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Theorem frgrareggt1 25243
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 25056 . . . . 5  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K ) )
2 2z 10917 . . . . . . . . . . . 12  |-  2  e.  ZZ
32a1i 11 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  2  <  K )  -> 
2  e.  ZZ )
4 nn0z 10908 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  K  e.  ZZ )
5 peano2zm 10928 . . . . . . . . . . . . 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 10937 . . . . . . . . . . . . 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 11112 . . . . . . . . . . 11  |-  ( ( K  -  1 )  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  ( K  -  1 )  e.  ZZ  /\  2  <_ 
( K  -  1 ) ) )
123, 7, 10, 11syl3anbrc 1180 . . . . . . . . . 10  |-  ( ( K  e.  NN0  /\  2  <  K )  -> 
( K  -  1 )  e.  ( ZZ>= ` 
2 ) )
13 exprmfct 14263 . . . . . . . . . 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 996 . . . . . . . . . . . . . . 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 1014 . . . . . . . . . . . . . . 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 25241 . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . 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 25119 . . . . . . . . . . . . . . . . . . 19  |-  ( V FriendGrph  E  ->  V USGrph  E )
25243ad2ant1 1017 . . . . . . . . . . . . . . . . . 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 1003 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( p  e. 
Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  /\  ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) ) )  ->  V  e.  Fin )
28 simplll 759 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( p  e. 
Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  /\  ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) ) )  ->  p  e.  Prime )
29 numclwwlk8 25242 . . . . . . . . . . . . . . . . 17  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  p  e.  Prime )  ->  ( ( # `
 ( ( V ClWWalksN  E ) `  p
) )  mod  p
)  =  0 )
3026, 27, 28, 29syl3anc 1228 . . . . . . . . . . . . . . . 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 2461 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  (
( V ClWWalksN  E ) `  p ) )  mod  p )  =  0  ->  ( ( (
# `  ( ( V ClWWalksN  E ) `  p
) )  mod  p
)  =  1  <->  0  =  1 ) )
32 ax-1ne0 9578 . . . . . . . . . . . . . . . . . . 19  |-  1  =/=  0
3332nesymi 2730 . . . . . . . . . . . . . . . . . 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 2945 . . . . . . . . 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 1018 . . . . 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 9628 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  1  e.  RR )
51 nn0re 10825 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  K  e.  RR )
5250, 51ltnled 9749 . . . . . . . . 9  |-  ( K  e.  NN0  ->  ( 1  <  K  <->  -.  K  <_  1 ) )
53 1e2m1 10672 . . . . . . . . . . . 12  |-  1  =  ( 2  -  1 )
5453a1i 11 . . . . . . . . . . 11  |-  ( K  e.  NN0  ->  1  =  ( 2  -  1 ) )
5554breq2d 4468 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  ( K  <_  1  <->  K  <_  ( 2  -  1 ) ) )
5655notbid 294 . . . . . . . . 9  |-  ( K  e.  NN0  ->  ( -.  K  <_  1  <->  -.  K  <_  ( 2  -  1 ) ) )
57 zltlem1 10937 . . . . . . . . . . . 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 10626 . . . . . . . . . 10  |-  2  e.  RR
63 lttri3 9685 . . . . . . . . . . 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 1018 . . . . . 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   (/)c0 3793   <.cop 4038   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Fincfn 7535   RRcr 9508   0cc0 9509   1c1 9510    < clt 9645    <_ cle 9646    - cmin 9824   2c2 10606   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106    mod cmo 11999   #chash 12408    || cdvds 13998   Primecprime 14229   USGrph cusg 24457   ClWWalksN cclwwlkn 24876   VDeg cvdg 25020   RegUSGrph crusgra 25050   FriendGrph cfrgra 25115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-ot 4041  df-uni 4252  df-int 4289  df-iun 4334  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-ec 7331  df-qs 7335  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-xadd 11344  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-hash 12409  df-word 12546  df-lsw 12547  df-concat 12548  df-s1 12549  df-substr 12550  df-reps 12553  df-csh 12772  df-s2 12825  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-sum 13521  df-dvds 13999  df-gcd 14157  df-prm 14230  df-phi 14308  df-usgra 24460  df-nbgra 24547  df-wlk 24635  df-trail 24636  df-pth 24637  df-spth 24638  df-wlkon 24641  df-spthon 24644  df-wwlk 24806  df-wwlkn 24807  df-clwwlk 24878  df-clwwlkn 24879  df-2wlkonot 24985  df-2spthonot 24987  df-2spthsot 24988  df-vdgr 25021  df-rgra 25051  df-rusgra 25052  df-frgra 25116
This theorem is referenced by:  frgrareg  25244
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