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Theorem frgraogt3nreg 25693
Description: If a finite friendship graph has an order greater than 3, it cannot be k-regular for any k. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
Assertion
Ref Expression
frgraogt3nreg  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  3  <  (
# `  V )
)  ->  A. k  e.  NN0  -.  <. V ,  E >. RegUSGrph  k )
Distinct variable groups:    k, E    k, V

Proof of Theorem frgraogt3nreg
StepHypRef Expression
1 simp1 1005 . . . . . . . 8  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  3  <  (
# `  V )
)  ->  V FriendGrph  E )
2 simp2 1006 . . . . . . . 8  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  3  <  (
# `  V )
)  ->  V  e.  Fin )
3 hashcl 12535 . . . . . . . . . . 11  |-  ( V  e.  Fin  ->  ( # `
 V )  e. 
NN0 )
4 0red 9643 . . . . . . . . . . . . . 14  |-  ( (
# `  V )  e.  NN0  ->  0  e.  RR )
5 3re 10683 . . . . . . . . . . . . . . . . . . 19  |-  3  e.  RR
65a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  V )  e.  NN0  ->  3  e.  RR )
7 nn0re 10878 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  V )  e.  NN0  ->  ( # `  V
)  e.  RR )
84, 6, 73jca 1185 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  V )  e.  NN0  ->  ( 0  e.  RR  /\  3  e.  RR  /\  ( # `  V )  e.  RR ) )
98adantr 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  (
0  e.  RR  /\  3  e.  RR  /\  ( # `
 V )  e.  RR ) )
10 3pos 10703 . . . . . . . . . . . . . . . . 17  |-  0  <  3
1110a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  0  <  3 )
12 simpr 462 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  3  <  ( # `  V
) )
13 lttr 9709 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  RR  /\  3  e.  RR  /\  ( # `
 V )  e.  RR )  ->  (
( 0  <  3  /\  3  <  ( # `  V ) )  -> 
0  <  ( # `  V
) ) )
1413imp 430 . . . . . . . . . . . . . . . 16  |-  ( ( ( 0  e.  RR  /\  3  e.  RR  /\  ( # `  V )  e.  RR )  /\  ( 0  <  3  /\  3  <  ( # `  V ) ) )  ->  0  <  ( # `
 V ) )
159, 11, 12, 14syl12anc 1262 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  0  <  ( # `  V
) )
1615ex 435 . . . . . . . . . . . . . 14  |-  ( (
# `  V )  e.  NN0  ->  ( 3  <  ( # `  V
)  ->  0  <  (
# `  V )
) )
17 ltne 9729 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR  /\  0  <  ( # `  V
) )  ->  ( # `
 V )  =/=  0 )
184, 16, 17syl6an 547 . . . . . . . . . . . . 13  |-  ( (
# `  V )  e.  NN0  ->  ( 3  <  ( # `  V
)  ->  ( # `  V
)  =/=  0 ) )
19 hasheq0 12541 . . . . . . . . . . . . . . 15  |-  ( V  e.  Fin  ->  (
( # `  V )  =  0  <->  V  =  (/) ) )
2019necon3bid 2689 . . . . . . . . . . . . . 14  |-  ( V  e.  Fin  ->  (
( # `  V )  =/=  0  <->  V  =/=  (/) ) )
2120biimpcd 227 . . . . . . . . . . . . 13  |-  ( (
# `  V )  =/=  0  ->  ( V  e.  Fin  ->  V  =/=  (/) ) )
2218, 21syl6 34 . . . . . . . . . . . 12  |-  ( (
# `  V )  e.  NN0  ->  ( 3  <  ( # `  V
)  ->  ( V  e.  Fin  ->  V  =/=  (/) ) ) )
2322com23 81 . . . . . . . . . . 11  |-  ( (
# `  V )  e.  NN0  ->  ( V  e.  Fin  ->  ( 3  <  ( # `  V
)  ->  V  =/=  (/) ) ) )
243, 23mpcom 37 . . . . . . . . . 10  |-  ( V  e.  Fin  ->  (
3  <  ( # `  V
)  ->  V  =/=  (/) ) )
2524a1i 11 . . . . . . . . 9  |-  ( V FriendGrph  E  ->  ( V  e. 
