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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 11223 | . . . . 5 ⊢ 2 ∈ ℕ | |
2 | frgrhash2wsp.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | wspniunwspnon 26888 | . . . . 5 ⊢ ((2 ∈ ℕ ∧ 𝐺 ∈ FriendGraph ) → (2 WSPathsN 𝐺) = ∪ 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏)) |
4 | 1, 3 | mpan 706 | . . . 4 ⊢ (𝐺 ∈ FriendGraph → (2 WSPathsN 𝐺) = ∪ 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏)) |
5 | 4 | fveq2d 6233 | . . 3 ⊢ (𝐺 ∈ FriendGraph → (#‘(2 WSPathsN 𝐺)) = (#‘∪ 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏))) |
6 | 5 | adantr 480 | . 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (#‘(2 WSPathsN 𝐺)) = (#‘∪ 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏))) |
7 | simpr 476 | . . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → 𝑉 ∈ Fin) | |
8 | eqid 2651 | . . 3 ⊢ (𝑉 ∖ {𝑎}) = (𝑉 ∖ {𝑎}) | |
9 | 2 | eleq1i 2721 | . . . . . 6 ⊢ (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin) |
10 | wspthnonfi 26887 | . . . . . 6 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑎(2 WSPathsNOn 𝐺)𝑏) ∈ Fin) | |
11 | 9, 10 | sylbi 207 | . . . . 5 ⊢ (𝑉 ∈ Fin → (𝑎(2 WSPathsNOn 𝐺)𝑏) ∈ Fin) |
12 | 11 | adantl 481 | . . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (𝑎(2 WSPathsNOn 𝐺)𝑏) ∈ Fin) |
13 | 12 | 3ad2ant1 1102 | . . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑎(2 WSPathsNOn 𝐺)𝑏) ∈ Fin) |
14 | 2wspiundisj 26930 | . . . 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 26929 | . . . 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 813 | . . . . 5 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → 𝐺 ∈ FriendGraph ) | |
19 | simpr 476 | . . . . . 6 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ 𝑉) | |
20 | eldifi 3765 | . . . . . 6 ⊢ (𝑏 ∈ (𝑉 ∖ {𝑎}) → 𝑏 ∈ 𝑉) | |
21 | 19, 20 | anim12i 589 | . . . . 5 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) |
22 | eldifsni 4353 | . . . . . . 7 ⊢ (𝑏 ∈ (𝑉 ∖ {𝑎}) → 𝑏 ≠ 𝑎) | |
23 | 22 | necomd 2878 | . . . . . 6 ⊢ (𝑏 ∈ (𝑉 ∖ {𝑎}) → 𝑎 ≠ 𝑏) |
24 | 23 | adantl 481 | . . . . 5 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → 𝑎 ≠ 𝑏) |
25 | 2 | frgr2wsp1 27310 | . . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑎 ≠ 𝑏) → (#‘(𝑎(2 WSPathsNOn 𝐺)𝑏)) = 1) |
26 | 18, 21, 24, 25 | syl3anc 1366 | . . . 4 ⊢ ((((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (#‘(𝑎(2 WSPathsNOn 𝐺)𝑏)) = 1) |
27 | 26 | 3impa 1278 | . . 3 ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ (𝑉 ∖ {𝑎})) → (#‘(𝑎(2 WSPathsNOn 𝐺)𝑏)) = 1) |
28 | 7, 8, 13, 15, 17, 27 | hash2iun1dif1 14600 | . 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (#‘∪ 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏)) = ((#‘𝑉) · ((#‘𝑉) − 1))) |
29 | 6, 28 | eqtrd 2685 | 1 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (#‘(2 WSPathsN 𝐺)) = ((#‘𝑉) · ((#‘𝑉) − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∖ cdif 3604 {csn 4210 ∪ ciun 4552 Disj wdisj 4652 ‘cfv 5926 (class class class)co 6690 Fincfn 7997 1c1 9975 · cmul 9979 − cmin 10304 ℕcn 11058 2c2 11108 #chash 13157 Vtxcvtx 25919 WSPathsN cwwspthsn 26776 WSPathsNOn cwwspthsnon 26777 FriendGraph cfrgr 27236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-ac2 9323 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-ifp 1033 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-disj 4653 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-oi 8456 df-card 8803 df-ac 8977 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-xnn0 11402 df-z 11416 df-uz 11726 df-rp 11871 df-fz 12365 df-fzo 12505 df-seq 12842 df-exp 12901 df-hash 13158 df-word 13331 df-concat 13333 df-s1 13334 df-s2 13639 df-s3 13640 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-clim 14263 df-sum 14461 df-edg 25985 df-uhgr 25998 df-upgr 26022 df-umgr 26023 df-uspgr 26090 df-usgr 26091 df-wlks 26551 df-wlkson 26552 df-trls 26645 df-trlson 26646 df-pths 26668 df-spths 26669 df-pthson 26670 df-spthson 26671 df-wwlks 26778 df-wwlksn 26779 df-wwlksnon 26780 df-wspthsn 26781 df-wspthsnon 26782 df-frgr 27237 |
This theorem is referenced by: frrusgrord0 27320 |
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