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Theorem frgrareggt1 30858
Description: If a finite friendship graph is k-regular with k > 1, then k must be 2. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
Assertion
Ref Expression
frgrareggt1  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( ( <. V ,  E >. RegUSGrph  K  /\  1  <  K )  ->  K  =  2 ) )

Proof of Theorem frgrareggt1
Dummy variables  p  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rusgraprop 30695 . . . . 5  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K ) )
2 2z 10790 . . . . . . . . . . . 12  |-  2  e.  ZZ
32a1i 11 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  2  <  K )  -> 
2  e.  ZZ )
4 nn0z 10781 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  K  e.  ZZ )
5 peano2zm 10800 . . . . . . . . . . . . 13  |-  ( K  e.  ZZ  ->  ( K  -  1 )  e.  ZZ )
64, 5syl 16 . . . . . . . . . . . 12  |-  ( K  e.  NN0  ->  ( K  -  1 )  e.  ZZ )
76adantr 465 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  2  <  K )  -> 
( K  -  1 )  e.  ZZ )
8 zltlem1 10809 . . . . . . . . . . . . 13  |-  ( ( 2  e.  ZZ  /\  K  e.  ZZ )  ->  ( 2  <  K  <->  2  <_  ( K  - 
1 ) ) )
92, 4, 8sylancr 663 . . . . . . . . . . . 12  |-  ( K  e.  NN0  ->  ( 2  <  K  <->  2  <_  ( K  -  1 ) ) )
109biimpa 484 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  2  <  K )  -> 
2  <_  ( K  -  1 ) )
11 eluz2 10979 . . . . . . . . . . 11  |-  ( ( K  -  1 )  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  ( K  -  1 )  e.  ZZ  /\  2  <_ 
( K  -  1 ) ) )
123, 7, 10, 11syl3anbrc 1172 . . . . . . . . . 10  |-  ( ( K  e.  NN0  /\  2  <  K )  -> 
( K  -  1 )  e.  ( ZZ>= ` 
2 ) )
13 exprmfct 13915 . . . . . . . . . 10  |-  ( ( K  -  1 )  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  ( K  -  1 ) )
1412, 13syl 16 . . . . . . . . 9  |-  ( ( K  e.  NN0  /\  2  <  K )  ->  E. p  e.  Prime  p 
||  ( K  - 
1 ) )
15 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( ( p  e.  Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  ->  <. V ,  E >. RegUSGrph  K )
16 simp1 988 . . . . . . . . . . . . . . 15  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  V FriendGrph  E )
1715, 16anim12i 566 . . . . . . . . . . . . . 14  |-  ( ( ( ( p  e. 
Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  /\  ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) ) )  ->  ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E ) )
18 pm3.22 449 . . . . . . . . . . . . . . . 16  |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  ( V  =/=  (/)  /\  V  e. 
Fin ) )
19183adant1 1006 . . . . . . . . . . . . . . 15  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( V  =/=  (/)  /\  V  e. 
Fin ) )
2019adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ( ( p  e. 
Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  /\  ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) ) )  ->  ( V  =/=  (/)  /\  V  e. 
Fin ) )
21 simpll 753 . . . . . . . . . . . . . 14  |-  ( ( ( ( p  e. 
Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  /\  ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) ) )  ->  (
p  e.  Prime  /\  p  ||  ( K  -  1 ) ) )
22 numclwwlk7 30856 . . . . . . . . . . . . . 14  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  (
p  e.  Prime  /\  p  ||  ( K  -  1 ) ) )  -> 
( ( # `  (
( V ClWWalksN  E ) `  p ) )  mod  p )  =  1 )
2317, 20, 21, 22syl3anc 1219 . . . . . . . . . . . . 13  |-  ( ( ( ( p  e. 
Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  /\  ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) ) )  ->  (
( # `  ( ( V ClWWalksN  E ) `  p
) )  mod  p
)  =  1 )
24 frisusgra 30733 . . . . . . . . . . . . . . . . . . 19  |-  ( V FriendGrph  E  ->  V USGrph  E )
25243ad2ant1 1009 . . . . . . . . . . . . . . . . . 18  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  V USGrph  E )
2625adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( p  e. 
Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  /\  ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) ) )  ->  V USGrph  E )
27 simpr2 995 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( p  e. 
Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  /\  ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) ) )  ->  V  e.  Fin )
28 simplll 757 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( p  e. 
Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  /\  ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) ) )  ->  p  e.  Prime )
29 numclwwlk8 30857 . . . . . . . . . . . . . . . . 17  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  p  e.  Prime )  ->  ( ( # `
 ( ( V ClWWalksN  E ) `  p
) )  mod  p
)  =  0 )
3026, 27, 28, 29syl3anc 1219 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( p  e. 
Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  /\  ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) ) )  ->  (
( # `  ( ( V ClWWalksN  E ) `  p
) )  mod  p
)  =  0 )
31 eqeq1 2458 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  (
( V ClWWalksN  E ) `  p ) )  mod  p )  =  0  ->  ( ( (
# `  ( ( V ClWWalksN  E ) `  p
) )  mod  p
)  =  1  <->  0  =  1 ) )
32 ax-1ne0 9463 . . . . . . . . . . . . . . . . . . 19  |-  1  =/=  0
3332nesymi 2725 . . . . . . . . . . . . . . . . . 18  |-  -.  0  =  1
3433pm2.21i 131 . . . . . . . . . . . . . . . . 17  |-  ( 0  =  1  ->  K  =  2 )
3531, 34syl6bi 228 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  (
( V ClWWalksN  E ) `  p ) )  mod  p )  =  0  ->  ( ( (
# `  ( ( V ClWWalksN  E ) `  p
) )  mod  p
)  =  1  ->  K  =  2 ) )
3630, 35syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ( p  e. 
Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  /\  ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) ) )  ->  (
( ( # `  (
( V ClWWalksN  E ) `  p ) )  mod  p )  =  1  ->  K  =  2 ) )
3736a1i 11 . . . . . . . . . . . . . 14  |-  ( ( K  e.  NN0  /\  2  <  K )  -> 
( ( ( ( p  e.  Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  /\  ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) ) )  -> 
( ( ( # `  ( ( V ClWWalksN  E ) `
 p ) )  mod  p )  =  1  ->  K  = 
2 ) ) )
3837com13 80 . . . . . . . . . . . . 13  |-  ( ( ( # `  (
( V ClWWalksN  E ) `  p ) )  mod  p )  =  1  ->  ( ( ( ( p  e.  Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  /\  ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) ) )  ->  (
( K  e.  NN0  /\  2  <  K )  ->  K  =  2 ) ) )
3923, 38mpcom 36 . . . . . . . . . . . 12  |-  ( ( ( ( p  e. 
Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  /\  ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) ) )  ->  (
( K  e.  NN0  /\  2  <  K )  ->  K  =  2 ) )
4039exp31 604 . . . . . . . . . . 11  |-  ( ( p  e.  Prime  /\  p  ||  ( K  -  1 ) )  ->  ( <. V ,  E >. RegUSGrph  K  ->  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  ->  (
( K  e.  NN0  /\  2  <  K )  ->  K  =  2 ) ) ) )
4140com24 87 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  p  ||  ( K  -  1 ) )  ->  (
( K  e.  NN0  /\  2  <  K )  ->  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  -> 
( <. V ,  E >. RegUSGrph  K  ->  K  =  2 ) ) ) )
4241rexlimiva 2942 . . . . . . . . 9  |-  ( E. p  e.  Prime  p  ||  ( K  -  1 )  ->  ( ( K  e.  NN0  /\  2  <  K )  ->  (
( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  ->  ( <. V ,  E >. RegUSGrph  K  ->  K  =  2 ) ) ) )
4314, 42mpcom 36 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  2  <  K )  -> 
( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  ->  ( <. V ,  E >. RegUSGrph  K  ->  K  =  2 ) ) )
4443ex 434 . . . . . . 7  |-  ( K  e.  NN0  ->  ( 2  <  K  ->  (
( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  ->  ( <. V ,  E >. RegUSGrph  K  ->  K  =  2 ) ) ) )
4544com24 87 . . . . . 6  |-  ( K  e.  NN0  ->  ( <. V ,  E >. RegUSGrph  K  ->  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  ->  (
2  <  K  ->  K  =  2 ) ) ) )
46453ad2ant2 1010 . . . . 5  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  K )  ->  ( <. V ,  E >. RegUSGrph  K  ->  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( 2  <  K  ->  K  =  2 ) ) ) )
471, 46mpcom 36 . . . 4  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  ->  (
2  <  K  ->  K  =  2 ) ) )
4847adantr 465 . . 3  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  1  <  K )  ->  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  -> 
( 2  <  K  ->  K  =  2 ) ) )
4948com13 80 . 2  |-  ( 2  <  K  ->  (
( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  ->  ( (
<. V ,  E >. RegUSGrph  K  /\  1  <  K )  ->  K  =  2 ) ) )
50 1red 9513 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  1  e.  RR )
51 nn0re 10700 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  K  e.  RR )
5250, 51ltnled 9633 . . . . . . . . 9  |-  ( K  e.  NN0  ->  ( 1  <  K  <->  -.  K  <_  1 ) )
