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Theorem frgraogt3nreg 24825
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 996 . . . . . . . 8  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  3  <  (
# `  V )
)  ->  V FriendGrph  E )
2 simp2 997 . . . . . . . 8  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  3  <  (
# `  V )
)  ->  V  e.  Fin )
3 hashcl 12396 . . . . . . . . . . 11  |-  ( V  e.  Fin  ->  ( # `
 V )  e. 
NN0 )
4 0red 9597 . . . . . . . . . . . . . 14  |-  ( (
# `  V )  e.  NN0  ->  0  e.  RR )
5 3re 10609 . . . . . . . . . . . . . . . . . . 19  |-  3  e.  RR
65a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  V )  e.  NN0  ->  3  e.  RR )
7 nn0re 10804 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  V )  e.  NN0  ->  ( # `  V
)  e.  RR )
84, 6, 73jca 1176 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  V )  e.  NN0  ->  ( 0  e.  RR  /\  3  e.  RR  /\  ( # `  V )  e.  RR ) )
98adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  (
0  e.  RR  /\  3  e.  RR  /\  ( # `
 V )  e.  RR ) )
10 3pos 10629 . . . . . . . . . . . . . . . . 17  |-  0  <  3
1110a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  0  <  3 )
12 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  3  <  ( # `  V
) )
13 lttr 9661 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  RR  /\  3  e.  RR  /\  ( # `
 V )  e.  RR )  ->  (
( 0  <  3  /\  3  <  ( # `  V ) )  -> 
0  <  ( # `  V
) ) )
1413imp 429 . . . . . . . . . . . . . . . 16  |-  ( ( ( 0  e.  RR  /\  3  e.  RR  /\  ( # `  V )  e.  RR )  /\  ( 0  <  3  /\  3  <  ( # `  V ) ) )  ->  0  <  ( # `
 V ) )
159, 11, 12, 14syl12anc 1226 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  0  <  ( # `  V
) )
1615ex 434 . . . . . . . . . . . . . 14  |-  ( (
# `  V )  e.  NN0  ->  ( 3  <  ( # `  V
)  ->  0  <  (
# `  V )
) )
17 ltne 9681 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR  /\  0  <  ( # `  V
) )  ->  ( # `
 V )  =/=  0 )
184, 16, 17syl6an 545 . . . . . . . . . . . . 13  |-  ( (
# `  V )  e.  NN0  ->  ( 3  <  ( # `  V
)  ->  ( # `  V
)  =/=  0 ) )
19 hasheq0 12401 . . . . . . . . . . . . . . 15  |-  ( V  e.  Fin  ->  (
( # `  V )  =  0  <->  V  =  (/) ) )
2019necon3bid 2725 . . . . . . . . . . . . . 14  |-  ( V  e.  Fin  ->  (
( # `  V )  =/=  0  <->  V  =/=  (/) ) )
2120biimpcd 224 . . . . . . . . . . . . 13  |-  ( (
# `  V )  =/=  0  ->  ( V  e.  Fin  ->  V  =/=  (/) ) )
2218, 21syl6 33 . . . . . . . . . . . 12  |-  ( (
# `  V )  e.  NN0  ->  ( 3  <  ( # `  V
)  ->  ( V  e.  Fin  ->  V  =/=  (/) ) ) )
2322com23 78 . . . . . . . . . . 11  |-  ( (
# `  V )  e.  NN0  ->  ( V  e.  Fin  ->  ( 3  <  ( # `  V
)  ->  V  =/=  (/) ) ) )
243, 23mpcom 36 . . . . . . . . . 10  |-  ( V  e.  Fin  ->  (
3  <  ( # `  V
)  ->  V  =/=  (/) ) )
2524a1i 11 . . . . . . . . 9  |-  ( V FriendGrph  E  ->  ( V  e. 
