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Theorem frghash2spot 25870
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, the order of vertices is not respected by Huneke, so he only counts half of the paths which are existing when respecting the order as it is the case for simple paths represented by ordered triples. (Contributed by Alexander van der Vekens, 6-Mar-2018.)
Assertion
Ref Expression
frghash2spot  |-  ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  -> 
( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) ) )

Proof of Theorem frghash2spot
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frisusgra 25799 . . . . 5  |-  ( V FriendGrph  E  ->  V USGrph  E )
2 usgrav 25144 . . . . 5  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
3 2spot2iun2spont 25698 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V 2SPathOnOt  E )  =  U_ a  e.  V  U_ b  e.  ( V 
\  { a } ) ( a ( V 2SPathOnOt  E ) b ) )
41, 2, 33syl 18 . . . 4  |-  ( V FriendGrph  E  ->  ( V 2SPathOnOt  E )  =  U_ a  e.  V  U_ b  e.  ( V  \  {
a } ) ( a ( V 2SPathOnOt  E ) b ) )
54fveq2d 5883 . . 3  |-  ( V FriendGrph  E  ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( # `  U_ a  e.  V  U_ b  e.  ( V  \  {
a } ) ( a ( V 2SPathOnOt  E ) b ) ) )
65adantr 472 . 2  |-  ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  -> 
( # `  ( V 2SPathOnOt  E ) )  =  ( # `  U_ a  e.  V  U_ b  e.  ( V  \  {
a } ) ( a ( V 2SPathOnOt  E ) b ) ) )
7 simpl 464 . . . 4  |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  V  e.  Fin )
87adantl 473 . . 3  |-  ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  ->  V  e.  Fin )
9 diffi 7821 . . . . . . 7  |-  ( V  e.  Fin  ->  ( V  \  { a } )  e.  Fin )
109adantr 472 . . . . . 6  |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  ( V  \  { a } )  e.  Fin )
1110adantl 473 . . . . 5  |-  ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  -> 
( V  \  {
a } )  e. 
Fin )
1211adantr 472 . . . 4  |-  ( ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  a  e.  V
)  ->  ( V  \  { a } )  e.  Fin )
13 simpr 468 . . . . . . . . . 10  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  E  e.  _V )
141, 2, 133syl 18 . . . . . . . . 9  |-  ( V FriendGrph  E  ->  E  e.  _V )
1514, 7anim12ci 577 . . . . . . . 8  |-  ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  -> 
( V  e.  Fin  /\  E  e.  _V )
)
1615adantr 472 . . . . . . 7  |-  ( ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  a  e.  V
)  ->  ( V  e.  Fin  /\  E  e. 
_V ) )
1716adantr 472 . . . . . 6  |-  ( ( ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  a  e.  V
)  /\  b  e.  ( V  \  { a } ) )  -> 
( V  e.  Fin  /\  E  e.  _V )
)
18 simpr 468 . . . . . . 7  |-  ( ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  a  e.  V
)  ->  a  e.  V )
19 eldifi 3544 . . . . . . 7  |-  ( b  e.  ( V  \  { a } )  ->  b  e.  V
)
2018, 19anim12i 576 . . . . . 6  |-  ( ( ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  a  e.  V
)  /\  b  e.  ( V  \  { a } ) )  -> 
( a  e.  V  /\  b  e.  V
) )
21 2spotfi 25699 . . . . . 6  |-  ( ( ( V  e.  Fin  /\  E  e.  _V )  /\  ( a  e.  V  /\  b  e.  V
) )  ->  (
a ( V 2SPathOnOt  E ) b )  e.  Fin )
2217, 20, 21syl2anc 673 . . . . 5  |-  ( ( ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  a  e.  V
)  /\  b  e.  ( V  \  { a } ) )  -> 
( a ( V 2SPathOnOt  E ) b )  e.  Fin )
2322ralrimiva 2809 . . . 4  |-  ( ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  a  e.  V
)  ->  A. b  e.  ( V  \  {
a } ) ( a ( V 2SPathOnOt  E ) b )  e.  Fin )
24 iunfi 7880 . . . 4  |-  ( ( ( V  \  {
a } )  e. 
