((♯‘𝑉) · (𝐾 · (𝐾 − 1))))) |
14 | 3 | frrusgrord0lem 30368 | . . . . 5 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐾 ∈ ℂ ∧ (♯‘𝑉) ∈ ℂ ∧ (♯‘𝑉) ≠ 0)) |
15 | peano2cnm 11573 | . . . . . . . 8 ⊢ ((♯‘𝑉) ∈ ℂ → ((♯‘𝑉) − 1) ∈ ℂ) | |
16 | 15 | 3ad2ant2 1133 | . . . . . . 7 ⊢ ((𝐾 ∈ ℂ ∧ (♯‘𝑉) ∈ ℂ ∧ (♯‘𝑉) ≠ 0) → ((♯‘𝑉) − 1) ∈ ℂ) |
17 | kcnktkm1cn 11692 | . . . . . . . 8 ⊢ (𝐾 ∈ ℂ → (𝐾 · (𝐾 − 1)) ∈ ℂ) | |
18 | 17 | 3ad2ant1 1132 | . . . . . . 7 ⊢ ((𝐾 ∈ ℂ ∧ (♯‘𝑉) ∈ ℂ ∧ (♯‘𝑉) ≠ 0) → (𝐾 · (𝐾 − 1)) ∈ ℂ) |
19 | simp2 1136 | . . . . . . 7 ⊢ ((𝐾 ∈ ℂ ∧ (♯‘𝑉) ∈ ℂ ∧ (♯‘𝑉) ≠ 0) → (♯‘𝑉) ∈ ℂ) | |
20 | simp3 1137 | . . . . . . 7 ⊢ ((𝐾 ∈ ℂ ∧ (♯‘𝑉) ∈ ℂ ∧ (♯‘𝑉) ≠ 0) → (♯‘𝑉) ≠ 0) | |
21 | 16, 18, 19, 20 | mulcand 11894 | . . . . . 6 ⊢ ((𝐾 ∈ ℂ ∧ (♯‘𝑉) ∈ ℂ ∧ (♯‘𝑉) ≠ 0) → (((♯‘𝑉) · ((♯‘𝑉) − 1)) = ((♯‘𝑉) · (𝐾 · (𝐾 − 1))) ↔ ((♯‘𝑉) − 1) = (𝐾 · (𝐾 − 1)))) |
22 | npcan1 11686 | . . . . . . . . 9 ⊢ ((♯‘𝑉) ∈ ℂ → (((♯‘𝑉) − 1) + 1) = (♯‘𝑉)) | |
23 | oveq1 7438 | . . . . . . . . 9 ⊢ (((♯‘𝑉) − 1) = (𝐾 · (𝐾 − 1)) → (((♯‘𝑉) − 1) + 1) = ((𝐾 · (𝐾 − 1)) + 1)) | |
24 | 22, 23 | sylan9req 2796 | . . . . . . . 8 ⊢ (((♯‘𝑉) ∈ ℂ ∧ ((♯‘𝑉) − 1) = (𝐾 · (𝐾 − 1))) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1)) |
25 | 24 | ex 412 | . . . . . . 7 ⊢ ((♯‘𝑉) ∈ ℂ → (((♯‘𝑉) − 1) = (𝐾 · (𝐾 − 1)) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) |
26 | 25 | 3ad2ant2 1133 | . . . . . 6 ⊢ ((𝐾 ∈ ℂ ∧ (♯‘𝑉) ∈ ℂ ∧ (♯‘𝑉) ≠ 0) → (((♯‘𝑉) − 1) = (𝐾 · (𝐾 − 1)) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) |
27 | 21, 26 | sylbid 240 | . . . . 5 ⊢ ((𝐾 ∈ ℂ ∧ (♯‘𝑉) ∈ ℂ ∧ (♯‘𝑉) ≠ 0) → (((♯‘𝑉) · ((♯‘𝑉) − 1)) = ((♯‘𝑉) · (𝐾 · (𝐾 − 1))) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) |
28 | 14, 27 | syl 17 | . . . 4 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (((♯‘𝑉) · ((♯‘𝑉) − 1)) = ((♯‘𝑉) · (𝐾 · (𝐾 − 1))) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) |
29 | 13, 28 | sylbird 260 | . . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → ((♯‘(2 WSPathsN 𝐺)) = ((♯‘𝑉) · (𝐾 · (𝐾 − 1))) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) |
30 | 8, 29 | mpd 15 | . 2 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1)) |
31 | 30 | ex 412 | 1 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∅c0 4339 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 − cmin 11490 2c2 12319 ♯chash 14366 Vtxcvtx 29028 USGraphcusgr 29181 FinUSGraphcfusgr 29348 VtxDegcvtxdg 29498 WSPathsN cwwspthsn 29858 FriendGraph cfrgr 30287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-ac2 10501 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-dju 9939 df-card 9977 df-ac 10154 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-rp 13033 df-xadd 13153 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-s2 14884 df-s3 14885 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-vtx 29030 df-iedg 29031 df-edg 29080 df-uhgr 29090 df-ushgr 29091 df-upgr 29114 df-umgr 29115 df-uspgr 29182 df-usgr 29183 df-fusgr 29349 df-nbgr 29365 df-vtxdg 29499 df-wlks 29632 df-wlkson 29633 df-trls 29725 df-trlson 29726 df-pths 29749 df-spths 29750 df-pthson 29751 df-spthson 29752 df-wwlks 29860 df-wwlksn 29861 df-wwlksnon 29862 df-wspthsn 29863 df-wspthsnon 29864 df-frgr 30288 |
This theorem is referenced by: frrusgrord 30370 |
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