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Theorem frgregordn0 24744
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 24665 . . . . 5  |-  ( V FriendGrph  E  ->  V USGrph  E )
2 usgreghash2spot 24743 . . . . 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 1261 . . . 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 24737 . . . . . 6  |-  ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  -> 
( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) ) )
653impb 1192 . . . . 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 2471 . . . . . 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 12390 . . . . . . . . . . . . 13  |-  ( V  e.  Fin  ->  ( # `
 V )  e. 
NN0 )
109nn0cnd 10850 . . . . . . . . . . . 12  |-  ( V  e.  Fin  ->  ( # `
 V )  e.  CC )
11 ax-1cn 9546 . . . . . . . . . . . . 13  |-  1  e.  CC
1211a1i 11 . . . . . . . . . . . 12  |-  ( V  e.  Fin  ->  1  e.  CC )
1310, 12subcld 9926 . . . . . . . . . . 11  |-  ( V  e.  Fin  ->  (
( # `  V )  -  1 )  e.  CC )
14133ad2ant2 1018 . . . . . . . . . 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 24584 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  K  e.  NN0 ) )
171, 16syl3an1 1261 . . . . . . . . . . 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 10801 . . . . . . . . . . 11  |-  ( K  e.  NN0  ->  K  e.  CC )
20 kcnktkm1cn 9984 . . . . . . . . . . 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 1018 . . . . . . . . . 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 12395 . . . . . . . . . . . . . 14  |-  ( V  e.  Fin  ->  (
( # `  V )  =  0  <->  V  =  (/) ) )
2625biimpd 207 . . . . . . . . . . . . 13  |-  ( V  e.  Fin  ->  (
( # `  V )  =  0  ->  V  =  (/) ) )
2726necon3d 2691 . . . . . . . . . . . 12  |-  ( V  e.  Fin  ->  ( V  =/=  (/)  ->  ( # `  V
)  =/=  0 ) )
2827imp 429 . . . . . . . . . . 11  |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  ( # `
 V )  =/=  0 )
29283adant1 1014 . . . . . . . . . 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 10178 . . . . . . . 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 9820 . . . . . . . . . . 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 2476 . . . . . . . . . . 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 1228 . . . . . . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   (/)c0 3785   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Fincfn 7513   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    - cmin 9801   NN0cn0 10791   #chash 12367   USGrph cusg 24003   2SPathOnOt c2spthot 24529   VDeg cvdg 24566   FriendGrph cfrgra 24661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-xadd 11315  df-fz 11669  df-fzo 11789  df-seq 12071  df-exp 12130  df-hash 12368  df-word 12502  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-clim 13267  df-sum 13465  df-usgra 24006  df-nbgra 24093  df-wlk 24181  df-trail 24182  df-pth 24183  df-spth 24184  df-wlkon 24187  df-spthon 24190  df-2wlkonot 24531  df-2spthonot 24533  df-2spthsot 24534  df-vdgr 24567  df-frgra 24662
This theorem is referenced by:  frrusgraord  24745
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