Fin  /\  A. b  e.  ( V  \  {
a } ) ( a ( V 2SPathOnOt  E ) b )  e.  Fin )  ->  U_ b  e.  ( V  \  { a } ) ( a ( V 2SPathOnOt  E )
b )  e.  Fin )
2512, 23, 24syl2anc 673 . . 3  |-  ( ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  a  e.  V
)  ->  U_ b  e.  ( V  \  {
a } ) ( a ( V 2SPathOnOt  E ) b )  e.  Fin )
26 2spotiundisj 25869 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V )  -> Disj  a  e.  V  U_ b  e.  ( V  \  {
a } ) ( a ( V 2SPathOnOt  E ) b ) )
271, 2, 263syl 18 . . . 4  |-  ( V FriendGrph  E  -> Disj  a  e.  V  U_ b  e.  ( V  \  { a } ) ( a ( V 2SPathOnOt  E ) b ) )
2827adantr 472 . . 3  |-  ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  -> Disj  a  e.  V  U_ b  e.  ( V  \  {
a } ) ( a ( V 2SPathOnOt  E ) b ) )
298, 25, 28hashiun 13959 . 2  |-  ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  -> 
( # `  U_ a  e.  V  U_ b  e.  ( V  \  {
a } ) ( a ( V 2SPathOnOt  E ) b ) )  = 
sum_ a  e.  V  ( # `  U_ b  e.  ( V  \  {
a } ) ( a ( V 2SPathOnOt  E ) b ) ) )
30 2spotdisj 25868 . . . . . . 7  |-  ( ( ( V  e.  Fin  /\  E  e.  _V )  /\  a  e.  V
)  -> Disj  b  e.  ( V  \  { a } ) ( a ( V 2SPathOnOt  E )
b ) )
3115, 30sylan 479 . . . . . 6  |-  ( ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  a  e.  V
)  -> Disj  b  e.  ( V  \  { a } ) ( a ( V 2SPathOnOt  E )
b ) )
3212, 22, 31hashiun 13959 . . . . 5  |-  ( ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  a  e.  V
)  ->  ( # `  U_ b  e.  ( V  \  {
a } ) ( a ( V 2SPathOnOt  E ) b ) )  = 
sum_ b  e.  ( V  \  { a } ) ( # `  ( a ( V 2SPathOnOt  E ) b ) ) )
33 simplll 776 . . . . . . 7  |-  ( ( ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  a  e.  V
)  /\  b  e.  ( V  \  { a } ) )  ->  V FriendGrph  E )
34 eldifsni 4089 . . . . . . . . 9  |-  ( b  e.  ( V  \  { a } )  ->  b  =/=  a
)
3534necomd 2698 . . . . . . . 8  |-  ( b  e.  ( V  \  { a } )  ->  a  =/=  b
)
3635adantl 473 . . . . . . 7  |-  ( ( ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  a  e.  V
)  /\  b  e.  ( V  \  { a } ) )  -> 
a  =/=  b )
37 frg2spot1 25865 . . . . . . 7  |-  ( ( V FriendGrph  E  /\  (
a  e.  V  /\  b  e.  V )  /\  a  =/=  b
)  ->  ( # `  (
a ( V 2SPathOnOt  E ) b ) )  =  1 )
3833, 20, 36, 37syl3anc 1292 . . . . . 6  |-  ( ( ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  a  e.  V
)  /\  b  e.  ( V  \  { a } ) )  -> 
( # `  ( a ( V 2SPathOnOt  E )
b ) )  =  1 )
3938sumeq2dv 13846 . . . . 5  |-  ( ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  a  e.  V
)  ->  sum_ b  e.  ( V  \  {
a } ) (
# `  ( a
( V 2SPathOnOt  E ) b ) )  =  sum_ b  e.  ( V  \  { a } ) 1 )
40 ax-1cn 9615 . . . . . . . . . . 11  |-  1  e.  CC
419, 40jctir 547 . . . . . . . . . 10  |-  ( V  e.  Fin  ->  (
( V  \  {
a } )  e. 
Fin  /\  1  e.  CC ) )
4241adantr 472 . . . . . . . . 9  |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  (
( V  \  {
a } )  e. 
Fin  /\  1  e.  CC ) )
4342adantr 472 . . . . . . . 8  |-  ( ( ( V  e.  Fin  /\  V  =/=  (/) )  /\  a  e.  V )  ->  ( ( V  \  { a } )  e.  Fin  /\  1  e.  CC ) )
44 fsumconst 13928 . . . . . . . 8  |-  ( ( ( V  \  {
a } )  e. 
Fin  /\  1  e.  CC )  ->  sum_ b  e.  ( V  \  {
a } ) 1  =  ( ( # `  ( V  \  {
a } ) )  x.  1 ) )
4543, 44syl 17 . . . . . . 7  |-  ( ( ( V  e.  Fin  /\  V  =/=  (/) )  /\  a  e.  V )  -> 
sum_ b  e.  ( V  \  { a } ) 1  =  ( ( # `  ( V  \  { a } ) )  x.  1 ) )
46 hashdifsn 12629 . . . . . . . . 9  |-  ( ( V  e.  Fin  /\  a  e.  V )  ->  ( # `  ( V  \  { a } ) )  =  ( ( # `  V
)  -  1 ) )
4746adantlr 729 . . . . . . . 8  |-  ( ( ( V  e.  Fin  /\  V  =/=  (/) )  /\  a  e.  V )  ->  ( # `  ( V  \  { a } ) )  =  ( ( # `  V
)  -  1 ) )
4847oveq1d 6323 . . . . . . 7  |-  ( ( ( V  e.  Fin  /\  V  =/=  (/) )  /\  a  e.  V )  ->  ( ( # `  ( V  \  { a } ) )  x.  1 )  =  ( ( ( # `  V
)  -  1 )  x.  1 ) )
49 hashnncl 12585 . . . . . . . . . . . 12  |-  ( V  e.  Fin  ->  (
( # `  V )  e.  NN  <->  V  =/=  (/) ) )
5049biimpar 493 . . . . . . . . . . 11  |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  ( # `
 V )  e.  NN )
51 nnm1nn0 10935 . . . . . . . . . . 11  |-  ( (
# `  V )  e.  NN  ->  ( ( # `
 V )  - 
1 )  e.  NN0 )
5250, 51syl 17 . . . . . . . . . 10  |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  (
( # `  V )  -  1 )  e. 
