Theorem frgrhash2wsp 27196

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 ( ( n _C 2 ) = ( ( n x. ( 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.)

Hypothesis

Ref	Expression
frgrhash2wsp.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
frgrhash2wsp	|- ( ( G e. FriendGraph /\ V e. Fin ) -> ( # ` ( 2 WSPathsN G ) ) = ( ( # ` V ) x. ( ( # ` V ) - 1 ) ) )

Proof of Theorem frgrhash2wsp

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2nn 11185	. . . . 5 |- 2 e. NN
2		frgrhash2wsp.v	. . . . . 6 |- V = (Vtx ` G )
3	2	wspniunwspnon 26819	. . . . 5 |- ( ( 2 e. NN /\ G e. FriendGraph ) -> ( 2 WSPathsN G ) = U_ a e. V U_ b e. ( V \ { a } ) ( a ( 2 WSPathsNOn G ) b ) )
4	1, 3	mpan 706	. . . 4 |- ( G e. FriendGraph -> ( 2 WSPathsN G ) = U_ a e. V U_ b e. ( V \ { a } ) ( a ( 2 WSPathsNOn G ) b ) )
5	4	fveq2d 6195	. . 3 |- ( G e. FriendGraph -> ( # ` ( 2 WSPathsN G ) ) = ( # ` U_ a e. V U_ b e. ( V \ { a } ) ( a ( 2 WSPathsNOn G ) b ) ) )
6	5	adantr 481	. 2 |- ( ( G e. FriendGraph /\ V e. Fin ) -> ( # ` ( 2 WSPathsN G ) ) = ( # ` U_ a e. V U_ b e. ( V \ { a } ) ( a ( 2 WSPathsNOn G ) b ) ) )
7		simpr 477	. . 3 |- ( ( G e. FriendGraph /\ V e. Fin ) -> V e. Fin )
8		eqid 2622	. . 3 |- ( V \ { a } ) = ( V \ { a } )
9	2	eleq1i 2692	. . . . . 6 |- ( V e. Fin <-> (Vtx ` G ) e. Fin )
10		wspthnonfi 26818	. . . . . 6 |- ( (Vtx ` G ) e. Fin -> ( a ( 2 WSPathsNOn G ) b ) e. Fin )
11	9, 10	sylbi 207	. . . . 5 |- ( V e. Fin -> ( a ( 2 WSPathsNOn G ) b ) e. Fin )
12	11	adantl 482	. . . 4 |- ( ( G e. FriendGraph /\ V e. Fin ) -> ( a ( 2 WSPathsNOn G ) b ) e. Fin )
13	12	3ad2ant1 1082	. . 3 |- ( ( ( G e. FriendGraph /\ V e. Fin ) /\ a e. V /\ b e. ( V \ { a } ) ) -> ( a ( 2 WSPathsNOn G ) b ) e. Fin )
14		2wspiundisj 26856	. . . 4 |- Disj a e. V U_ b e. ( V \ { a } ) ( a ( 2 WSPathsNOn G ) b )
15	14	a1i 11	. . 3 |- ( ( G e. FriendGraph /\ V e. Fin ) -> Disj a e. V U_ b e. ( V \ { a } ) ( a ( 2 WSPathsNOn G ) b ) )
16		2wspdisj 26855	. . . 4 |- Disj b e. ( V \ { a } ) ( a ( 2 WSPathsNOn G ) b )
17	16	a1i 11	. . 3 |- ( ( ( G e. FriendGraph /\ V e. Fin ) /\ a e. V ) -> Disj b e. ( V \ { a } ) ( a ( 2 WSPathsNOn G ) b ) )
18		simplll 798	. . . . 5 |- ( ( ( ( G e. FriendGraph /\ V e. Fin ) /\ a e. V ) /\ b e. ( V \ { a } ) ) -> G e. FriendGraph )
19		simpr 477	. . . . . 6 |- ( ( ( G e. FriendGraph /\ V e. Fin ) /\ a e. V ) -> a e. V )
20		eldifi 3732	. . . . . 6 |- ( b e. ( V \ { a } ) -> b e. V )
21	19, 20	anim12i 590	. . . . 5 |- ( ( ( ( G e. FriendGraph /\ V e. Fin ) /\ a e. V ) /\ b e. ( V \ { a } ) ) -> ( a e. V /\ b e. V ) )
22		eldifsni 4320	. . . . . . 7 |- ( b e. ( V \ { a } ) -> b =/= a )
23	22	necomd 2849	. . . . . 6 |- ( b e. ( V \ { a } ) -> a =/= b )
24	23	adantl 482	. . . . 5 |- ( ( ( ( G e. FriendGraph /\ V e. Fin ) /\ a e. V ) /\ b e. ( V \ { a } ) ) -> a =/= b )
25	2	frgr2wsp1 27194	. . . . 5 |- ( ( G e. FriendGraph /\ ( a e. V /\ b e. V ) /\ a =/= b ) -> ( # ` ( a ( 2 WSPathsNOn G ) b ) ) = 1 )
26	18, 21, 24, 25	syl3anc 1326	. . . 4 |- ( ( ( ( G e. FriendGraph /\ V e. Fin ) /\ a e. V ) /\ b e. ( V \ { a } ) ) -> ( # ` ( a ( 2 WSPathsNOn G ) b ) ) = 1 )
27	26	3impa 1259	. . 3 |- ( ( ( G e. FriendGraph /\ V e. Fin ) /\ a e. V /\ b e. ( V \ { a } ) ) -> ( # ` ( a ( 2 WSPathsNOn G ) b ) ) = 1 )
28	7, 8, 13, 15, 17, 27	hash2iun1dif1 14556	. 2 |- ( ( G e. FriendGraph /\ V e. Fin ) -> ( # ` U_ a e. V U_ b e. ( V \ { a } ) ( a ( 2 WSPathsNOn G ) b ) ) = ( ( # ` V ) x. ( ( # ` V ) - 1 ) ) )
29	6, 28	eqtrd 2656	1 |- ( ( G e. FriendGraph /\ V e. Fin ) -> ( # ` ( 2 WSPathsN G ) ) = ( ( # ` V ) x. ( ( # ` V ) - 1 ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177   U_ciun 4520  Disj wdisj 4620   ` cfv 5888  (class class class)co 6650   Fincfn 7955   1c1 9937   x. cmul 9941   - cmin 10266   NNcn 11020   2c2 11070   #chash 13117  Vtxcvtx 25874   WSPathsN cwwspthsn 26720   WSPathsNOn cwwspthsnon 26721   FriendGraph cfrgr 27120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-umgr 25978  df-uspgr 26045  df-usgr 26046  df-wlks 26495  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-spths 26613  df-pthson 26614  df-spthson 26615  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724  df-wspthsn 26725  df-wspthsnon 26726  df-frgr 27121

This theorem is referenced by:  frrusgrord0  27204