Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frgregordn0 Structured version   Unicode version

Theorem frgregordn0 30660
Description: If a nonempty friendship graph is k-regular, its order is k(k-1)+1. This corresponds to the third claim in the proof of the friendship theorem in [Huneke] p. 2: "Next we claim that the number n of vertices in G is exactly k(k-1)+1.". (Contributed by Alexander van der Vekens, 11-Mar-2018.)
Assertion
Ref Expression
frgregordn0  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) )
Distinct variable groups:    v, E    v, K    v, V

Proof of Theorem frgregordn0
StepHypRef Expression
1 frisusgra 30581 . . . . 5  |-  ( V FriendGrph  E  ->  V USGrph  E )
2 usgreghash2spot 30659 . . . . 5  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( K  x.  ( K  - 
1 ) ) ) ) )
31, 2syl3an1 1251 . . . 4  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( K  x.  ( K  - 
1 ) ) ) ) )
43imp 429 . . 3  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( K  x.  ( K  - 
1 ) ) ) )
5 frghash2spot 30653 . . . . . 6  |-  ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  -> 
( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) ) )
653impb 1183 . . . . 5  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) ) )
76adantr 465 . . . 4  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) ) )
8 eqeq1 2447 . . . . . 6  |-  ( (
# `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) )  ->  ( ( # `  ( V 2SPathOnOt  E )
)  =  ( (
# `  V )  x.  ( K  x.  ( K  -  1 ) ) )  <->  ( ( # `
 V )  x.  ( ( # `  V
)  -  1 ) )  =  ( (
# `  V )  x.  ( K  x.  ( K  -  1 ) ) ) ) )
9 hashcl 12124 . . . . . . . . . . . . 13  |-  ( V  e.  Fin  ->  ( # `
 V )  e. 
NN0 )
109nn0cnd 10636 . . . . . . . . . . . 12  |-  ( V  e.  Fin  ->  ( # `
 V )  e.  CC )
11 ax-1cn 9338 . . . . . . . . . . . . 13  |-  1  e.  CC
1211a1i 11 . . . . . . . . . . . 12  |-  ( V  e.  Fin  ->  1  e.  CC )
1310, 12subcld 9717 . . . . . . . . . . 11  |-  ( V  e.  Fin  ->  (
( # `  V )  -  1 )  e.  CC )
14133ad2ant2 1010 . . . . . . . . . 10  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( ( # `
 V )  - 
1 )  e.  CC )
1514adantr 465 . . . . . . . . 9  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( ( # `
 V )  - 
1 )  e.  CC )
16 usgfiregdegfi 30525 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  K  e.  NN0 ) )
171, 16syl3an1 1251 . . . . . . . . . . 11  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  K  e.  NN0 ) )
1817imp 429 . . . . . . . . . 10  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  K  e.  NN0 )
19 nn0cn 10587 . . . . . . . . . . 11  |-  ( K  e.  NN0  ->  K  e.  CC )
20 kcnktkm1cn 30167 . . . . . . . . . . 11  |-  ( K  e.  CC  ->  ( K  x.  ( K  -  1 ) )  e.  CC )
2119, 20syl 16 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  ( K  x.  ( K  - 
1 ) )  e.  CC )
2218, 21syl 16 . . . . . . . . 9  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( K  x.  ( K  -  1 ) )  e.  CC )
23103ad2ant2 1010 . . . . . . . . . 10  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( # `  V
)  e.  CC )
2423adantr 465 . . . . . . . . 9  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( # `  V
)  e.  CC )
25 hasheq0 12129 . . . . . . . . . . . . . 14  |-  ( V  e.  Fin  ->  (
( # `  V )  =  0  <->  V  =  (/) ) )
2625biimpd 207 . . . . . . . . . . . . 13  |-  ( V  e.  Fin  ->  (
( # `  V )  =  0  ->  V  =  (/) ) )
2726necon3d 2644 . . . . . . . . . . . 12  |-  ( V  e.  Fin  ->  ( V  =/=  (/)  ->  ( # `  V
)  =/=  0 ) )
2827imp 429 . . . . . . . . . . 11  |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  ( # `
 V )  =/=  0 )
29283adant1 1006 . . . . . . . . . 10  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( # `  V
)  =/=  0 )
3029adantr 465 . . . . . . . . 9  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( # `  V
)  =/=  0 )
3115, 22, 24, 30mulcand 9967 . . . . . . . 8  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( (