NN0 )
5352nn0red 10950 . . . . . . . . 9  |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  (
( # `  V )  -  1 )  e.  RR )
54 ax-1rid 9627 . . . . . . . . 9  |-  ( ( ( # `  V
)  -  1 )  e.  RR  ->  (
( ( # `  V
)  -  1 )  x.  1 )  =  ( ( # `  V
)  -  1 ) )
5553, 54syl 17 . . . . . . . 8  |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  (
( ( # `  V
)  -  1 )  x.  1 )  =  ( ( # `  V
)  -  1 ) )
5655adantr 472 . . . . . . 7  |-  ( ( ( V  e.  Fin  /\  V  =/=  (/) )  /\  a  e.  V )  ->  ( ( ( # `  V )  -  1 )  x.  1 )  =  ( ( # `  V )  -  1 ) )
5745, 48, 563eqtrd 2509 . . . . . 6  |-  ( ( ( V  e.  Fin  /\  V  =/=  (/) )  /\  a  e.  V )  -> 
sum_ b  e.  ( V  \  { a } ) 1  =  ( ( # `  V
)  -  1 ) )
5857adantll 728 . . . . 5  |-  ( ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  a  e.  V
)  ->  sum_ b  e.  ( V  \  {
a } ) 1  =  ( ( # `  V )  -  1 ) )
5932, 39, 583eqtrd 2509 . . . 4  |-  ( ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  a  e.  V
)  ->  ( # `  U_ b  e.  ( V  \  {
a } ) ( a ( V 2SPathOnOt  E ) b ) )  =  ( ( # `  V
)  -  1 ) )
6059sumeq2dv 13846 . . 3  |-  ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  ->  sum_ a  e.  V  (
# `  U_ b  e.  ( V  \  {
a } ) ( a ( V 2SPathOnOt  E ) b ) )  = 
sum_ a  e.  V  ( ( # `  V
)  -  1 ) )
61 hashcl 12576 . . . . . . . 8  |-  ( V  e.  Fin  ->  ( # `
 V )  e. 
NN0 )
6261nn0cnd 10951 . . . . . . 7  |-  ( V  e.  Fin  ->  ( # `
 V )  e.  CC )
63 1cnd 9677 . . . . . . 7  |-  ( V  e.  Fin  ->  1  e.  CC )
6462, 63subcld 10005 . . . . . 6  |-  ( V  e.  Fin  ->  (
( # `  V )  -  1 )  e.  CC )
65 fsumconst 13928 . . . . . 6  |-  ( ( V  e.  Fin  /\  ( ( # `  V
)  -  1 )  e.  CC )  ->  sum_ a  e.  V  ( ( # `  V
)  -  1 )  =  ( ( # `  V )  x.  (
( # `  V )  -  1 ) ) )
6664, 65mpdan 681 . . . . 5  |-  ( V  e.  Fin  ->  sum_ a  e.  V  ( ( # `
 V )  - 
1 )  =  ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) ) )
6766adantr 472 . . . 4  |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  sum_ a  e.  V  ( ( # `
 V )  - 
1 )  =  ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) ) )
6867adantl 473 . . 3  |-  ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  ->  sum_ a  e.  V  ( ( # `  V
)  -  1 )  =  ( ( # `  V )  x.  (
( # `  V )  -  1 ) ) )
6960, 68eqtrd 2505 . 2  |-  ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  ->  sum_ a  e.  V  (
# `  U_ b  e.  ( V  \  {
a } ) ( a ( V 2SPathOnOt  E ) b ) )  =  ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) ) )
706, 29, 693eqtrd 2509 1  |-  ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  -> 
( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   _Vcvv 3031    \ cdif 3387   (/)c0 3722   {csn 3959   U_ciun 4269  Disj wdisj 4366   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Fincfn 7587   CCcc 9555   RRcr 9556   1c1 9558    x. cmul 9562    - cmin 9880   NNcn 10631   NN0cn0 10893   #chash 12553   sum_csu 13829   USGrph cusg 25136   2SPathOnOt c2spthot 25663   2SPathOnOt c2pthonot 25664   FriendGrph cfrgra 25795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-ot 3968  df-uni 4191  df-int 4227  df-iun 4271  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-word 12711  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-usgra 25139  df-wlk 25315  df-trail 25316  df-pth 25317  df-spth 25318  df-wlkon 25321  df-spthon 25324  df-2wlkonot 25665  df-2spthonot 25667  df-2spthsot 25668  df-frgra 25796
This theorem is referenced by:  frgregordn0  25877
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