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Theorem frgrareggt1 25842
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 25655 . . . . 5  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K ) )
2 2z 10976 . . . . . . . . . . . 12  |-  2  e.  ZZ
32a1i 11 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  2  <  K )  -> 
2  e.  ZZ )
4 nn0z 10967 . . . . . . . . . . . . 13  |-  ( K  e.  NN0  ->  K  e.  ZZ )
5 peano2zm 10987 . . . . . . . . . . . . 13  |-  ( K  e.  ZZ  ->  ( K  -  1 )  e.  ZZ )
64, 5syl 17 . . . . . . . . . . . 12  |-  ( K  e.  NN0  ->  ( K  -  1 )  e.  ZZ )
76adantr 466 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  2  <  K )  -> 
( K  -  1 )  e.  ZZ )
8 zltlem1 10996 . . . . . . . . . . . . 13  |-  ( ( 2  e.  ZZ  /\  K  e.  ZZ )  ->  ( 2  <  K  <->  2  <_  ( K  - 
1 ) ) )
92, 4, 8sylancr 667 . . . . . . . . . . . 12  |-  ( K  e.  NN0  ->  ( 2  <  K  <->  2  <_  ( K  -  1 ) ) )
109biimpa 486 . . . . . . . . . . 11  |-  ( ( K  e.  NN0  /\  2  <  K )  -> 
2  <_  ( K  -  1 ) )
11 eluz2 11172 . . . . . . . . . . 11  |-  ( ( K  -  1 )  e.  ( ZZ>= `  2
)  <->  ( 2  e.  ZZ  /\  ( K  -  1 )  e.  ZZ  /\  2  <_ 
( K  -  1 ) ) )
123, 7, 10, 11syl3anbrc 1189 . . . . . . . . . 10  |-  ( ( K  e.  NN0  /\  2  <  K )  -> 
( K  -  1 )  e.  ( ZZ>= ` 
2 ) )
13 exprmfct 14647 . . . . . . . . . 10  |-  ( ( K  -  1 )  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  ( K  -  1 ) )
1412, 13syl 17 . . . . . . . . 9  |-  ( ( K  e.  NN0  /\  2  <  K )  ->  E. p  e.  Prime  p 
||  ( K  - 
1 ) )
15 simpr 462 . . . . . . . . . . . . . . 15  |-  ( ( ( p  e.  Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  ->  <. V ,  E >. RegUSGrph  K )
16 simp1 1005 . . . . . . . . . . . . . . 15  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  V FriendGrph  E )
1715, 16anim12i 568 . . . . . . . . . . . . . 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 pm3.22 450 . . . . . . . . . . . . . . . 16  |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  ( V  =/=  (/)  /\  V  e. 
Fin ) )
19183adant1 1023 . . . . . . . . . . . . . . 15  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( V  =/=  (/)  /\  V  e. 
Fin ) )
2019adantl 467 . . . . . . . . . . . . . 14  |-  ( ( ( ( p  e. 
Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  /\  ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) ) )  ->  ( V  =/=  (/)  /\  V  e. 
Fin ) )
21 simpll 758 . . . . . . . . . . . . . 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 25840 . . . . . . . . . . . . . 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 1264 . . . . . . . . . . . . 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 25718 . . . . . . . . . . . . . . . . . . 19  |-  ( V FriendGrph  E  ->  V USGrph  E )
25243ad2ant1 1026 . . . . . . . . . . . . . . . . . 18  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  V USGrph  E )
2625adantl 467 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( p  e. 
Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  /\  ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) ) )  ->  V USGrph  E )
27 simpr2 1012 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( p  e. 
Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  /\  ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) ) )  ->  V  e.  Fin )
28 simplll 766 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( p  e. 
