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Theorem frgrareggt1 24779
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 24591 . . . . 5  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K ) )
2 2z 10885 . . . . . . . . . . . 12  |-  2  e.  ZZ
32a1i 11 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  2  <  K )  -> 
2  e.  ZZ )
4 nn0z 10876 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  K  e.  ZZ )
5 peano2zm 10895 . . . . . . . . . . . . 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 10904 . . . . . . . . . . . . 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 11077 . . . . . . . . . . 11  |-  ( ( K  -  1 )  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  ( K  -  1 )  e.  ZZ  /\  2  <_ 
( K  -  1 ) ) )
123, 7, 10, 11syl3anbrc 1175 . . . . . . . . . 10  |-  ( ( K  e.  NN0  /\  2  <  K )  -> 
( K  -  1 )  e.  ( ZZ>= ` 
2 ) )
13 exprmfct 14099 . . . . . . . . . 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 991 . . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . . . 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 24777 . . . . . . . . . . . . . 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 1223 . . . . . . . . . . . . 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 24654 . . . . . . . . . . . . . . . . . . 19  |-  ( V FriendGrph  E  ->  V USGrph  E )
25243ad2ant1 1012 . . . . . . . . . . . . . . . . . 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 998 . . . . . . . . . . . . . . . . 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 24778 . . . . . . . . . . . . . . . . 17  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  p  e.  Prime )  ->  ( ( # `
 ( ( V ClWWalksN  E ) `  p
) )  mod  p
)  =  0 )
3026, 27, 28, 29syl3anc 1223 . . . . . . . . . . . . . . . 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 2464 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  (
( V ClWWalksN  E ) `  p ) )  mod  p )  =  0  ->  ( ( (
# `  ( ( V ClWWalksN  E ) `  p
) )  mod  p
)  =  1  <->  0  =  1 ) )
32 ax-1ne0 9550 . . . . . . . . . . . . . . . . . . 19  |-  1  =/=  0
3332nesymi 2733 . . . . . . . . . . . . . . . . . 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 2944 . . . . . . . . 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 1013 . . . . 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 9600 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  1  e.  RR )
51 nn0re 10793 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  K  e.  RR )
5250, 51ltnled 9720 . . . . . . . . 9  |-  ( K  e.  NN0  ->  ( 1  <  K  <->  -.  K  <_  1 ) )
53 1e2m1 10640 . . . . . . . . . . . 12  |-  1  =  ( 2  -  1 )
5453a1i 11 . . . . . . . . . . 11  |-  ( K  e.  NN0  ->  1  =  ( 2  -  1 ) )
5554breq2d 4452 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  ( K  <_  1  <->  K  <_  ( 2  -  1 ) ) )
5655notbid 294 . . . . . . . . 9  |-  ( K  e.  NN0  ->  ( -.  K  <_  1  <->  -.  K  <_  ( 2  -  1 ) ) )
57 zltlem1 10904 . . . . . . . . . . . 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 10594 . . . . . . . . . 10  |-  2  e.  RR
63 lttri3 9657 . . . . . . . . . . 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 1013 . . . . . 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 968    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   E.wrex 2808   (/)c0 3778   <.cop 4026   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Fincfn 7506   RRcr 9480   0cc0 9481   1c1 9482    < clt 9617    <_ cle 9618    - cmin 9794   2c2 10574   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071    mod cmo 11952   #chash 12360    || cdivides 13836   Primecprime 14065   USGrph cusg 23993   ClWWalksN cclwwlkn 24411   VDeg cvdg 24555   RegUSGrph crusgra 24585   FriendGrph cfrgra 24650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-ot 4029  df-uni 4239  df-int 4276  df-iun 4320  df-disj 4411  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-ec 7303  df-qs 7307  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-xadd 11308  df-fz 11662  df-fzo 11782  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-hash 12361  df-word 12495  df-lsw 12496  df-concat 12497  df-s1 12498  df-substr 12499  df-reps 12502  df-csh 12710  df-s2 12763  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-dvds 13837  df-gcd 13993  df-prm 14066  df-phi 14144  df-usgra 23996  df-nbgra 24082  df-wlk 24170  df-trail 24171  df-pth 24172  df-spth 24173  df-wlkon 24176  df-spthon 24179  df-wwlk 24341  df-wwlkn 24342  df-clwwlk 24413  df-clwwlkn 24414  df-2wlkonot 24520  df-2spthonot 24522  df-2spthsot 24523  df-vdgr 24556  df-rgra 24586  df-rusgra 24587  df-frgra 24651
This theorem is referenced by:  frgrareg  24780
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