( # `  V )  x.  ( ( # `  V )  -  1 ) )  =  ( ( # `  V
)  x.  ( K  x.  ( K  - 
1 ) ) )  <-> 
( ( # `  V
)  -  1 )  =  ( K  x.  ( K  -  1
) ) ) )
3211a1i 11 . . . . . . . . 9  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  1  e.  CC )
33 subadd2 9612 . . . . . . . . . . 11  |-  ( ( ( # `  V
)  e.  CC  /\  1  e.  CC  /\  ( K  x.  ( K  -  1 ) )  e.  CC )  -> 
( ( ( # `  V )  -  1 )  =  ( K  x.  ( K  - 
1 ) )  <->  ( ( K  x.  ( K  -  1 ) )  +  1 )  =  ( # `  V
) ) )
34 eqcom 2443 . . . . . . . . . . 11  |-  ( ( ( K  x.  ( K  -  1 ) )  +  1 )  =  ( # `  V
)  <->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) )
3533, 34syl6bb 261 . . . . . . . . . 10  |-  ( ( ( # `  V
)  e.  CC  /\  1  e.  CC  /\  ( K  x.  ( K  -  1 ) )  e.  CC )  -> 
( ( ( # `  V )  -  1 )  =  ( K  x.  ( K  - 
1 ) )  <->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) )
3635biimpd 207 . . . . . . . . 9  |-  ( ( ( # `  V
)  e.  CC  /\  1  e.  CC  /\  ( K  x.  ( K  -  1 ) )  e.  CC )  -> 
( ( ( # `  V )  -  1 )  =  ( K  x.  ( K  - 
1 ) )  -> 
( # `  V )  =  ( ( K  x.  ( K  - 
1 ) )  +  1 ) ) )
3724, 32, 22, 36syl3anc 1218 . . . . . . . 8  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( (
( # `  V )  -  1 )  =  ( K  x.  ( K  -  1 ) )  ->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) )
3831, 37sylbid 215 . . . . . . 7  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( (
( # `  V )  x.  ( ( # `  V )  -  1 ) )  =  ( ( # `  V
)  x.  ( K  x.  ( K  - 
1 ) ) )  ->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) )
3938com12 31 . . . . . 6  |-  ( ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) )  =  ( ( # `  V )  x.  ( K  x.  ( K  -  1 ) ) )  ->  ( (
( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  K )  ->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) )
408, 39syl6bi 228 . . . . 5  |-  ( (
# `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) )  ->  ( ( # `  ( V 2SPathOnOt  E )
)  =  ( (
# `  V )  x.  ( K  x.  ( K  -  1 ) ) )  ->  (
( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) ) )
4140com23 78 . . . 4  |-  ( (
# `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) )  ->  ( ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  K )  ->  ( ( # `  ( V 2SPathOnOt  E )
)  =  ( (
# `  V )  x.  ( K  x.  ( K  -  1 ) ) )  ->  ( # `
 V )  =  ( ( K  x.  ( K  -  1
) )  +  1 ) ) ) )
427, 41mpcom 36 . . 3  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( ( # `
 ( V 2SPathOnOt  E ) )  =  ( (
# `  V )  x.  ( K  x.  ( K  -  1 ) ) )  ->  ( # `
 V )  =  ( ( K  x.  ( K  -  1
) )  +  1 ) ) )
434, 42mpd 15 . 2  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) )
4443ex 434 1  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2604   A.wral 2713   (/)c0 3635   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   Fincfn 7308   CCcc 9278   0cc0 9280   1c1 9281    + caddc 9283    x. cmul 9285    - cmin 9593   NN0cn0 10577   #chash 12101   USGrph cusg 23262   VDeg cvdg 23561   2SPathOnOt c2spthot 30372   FriendGrph cfrgra 30577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-ot 3884  df-uni 4090  df-int 4127  df-iun 4171  df-disj 4261  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-er 7099  df-map 7214  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-sup 7689  df-oi 7722  df-card 8107  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-n0 10578  df-z 10645  df-uz 10860  df-rp 10990  df-xadd 11088  df-fz 11436  df-fzo 11547  df-seq 11805  df-exp 11864  df-hash 12102  df-word 12227  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-clim 12964  df-sum 13162  df-usgra 23264  df-nbgra 23330  df-wlk 23413  df-trail 23414  df-pth 23415  df-spth 23416  df-wlkon 23419  df-spthon 23422  df-vdgr 23562  df-2wlkonot 30374  df-2spthonot 30376  df-2spthsot 30377  df-frgra 30578
This theorem is referenced by:  frrusgraord  30661
  Copyright terms: Public domain W3C validator