53 1e2m1 10549 . . . . . . . . . . . 12  |-  1  =  ( 2  -  1 )
5453a1i 11 . . . . . . . . . . 11  |-  ( K  e.  NN0  ->  1  =  ( 2  -  1 ) )
5554breq2d 4413 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  ( K  <_  1  <->  K  <_  ( 2  -  1 ) ) )
5655notbid 294 . . . . . . . . 9  |-  ( K  e.  NN0  ->  ( -.  K  <_  1  <->  -.  K  <_  ( 2  -  1 ) ) )
57 zltlem1 10809 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  2  e.  ZZ )  ->  ( K  <  2  <->  K  <_  ( 2  -  1 ) ) )
584, 2, 57sylancl 662 . . . . . . . . . . 11  |-  ( K  e.  NN0  ->  ( K  <  2  <->  K  <_  ( 2  -  1 ) ) )
5958bicomd 201 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  ( K  <_  ( 2  -  1 )  <->  K  <  2 ) )
6059notbid 294 . . . . . . . . 9  |-  ( K  e.  NN0  ->  ( -.  K  <_  ( 2  -  1 )  <->  -.  K  <  2 ) )
6152, 56, 603bitrd 279 . . . . . . . 8  |-  ( K  e.  NN0  ->  ( 1  <  K  <->  -.  K  <  2 ) )
62 2re 10503 . . . . . . . . . 10  |-  2  e.  RR
63 lttri3 9570 . . . . . . . . . . 11  |-  ( ( K  e.  RR  /\  2  e.  RR )  ->  ( K  =  2  <-> 
( -.  K  <  2  /\  -.  2  <  K ) ) )
6463biimprd 223 . . . . . . . . . 10  |-  ( ( K  e.  RR  /\  2  e.  RR )  ->  ( ( -.  K  <  2  /\  -.  2  <  K )  ->  K  =  2 ) )
6551, 62, 64sylancl 662 . . . . . . . . 9  |-  ( K  e.  NN0  ->  ( ( -.  K  <  2  /\  -.  2  <  K
)  ->  K  = 
2 ) )
6665expd 436 . . . . . . . 8  |-  ( K  e.  NN0  ->  ( -.  K  <  2  -> 
( -.  2  < 
K  ->  K  = 
2 ) ) )
6761, 66sylbid 215 . . . . . . 7  |-  ( K  e.  NN0  ->  ( 1  <  K  ->  ( -.  2  <  K  ->  K  =  2 ) ) )
68673ad2ant2 1010 . . . . . 6  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  K )  ->  ( 1  < 
K  ->  ( -.  2  <  K  ->  K  =  2 ) ) )
691, 68syl 16 . . . . 5  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( 1  <  K  ->  ( -.  2  < 
K  ->  K  = 
2 ) ) )
7069imp 429 . . . 4  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  1  <  K )  ->  ( -.  2  <  K  ->  K  = 
2 ) )
7170com12 31 . . 3  |-  ( -.  2  <  K  -> 
( ( <. V ,  E >. RegUSGrph  K  /\  1  < 
K )  ->  K  =  2 ) )
7271a1d 25 . 2  |-  ( -.  2  <  K  -> 
( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  ->  (
( <. V ,  E >. RegUSGrph  K  /\  1  <  K
)  ->  K  = 
2 ) ) )
7349, 72pm2.61i 164 1  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( ( <. V ,  E >. RegUSGrph  K  /\  1  <  K )  ->  K  =  2 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800   (/)c0 3746   <.cop 3992   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   Fincfn 7421   RRcr 9393   0cc0 9394   1c1 9395    < clt 9530    <_ cle 9531    - cmin 9707   2c2 10483   NN0cn0 10691   ZZcz 10758   ZZ>=cuz 10973    mod cmo 11826   #chash 12221    || cdivides 13654   Primecprime 13882   USGrph cusg 23417   VDeg cvdg 23716   ClWWalksN cclwwlkn 30563   RegUSGrph crusgra 30689   FriendGrph cfrgra 30729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-ot 3995  df-uni 4201  df-int 4238  df-iun 4282  df-disj 4372  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-2o 7032  df-oadd 7035  df-er 7212  df-ec 7214  df-qs 7218  df-map 7327  df-pm 7328  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-sup 7803  df-oi 7836  df-card 8221  df-cda 8449  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-n0 10692  df-z 10759  df-uz 10974  df-rp 11104  df-xadd 11202  df-fz 11556  df-fzo 11667  df-fl 11760  df-mod 11827  df-seq 11925  df-exp 11984  df-hash 12222  df-word 12348  df-lsw 12349  df-concat 12350  df-s1 12351  df-substr 12352  df-reps 12355  df-csh 12545  df-s2 12594  df-cj 12707  df-re 12708  df-im 12709  df-sqr 12843  df-abs 12844  df-clim 13085  df-sum 13283  df-dvds 13655  df-gcd 13810  df-prm 13883  df-phi 13960  df-usgra 23419  df-nbgra 23485  df-wlk 23568  df-trail 23569  df-pth 23570  df-spth 23571  df-wlkon 23574  df-spthon 23577  df-vdgr 23717  df-wwlk 30462  df-wwlkn 30463  df-2wlkonot 30526  df-2spthonot 30528  df-2spthsot 30529  df-clwwlk 30565  df-clwwlkn 30566  df-rgra 30690  df-rusgra 30691  df-frgra 30730
This theorem is referenced by:  frgrareg  30859
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