Fin  ->  ( 3  < 
( # `  V )  ->  V  =/=  (/) ) ) )
26253imp 1190 . . . . . . . 8  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  3  <  (
# `  V )
)  ->  V  =/=  (/) )
271, 2, 263jca 1176 . . . . . . 7  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  3  <  (
# `  V )
)  ->  ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) ) )
2827ad2antrl 727 . . . . . 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 457 . . . . . 6  |-  ( (
<. V ,  E >. RegUSGrph  k  /\  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  3  <  ( # `  V
) )  /\  k  e.  NN0 ) )  ->  <. V ,  E >. RegUSGrph  k
)
30 frgraregord13 24824 . . . . . 6  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  <. V ,  E >. RegUSGrph  k )  ->  ( ( # `  V )  =  1  \/  ( # `  V
)  =  3 ) )
3128, 29, 30syl2anc 661 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  k  /\  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  3  <  ( # `  V
) )  /\  k  e.  NN0 ) )  -> 
( ( # `  V
)  =  1  \/  ( # `  V
)  =  3 ) )
32 1red 9611 . . . . . . . . . . . . 13  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  1  e.  RR )
335a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  3  e.  RR )
347adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  ( # `
 V )  e.  RR )
35 1lt3 10704 . . . . . . . . . . . . . . 15  |-  1  <  3
3635a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  1  <  3 )
3732, 33, 34, 36, 12lttrd 9742 . . . . . . . . . . . . 13  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  1  <  ( # `  V
) )
3832, 37gtned 9719 . . . . . . . . . . . 12  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  ( # `
 V )  =/=  1 )
39 eqneqall 2674 . . . . . . . . . . . 12  |-  ( (
# `  V )  =  1  ->  (
( # `  V )  =/=  1  ->  -.  <. V ,  E >. RegUSGrph  k
) )
4038, 39syl5com 30 . . . . . . . . . . 11  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  (
( # `  V )  =  1  ->  -.  <. V ,  E >. RegUSGrph  k
) )
41 ltne 9681 . . . . . . . . . . . . 13  |-  ( ( 3  e.  RR  /\  3  <  ( # `  V
) )  ->  ( # `
 V )  =/=  3 )
426, 41sylan 471 . . . . . . . . . . . 12  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  ( # `
 V )  =/=  3 )
43 eqneqall 2674 . . . . . . . . . . . 12  |-  ( (
# `  V )  =  3  ->  (
( # `  V )  =/=  3  ->  -.  <. V ,  E >. RegUSGrph  k
) )
4442, 43syl5com 30 . . . . . . . . . . 11  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  (
( # `  V )  =  3  ->  -.  <. V ,  E >. RegUSGrph  k
) )
4540, 44jaod 380 . . . . . . . . . 10  |-  ( ( ( # `  V
)  e.  NN0  /\  3  <  ( # `  V
) )  ->  (
( ( # `  V
)  =  1  \/  ( # `  V
)  =  3 )  ->  -.  <. V ,  E >. RegUSGrph  k ) )
4645ex 434 . . . . . . . . 9  |-  ( (
# `  V )  e.  NN0  ->  ( 3  <  ( # `  V
)  ->  ( (
( # `  V )  =  1  \/  ( # `
 V )  =  3 )  ->  -.  <. V ,  E >. RegUSGrph  k
) ) )
473, 46syl 16 . . . . . . . 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 1190 . . . . . 6  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  3  <  (
# `  V )
)  ->  ( (
( # `  V )  =  1  \/  ( # `
 V )  =  3 )  ->  -.  <. V ,  E >. RegUSGrph  k
) )
5049ad2antrl 727 . . . . 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 434 . . 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 164 . 2  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  3  <  ( # `  V
) )  /\  k  e.  NN0 )  ->  -.  <. V ,  E >. RegUSGrph  k
)
5554ralrimiva 2878 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 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   (/)c0 3785   <.cop 4033   class class class wbr 4447   ` cfv 5588   Fincfn 7516   RRcr 9491   0cc0 9492   1c1 9493    < clt 9628   3c3 10586   NN0cn0 10795   #chash 12373   RegUSGrph crusgra 24627   FriendGrph cfrgra 24692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-ec 7313  df-qs 7317  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-xadd 11319  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-hash 12374  df-word 12508  df-lsw 12509  df-concat 12510  df-s1 12511  df-substr 12512  df-reps 12515  df-csh 12723  df-s2 12776  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-sum 13472  df-dvds 13848  df-gcd 14004  df-prm 14077  df-phi 14155  df-usgra 24037  df-nbgra 24124  df-wlk 24212  df-trail 24213  df-pth 24214  df-spth 24215  df-wlkon 24218  df-spthon 24221  df-wwlk 24383  df-wwlkn 24384  df-clwwlk 24455  df-clwwlkn 24456  df-2wlkonot 24562  df-2spthonot 24564  df-2spthsot 24565  df-vdgr 24598  df-rgra 24628  df-rusgra 24629  df-frgra 24693
This theorem is referenced by:  friendshipgt3  24826
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