Prime  /\  p  ||  ( K  -  1 ) )  /\  <. V ,  E >. RegUSGrph  K )  /\  ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) ) )  ->  p  e.  Prime )
29 numclwwlk8 25841 . . . . . . . . . . . . . . . . 17  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  p  e.  Prime )  ->  ( ( # `
 ( ( V ClWWalksN  E ) `  p
) )  mod  p
)  =  0 )
3026, 27, 28, 29syl3anc 1264 . . . . . . . . . . . . . . . 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 2426 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  (
( V ClWWalksN  E ) `  p ) )  mod  p )  =  0  ->  ( ( (
# `  ( ( V ClWWalksN  E ) `  p
) )  mod  p
)  =  1  <->  0  =  1 ) )
32 ax-1ne0 9615 . . . . . . . . . . . . . . . . . . 19  |-  1  =/=  0
3332nesymi 2693 . . . . . . . . . . . . . . . . . 18  |-  -.  0  =  1
3433pm2.21i 134 . . . . . . . . . . . . . . . . 17  |-  ( 0  =  1  ->  K  =  2 )
3531, 34syl6bi 231 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  (
( V ClWWalksN  E ) `  p ) )  mod  p )  =  0  ->  ( ( (
# `  ( ( V ClWWalksN  E ) `  p
) )  mod  p
)  =  1  ->  K  =  2 ) )
3630, 35syl 17 . . . . . . . . . . . . . . 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 83 . . . . . . . . . . . . 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 37 . . . . . . . . . . . 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 607 . . . . . . . . . . 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 90 . . . . . . . . . 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 2910 . . . . . . . . 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 37 . . . . . . . 8  |-  ( ( K  e.  NN0  /\  2  <  K )  -> 
( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  ->  ( <. V ,  E >. RegUSGrph  K  ->  K  =  2 ) ) )
4443ex 435 . . . . . . 7  |-  ( K  e.  NN0  ->  ( 2  <  K  ->  (
( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  ->  ( <. V ,  E >. RegUSGrph  K  ->  K  =  2 ) ) ) )
4544com24 90 . . . . . 6  |-  ( K  e.  NN0  ->  ( <. V ,  E >. RegUSGrph  K  ->  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  ->  (
2  <  K  ->  K  =  2 ) ) ) )
46453ad2ant2 1027 . . . . 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 37 . . . 4  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  ->  (
2  <  K  ->  K  =  2 ) ) )
4847adantr 466 . . 3  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  1  <  K )  ->  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  -> 
( 2  <  K  ->  K  =  2 ) ) )
4948com13 83 . 2  |-  ( 2  <  K  ->  (
( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  ->  ( (
<. V ,  E >. RegUSGrph  K  /\  1  <  K )  ->  K  =  2 ) ) )
50 1red 9665 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  1  e.  RR )
51 nn0re 10885 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  K  e.  RR )
5250, 51ltnled 9789 . . . . . . . . 9  |-  ( K  e.  NN0  ->  ( 1  <  K  <->  -.  K  <_  1 ) )
53 1e2m1 10732 . . . . . . . . . . . 12  |-  1  =  ( 2  -  1 )
5453a1i 11 . . . . . . . . . . 11  |-  ( K  e.  NN0  ->  1  =  ( 2  -  1 ) )
5554breq2d 4435 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  ( K  <_  1  <->  K  <_  ( 2  -  1 ) ) )
5655notbid 295 . . . . . . . . 9  |-  ( K  e.  NN0  ->  ( -.  K  <_  1  <->  -.  K  <_  ( 2  -  1 ) ) )
57 zltlem1 10996 . . . . . . . . . . . 12  |-  ( ( K  e.  ZZ  /\  2  e.  ZZ )  ->  ( K  <  2  <->  K  <_  ( 2  -  1 ) ) )
584, 2, 57sylancl 666 . . . . . . . . . . 11  |-  ( K  e.  NN0  ->  ( K  <  2  <->  K  <_  ( 2  -  1 ) ) )
