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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 11711 | . . . . 5 ⊢ 2 ∈ ℕ | |
2 | frgrhash2wsp.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | wspniunwspnon 27702 | . . . . 5 ⊢ ((2 ∈ ℕ ∧ 𝐺 ∈ FriendGraph ) → (2 WSPathsN 𝐺) = ∪ 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏)) |
4 | 1, 3 | mpan 688 | . . . 4 ⊢ (𝐺 ∈ FriendGraph → (2 WSPathsN 𝐺) = ∪ 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏)) |
5 | 4 | fveq2d 6674 | . . 3 ⊢ (𝐺 ∈ FriendGraph → (♯‘(2 WSPathsN 𝐺)) = (♯‘∪ 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏))) |
6 | 5 | adantr 483 | . 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (♯‘(2 WSPathsN 𝐺)) = (♯‘∪ 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏))) |
7 | simpr 487 | . . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → 𝑉 ∈ Fin) | |
8 | eqid 2821 | . . 3 ⊢ (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝑎}) | |
9 | 2 | eleq1i 2903 | . . . . . 6 ⊢ (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin) |
10 | wspthnonfi 27701 | . . . . . 6 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑎(2 WSPathsNOn 𝐺)𝑏) ∈ Fin) | |
11 | 9, 10 | sylbi 219 | . . . . 5 ⊢ (𝑉 ∈ Fin → (𝑎(2 WSPathsNOn 𝐺)𝑏) ∈ Fin) |
12 | 11 | adantl 484 | . . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (𝑎(2 WSPathsNOn 𝐺)𝑏) ∈ Fin) |
13 | 12 | 3ad2ant1 1129 | . . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑎(2 WSPathsNOn 𝐺)𝑏) ∈ Fin) |
14 | 2wspiundisj 27742 | . . . 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 27741 | . . . 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 773 | . . . . 5 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → 𝐺 ∈ FriendGraph ) | |
19 | simpr 487 | . . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ 𝑉) | |
20 | eldifi 4103 | . . . . . 6 ⊢ (𝑏 ∈ (𝑉 ∖ {𝑎}) → 𝑏 ∈ 𝑉) | |
21 | 19, 20 | anim12i 614 | . . . . 5 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) |
22 | eldifsni 4722 | . . . . . . 7 ⊢ (𝑏 ∈ (𝑉 ∖ {𝑎}) → 𝑏 ≠ 𝑎) | |
23 | 22 | necomd 3071 | . . . . . 6 ⊢ (𝑏 ∈ (𝑉 ∖ {𝑎}) → 𝑎 ≠ 𝑏) |
24 | 23 | adantl 484 | . . . . 5 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → 𝑎 ≠ 𝑏) |
25 | 2 | frgr2wsp1 28109 | . . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → (♯‘(𝑎(2 WSPathsNOn 𝐺)𝑏)) = 1) |
26 | 18, 21, 24, 25 | syl3anc 1367 | . . . 4 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (♯‘(𝑎(2 WSPathsNOn 𝐺)𝑏)) = 1) |
27 | 26 | 3impa 1106 | . . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (♯‘(𝑎(2 WSPathsNOn 𝐺)𝑏)) = 1) |
28 | 7, 8, 13, 15, 17, 27 | hash2iun1dif1 15179 | . 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (♯‘∪ 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏)) = ((♯‘𝑉) · ((♯‘𝑉) − 1))) |
29 | 6, 28 | eqtrd 2856 | 1 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (♯‘(2 WSPathsN 𝐺)) = ((♯‘𝑉) · ((♯‘𝑉) − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3933 {csn 4567 ∪ ciun 4919 Disj wdisj 5031 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 1c1 10538 · cmul 10542 − cmin 10870 ℕcn 11638 2c2 11693 ♯chash 13691 Vtxcvtx 26781 WSPathsN cwwspthsn 27606 WSPathsNOn cwwspthsnon 27607 FriendGraph cfrgr 28037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-ac2 9885 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-disj 5032 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-oi 8974 df-dju 9330 df-card 9368 df-ac 9542 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-s2 14210 df-s3 14211 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-edg 26833 df-uhgr 26843 df-upgr 26867 df-umgr 26868 df-uspgr 26935 df-usgr 26936 df-wlks 27381 df-wlkson 27382 df-trls 27474 df-trlson 27475 df-pths 27497 df-spths 27498 df-pthson 27499 df-spthson 27500 df-wwlks 27608 df-wwlksn 27609 df-wwlksnon 27610 df-wspthsn 27611 df-wspthsnon 27612 df-frgr 28038 |
This theorem is referenced by: frrusgrord0 28119 |
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