Fin  ->  ( 3  < 
( # `  V )  ->  V  =/=  (/) ) ) )
26253imp 1199 . . . . . . . 8  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  3  <  (
# `  V )
)  ->  V  =/=  (/) )
271, 2, 263jca 1185 . . . . . . 7  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  3  <  (
# `  V )
)  ->  ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) ) )
2827ad2antrl 732 . . . . . 6  |-  ( (
<. V ,  E >. RegUSGrph  k  /\  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  3  <  ( # `  V
) )  /\  k  e.  NN0 ) )  -> 
( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) ) )
29 simpl 458 . . . . . 6  |-  ( (
<. V ,  E >. RegUSGrph  k  /\  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  3  <  ( # `  V
) )  /\  k  e.  NN0 ) )  ->  <. V ,  E >. RegUSGrph  k
)
30 frgraregord13 25692 . . . . . 6  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  <. V ,  E >. RegUSGrph  k )  ->  ( ( # `  V )  =  1  \/  ( # `  V
)  =  3 ) )
3128, 29, 30syl2anc 665 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  k  /\  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  3  <  ( # `  V
) )  /\  k  e.  NN0 ) )  -> 
( ( # `  V
)  =  1  \/  ( # `  V
)  =  3 ) )
32 1red 9657 . . . . . . . . . . . . 13  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  1  e.  RR )
335a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  3  e.  RR )
347adantr 466 . . . . . . . . . . . . . 14  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  ( # `
 V )  e.  RR )
35 1lt3 10778 . . . . . . . . . . . . . . 15  |-  1  <  3
3635a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  1  <  3 )
3732, 33, 34, 36, 12lttrd 9795 . . . . . . . . . . . . 13  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  1  <  ( # `  V
) )
3832, 37gtned 9769 . . . . . . . . . . . 12  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  ( # `
 V )  =/=  1 )
39 eqneqall 2638 . . . . . . . . . . . 12  |-  ( (
# `  V )  =  1  ->  (
( # `  V )  =/=  1  ->  -.  <. V ,  E >. RegUSGrph  k
) )
4038, 39syl5com 31 . . . . . . . . . . 11  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  (
( # `  V )  =  1  ->  -.  <. V ,  E >. RegUSGrph  k
) )
41 ltne 9729 . . . . . . . . . . . . 13  |-  ( ( 3  e.  RR  /\  3  <  ( # `  V
) )  ->  ( # `
 V )  =/=  3 )
426, 41sylan 473 . . . . . . . . . . . 12  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  ( # `
 V )  =/=  3 )
43 eqneqall 2638 . . . . . . . . . . . 12  |-  ( (
# `  V )  =  3  ->  (
( # `  V )  =/=  3  ->  -.  <. V ,  E >. RegUSGrph  k
) )
4442, 43syl5com 31 . . . . . . . . . . 11  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  (
( # `  V )  =  3  ->  -.  <. V ,  E >. RegUSGrph  k
) )
4540, 44jaod 381 . . . . . . . . . 10  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  (
( ( # `  V
)  =  1  \/  ( # `  V
)  =  3 )  ->  -.  <. V ,  E >. RegUSGrph  k ) )
4645ex 435 . . . . . . . . 9  |-  ( (
# `  V )  e.  NN0  ->  ( 3  <  ( # `  V
)  ->  ( (
( # `  V )  =  1  \/  ( # `
 V )  =  3 )  ->  -.  <. V ,  E >. RegUSGrph  k
) ) )
473, 46syl 17 . . . . . . . 8  |-  ( V  e.  Fin  ->  (
3  <  ( # `  V
)  ->  ( (
( # `  V )  =  1  \/  ( # `
 V )  =  3 )  ->  -.  <. V ,  E >. RegUSGrph  k
) ) )
4847a1i 11 . . . . . . 7  |-  ( V FriendGrph  E  ->  ( V  e. 
Fin  ->  ( 3  < 
( # `  V )  ->  ( ( (
# `  V )  =  1  \/  ( # `
 V )  =  3 )  ->  -.  <. V ,  E >. RegUSGrph  k
) ) ) )
49483imp 1199 . . . . . 6  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  3  <  (
# `  V )
)  ->  ( (
( # `  V )  =  1  \/  ( # `
 V )  =  3 )  ->  -.  <. V ,  E >. RegUSGrph  k
) )
5049ad2antrl 732 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  k  /\  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  3  <  ( # `  V
) )  /\  k  e.  NN0 ) )  -> 
( ( ( # `  V )  =  1  \/  ( # `  V
)  =  3 )  ->  -.  <. V ,  E >. RegUSGrph  k ) )
5131, 50mpd 15 . . . 4  |-  ( (
<. V ,  E >. RegUSGrph  k  /\  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  3  <  ( # `  V
) )  /\  k  e.  NN0 ) )  ->  -.  <. V ,  E >. RegUSGrph 
k )
5251ex 435 . . 3  |-  ( <. V ,  E >. RegUSGrph  k  ->  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  3  <  ( # `  V ) )  /\  k  e.  NN0 )  ->  -.  <. V ,  E >. RegUSGrph 
k ) )
53 ax-1 6 . . 3  |-  ( -. 
<. V ,  E >. RegUSGrph  k  ->  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  3  <  ( # `  V ) )  /\  k  e.  NN0 )  ->  -.  <. V ,  E >. RegUSGrph 
k ) )
5452, 53pm2.61i 167 . 2  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  3  <  ( # `  V
) )  /\  k  e.  NN0 )  ->  -.  <. V ,  E >. RegUSGrph  k
)
5554ralrimiva 2846 1  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  3  <  (
# `  V )
)  ->  A. k  e.  NN0  -.  <. V ,  E >. RegUSGrph  k )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   (/)c0 3767   <.cop 4008   class class class wbr 4426   ` cfv 5601   Fincfn 7577   RRcr 9537   0cc0 9538   1c1 9539    < clt 9674   3c3 10660   NN0cn0 10869   #chash 12512   RegUSGrph crusgra 25496   FriendGrph cfrgra 25561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
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 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-ot 4011  df-uni 4223  df-int 4259  df-iun 4304  df-disj 4398  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  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 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-ec 7373  df-qs 7377  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-xadd 11410  df-ico 11641  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-hash 12513  df-word 12651  df-lsw 12652  df-concat 12653  df-s1 12654  df-substr 12655  df-reps 12658  df-csh 12876  df-s2 12929  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-dvds 14284  df-gcd 14443  df-prm 14594  df-phi 14683  df-usgra 24906  df-nbgra 24993  df-wlk 25081  df-trail 25082  df-pth 25083  df-spth 25084  df-wlkon 25087  df-spthon 25090  df-wwlk 25252  df-wwlkn 25253  df-clwwlk 25324  df-clwwlkn 25325  df-2wlkonot 25431  df-2spthonot 25433  df-2spthsot 25434  df-vdgr 25467  df-rgra 25497  df-rusgra 25498  df-frgra 25562
This theorem is referenced by:  friendshipgt3  25694
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