5958bicomd 204 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  ( K  <_  ( 2  -  1 )  <->  K  <  2 ) )
6059notbid 295 . . . . . . . . 9  |-  ( K  e.  NN0  ->  ( -.  K  <_  ( 2  -  1 )  <->  -.  K  <  2 ) )
6152, 56, 603bitrd 282 . . . . . . . 8  |-  ( K  e.  NN0  ->  ( 1  <  K  <->  -.  K  <  2 ) )
62 2re 10686 . . . . . . . . . 10  |-  2  e.  RR
63 lttri3 9724 . . . . . . . . . . 11  |-  ( ( K  e.  RR  /\  2  e.  RR )  ->  ( K  =  2  <-> 
( -.  K  <  2  /\  -.  2  <  K ) ) )
6463biimprd 226 . . . . . . . . . 10  |-  ( ( K  e.  RR  /\  2  e.  RR )  ->  ( ( -.  K  <  2  /\  -.  2  <  K )  ->  K  =  2 ) )
6551, 62, 64sylancl 666 . . . . . . . . 9  |-  ( K  e.  NN0  ->  ( ( -.  K  <  2  /\  -.  2  <  K
)  ->  K  = 
2 ) )
6665expd 437 . . . . . . . 8  |-  ( K  e.  NN0  ->  ( -.  K  <  2  -> 
( -.  2  < 
K  ->  K  = 
2 ) ) )
6761, 66sylbid 218 . . . . . . 7  |-  ( K  e.  NN0  ->  ( 1  <  K  ->  ( -.  2  <  K  ->  K  =  2 ) ) )
68673ad2ant2 1027 . . . . . 6  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  K )  ->  ( 1  < 
K  ->  ( -.  2  <  K  ->  K  =  2 ) ) )
691, 68syl 17 . . . . 5  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( 1  <  K  ->  ( -.  2  < 
K  ->  K  = 
2 ) ) )
7069imp 430 . . . 4  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  1  <  K )  ->  ( -.  2  <  K  ->  K  = 
2 ) )
7170com12 32 . . 3  |-  ( -.  2  <  K  -> 
( ( <. V ,  E >. RegUSGrph  K  /\  1  < 
K )  ->  K  =  2 ) )
7271a1d 26 . 2  |-  ( -.  2  <  K  -> 
( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  ->  (
( <. V ,  E >. RegUSGrph  K  /\  1  <  K
)  ->  K  = 
2 ) ) )
7349, 72pm2.61i 167 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   E.wrex 2772   (/)c0 3761   <.cop 4004   class class class wbr 4423   ` cfv 5601  (class class class)co 6305   Fincfn 7580   RRcr 9545   0cc0 9546   1c1 9547    < clt 9682    <_ cle 9683    - cmin 9867   2c2 10666   NN0cn0 10876   ZZcz 10944   ZZ>=cuz 11166    mod cmo 12102   #chash 12521    || cdvds 14304   Primecprime 14621   USGrph cusg 25055   ClWWalksN cclwwlkn 25475   VDeg cvdg 25619   RegUSGrph crusgra 25649   FriendGrph cfrgra 25714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-inf2 8155  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-ot 4007  df-uni 4220  df-int 4256  df-iun 4301  df-disj 4395  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-2o 7194  df-oadd 7197  df-er 7374  df-ec 7376  df-qs 7380  df-map 7485  df-pm 7486  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-sup 7965  df-inf 7966  df-oi 8034  df-card 8381  df-cda 8605  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-xadd 11417  df-ico 11648  df-fz 11792  df-fzo 11923  df-fl 12034  df-mod 12103  df-seq 12220  df-exp 12279  df-hash 12522  df-word 12668  df-lsw 12669  df-concat 12670  df-s1 12671  df-substr 12672  df-reps 12675  df-csh 12893  df-s2 12946  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13551  df-sum 13752  df-dvds 14305  df-gcd 14468  df-prm 14622  df-phi 14713  df-usgra 25058  df-nbgra 25146  df-wlk 25234  df-trail 25235  df-pth 25236  df-spth 25237  df-wlkon 25240  df-spthon 25243  df-wwlk 25405  df-wwlkn 25406  df-clwwlk 25477  df-clwwlkn 25478  df-2wlkonot 25584  df-2spthonot 25586  df-2spthsot 25587  df-vdgr 25620  df-rgra 25650  df-rusgra 25651  df-frgra 25715
This theorem is referenced by:  frgrareg  25843
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