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Mirrors > Home > MPE Home > Th. List > frgrhash2wsp | Structured version Visualization version GIF version |
Description: The number of simple paths of length 2 is n*(n-1) in a friendship graph with n vertices. This corresponds to the proof of claim 3 in [Huneke] p. 2: "... the paths of length two in G: by assumption there are ( n 2 ) such paths.". However, Huneke counts undirected paths, so obtains the result ((𝑛C2) = ((𝑛 · (𝑛 − 1)) / 2)), whereas we count directed paths, obtaining twice that number. (Contributed by Alexander van der Vekens, 6-Mar-2018.) (Revised by AV, 10-Jan-2022.) |
Ref | Expression |
---|---|
frgrhash2wsp.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
frgrhash2wsp | ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (♯‘(2 WSPathsN 𝐺)) = ((♯‘𝑉) · ((♯‘𝑉) − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn 11903 | . . . . 5 ⊢ 2 ∈ ℕ | |
2 | frgrhash2wsp.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | wspniunwspnon 28007 | . . . . 5 ⊢ ((2 ∈ ℕ ∧ 𝐺 ∈ FriendGraph ) → (2 WSPathsN 𝐺) = ∪ 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏)) |
4 | 1, 3 | mpan 690 | . . . 4 ⊢ (𝐺 ∈ FriendGraph → (2 WSPathsN 𝐺) = ∪ 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏)) |
5 | 4 | fveq2d 6721 | . . 3 ⊢ (𝐺 ∈ FriendGraph → (♯‘(2 WSPathsN 𝐺)) = (♯‘∪ 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏))) |
6 | 5 | adantr 484 | . 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (♯‘(2 WSPathsN 𝐺)) = (♯‘∪ 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏))) |
7 | simpr 488 | . . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → 𝑉 ∈ Fin) | |
8 | eqid 2737 | . . 3 ⊢ (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝑎}) | |
9 | 2 | eleq1i 2828 | . . . . . 6 ⊢ (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin) |
10 | wspthnonfi 28006 | . . . . . 6 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑎(2 WSPathsNOn 𝐺)𝑏) ∈ Fin) | |
11 | 9, 10 | sylbi 220 | . . . . 5 ⊢ (𝑉 ∈ Fin → (𝑎(2 WSPathsNOn 𝐺)𝑏) ∈ Fin) |
12 | 11 | adantl 485 | . . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (𝑎(2 WSPathsNOn 𝐺)𝑏) ∈ Fin) |
13 | 12 | 3ad2ant1 1135 | . . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑎(2 WSPathsNOn 𝐺)𝑏) ∈ Fin) |
14 | 2wspiundisj 28047 | . . . 4 ⊢ Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) | |
15 | 14 | a1i 11 | . . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏)) |
16 | 2wspdisj 28046 | . . . 4 ⊢ Disj 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) | |
17 | 16 | a1i 11 | . . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉) → Disj 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏)) |
18 | simplll 775 | . . . . 5 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → 𝐺 ∈ FriendGraph ) | |
19 | simpr 488 | . . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ 𝑉) | |
20 | eldifi 4041 | . . . . . 6 ⊢ (𝑏 ∈ (𝑉 ∖ {𝑎}) → 𝑏 ∈ 𝑉) | |
21 | 19, 20 | anim12i 616 | . . . . 5 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) |
22 | eldifsni 4703 | . . . . . . 7 ⊢ (𝑏 ∈ (𝑉 ∖ {𝑎}) → 𝑏 ≠ 𝑎) | |
23 | 22 | necomd 2996 | . . . . . 6 ⊢ (𝑏 ∈ (𝑉 ∖ {𝑎}) → 𝑎 ≠ 𝑏) |
24 | 23 | adantl 485 | . . . . 5 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → 𝑎 ≠ 𝑏) |
25 | 2 | frgr2wsp1 28413 | . . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → (♯‘(𝑎(2 WSPathsNOn 𝐺)𝑏)) = 1) |
26 | 18, 21, 24, 25 | syl3anc 1373 | . . . 4 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (♯‘(𝑎(2 WSPathsNOn 𝐺)𝑏)) = 1) |
27 | 26 | 3impa 1112 | . . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (♯‘(𝑎(2 WSPathsNOn 𝐺)𝑏)) = 1) |
28 | 7, 8, 13, 15, 17, 27 | hash2iun1dif1 15388 | . 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (♯‘∪ 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏)) = ((♯‘𝑉) · ((♯‘𝑉) − 1))) |
29 | 6, 28 | eqtrd 2777 | 1 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (♯‘(2 WSPathsN 𝐺)) = ((♯‘𝑉) · ((♯‘𝑉) − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 ∖ cdif 3863 {csn 4541 ∪ ciun 4904 Disj wdisj 5018 ‘cfv 6380 (class class class)co 7213 Fincfn 8626 1c1 10730 · cmul 10734 − cmin 11062 ℕcn 11830 2c2 11885 ♯chash 13896 Vtxcvtx 27087 WSPathsN cwwspthsn 27912 WSPathsNOn cwwspthsnon 27913 FriendGraph cfrgr 28341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-ac2 10077 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ifp 1064 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-disj 5019 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-oadd 8206 df-er 8391 df-map 8510 df-pm 8511 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-oi 9126 df-dju 9517 df-card 9555 df-ac 9730 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-xnn0 12163 df-z 12177 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-seq 13575 df-exp 13636 df-hash 13897 df-word 14070 df-concat 14126 df-s1 14153 df-s2 14413 df-s3 14414 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-sum 15250 df-edg 27139 df-uhgr 27149 df-upgr 27173 df-umgr 27174 df-uspgr 27241 df-usgr 27242 df-wlks 27687 df-wlkson 27688 df-trls 27780 df-trlson 27781 df-pths 27803 df-spths 27804 df-pthson 27805 df-spthson 27806 df-wwlks 27914 df-wwlksn 27915 df-wwlksnon 27916 df-wspthsn 27917 df-wspthsnon 27918 df-frgr 28342 |
This theorem is referenced by: frrusgrord0 28423 |
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