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Theorem frgregordn0 25196
Description: If a nonempty friendship graph is k-regular, its order is k(k-1)+1. This corresponds to claim 3 in [Huneke] p. 2: "Next we claim that the number n of vertices in G is exactly k(k-1)+1.". (Contributed by Alexander van der Vekens, 11-Mar-2018.)
Assertion
Ref Expression
frgregordn0  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) )
Distinct variable groups:    v, E    v, K    v, V

Proof of Theorem frgregordn0
StepHypRef Expression
1 frisusgra 25118 . . . . 5  |-  ( V FriendGrph  E  ->  V USGrph  E )
2 usgreghash2spot 25195 . . . . 5  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( K  x.  ( K  - 
1 ) ) ) ) )
31, 2syl3an1 1261 . . . 4  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( K  x.  ( K  - 
1 ) ) ) ) )
43imp 429 . . 3  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( K  x.  ( K  - 
1 ) ) ) )
5 frghash2spot 25189 . . . . . 6  |-  ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  -> 
( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) ) )
653impb 1192 . . . . 5  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) ) )
76adantr 465 . . . 4  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) ) )
8 eqeq1 2461 . . . . . 6  |-  ( (
# `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) )  ->  ( ( # `  ( V 2SPathOnOt  E )
)  =  ( (
# `  V )  x.  ( K  x.  ( K  -  1 ) ) )  <->  ( ( # `
 V )  x.  ( ( # `  V
)  -  1 ) )  =  ( (
# `  V )  x.  ( K  x.  ( K  -  1 ) ) ) ) )
9 hashcl 12430 . . . . . . . . . . . . 13  |-  ( V  e.  Fin  ->  ( # `
 V )  e. 
NN0 )
109nn0cnd 10875 . . . . . . . . . . . 12  |-  ( V  e.  Fin  ->  ( # `
 V )  e.  CC )
11 1cnd 9629 . . . . . . . . . . . 12  |-  ( V  e.  Fin  ->  1  e.  CC )
1210, 11subcld 9950 . . . . . . . . . . 11  |-  ( V  e.  Fin  ->  (
( # `  V )  -  1 )  e.  CC )
13123ad2ant2 1018 . . . . . . . . . 10  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( ( # `
 V )  - 
1 )  e.  CC )
1413adantr 465 . . . . . . . . 9  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( ( # `
 V )  - 
1 )  e.  CC )
15 usgfiregdegfi 25037 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  K  e.  NN0 ) )
161, 15syl3an1 1261 . . . . . . . . . . 11  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  K  e.  NN0 ) )
1716imp 429 . . . . . . . . . 10  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  K  e.  NN0 )
18 nn0cn 10826 . . . . . . . . . 10  |-  ( K  e.  NN0  ->  K  e.  CC )
19 kcnktkm1cn 10009 . . . . . . . . . 10  |-  ( K  e.  CC  ->  ( K  x.  ( K  -  1 ) )  e.  CC )
2017, 18, 193syl 20 . . . . . . . . 9  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( K  x.  ( K  -  1 ) )  e.  CC )
21103ad2ant2 1018 . . . . . . . . . 10  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( # `  V
)  e.  CC )
2221adantr 465 . . . . . . . . 9  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( # `  V
)  e.  CC )
23 hasheq0 12435 . . . . . . . . . . . . . 14  |-  ( V  e.  Fin  ->  (
( # `  V )  =  0  <->  V  =  (/) ) )
2423biimpd 207 . . . . . . . . . . . . 13  |-  ( V  e.  Fin  ->  (
( # `  V )  =  0  ->  V  =  (/) ) )
2524necon3d 2681 . . . . . . . . . . . 12  |-  ( V  e.  Fin  ->  ( V  =/=  (/)  ->  ( # `  V
)  =/=  0 ) )
2625imp 429 . . . . . . . . . . 11  |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  ( # `
 V )  =/=  0 )
27263adant1 1014 . . . . . . . . . 10  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( # `  V
)  =/=  0 )
2827adantr 465 . . . . . . . . 9  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( # `  V
)  =/=  0 )
2914, 20, 22, 28mulcand 10203 . . . . . . . 8  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( (
( # `  V )  x.  ( ( # `  V )  -  1 ) )  =  ( ( # `  V
)  x.  ( K  x.  ( K  - 
1 ) ) )  <-> 
( ( # `  V
)  -  1 )  =  ( K  x.  ( K  -  1
) ) ) )
30 1cnd 9629 . . . . . . . . 9  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  1  e.  CC )
31 subadd2 9843 . . . . . . . . . . 11  |-  ( ( ( # `  V
)  e.  CC  /\  1  e.  CC  /\  ( K  x.  ( K  -  1 ) )  e.  CC )  -> 
( ( ( # `  V )  -  1 )  =  ( K  x.  ( K  - 
1 ) )  <->  ( ( K  x.  ( K  -  1 ) )  +  1 )  =  ( # `  V
) ) )
32 eqcom 2466 . . . . . . . . . . 11  |-  ( ( ( K  x.  ( K  -  1 ) )  +  1 )  =  ( # `  V
)  <->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) )
3331, 32syl6bb 261 . . . . . . . . . 10  |-  ( ( ( # `  V
)  e.  CC  /\  1  e.  CC  /\  ( K  x.  ( K  -  1 ) )  e.  CC )  -> 
( ( ( # `  V )  -  1 )  =  ( K  x.  ( K  - 
1 ) )  <->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) )
3433biimpd 207 . . . . . . . . 9  |-  ( ( ( # `  V
)  e.  CC  /\  1  e.  CC  /\  ( K  x.  ( K  -  1 ) )  e.  CC )  -> 
( ( ( # `  V )  -  1 )  =  ( K  x.  ( K  - 
1 ) )  -> 
( # `  V )  =  ( ( K  x.  ( K  - 
1 ) )  +  1 ) ) )
3522, 30, 20, 34syl3anc 1228 . . . . . . . 8  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( (
( # `  V )  -  1 )  =  ( K  x.  ( K  -  1 ) )  ->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) )
3629, 35sylbid 215 . . . . . . 7  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( (
( # `  V )  x.  ( ( # `  V )  -  1 ) )  =  ( ( # `  V
)  x.  ( K  x.  ( K  - 
1 ) ) )  ->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) )
3736com12 31 . . . . . 6  |-  ( ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) )  =  ( ( # `  V )  x.  ( K  x.  ( K  -  1 ) ) )  ->  ( (
( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  K )  ->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) )
388, 37syl6bi 228 . . . . 5  |-  ( (
# `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) )  ->  ( ( # `  ( V 2SPathOnOt  E )
)  =  ( (
# `  V )  x.  ( K  x.  ( K  -  1 ) ) )  ->  (
( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) ) )
3938com23 78 . . . 4  |-  ( (
# `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V
)  x.  ( (
# `  V )  -  1 ) )  ->  ( ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  K )  ->  ( ( # `  ( V 2SPathOnOt  E )
)  =  ( (
# `  V )  x.  ( K  x.  ( K  -  1 ) ) )  ->  ( # `
 V )  =  ( ( K  x.  ( K  -  1
) )  +  1 ) ) ) )
407, 39mpcom 36 . . 3  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( ( # `
 ( V 2SPathOnOt  E ) )  =  ( (
# `  V )  x.  ( K  x.  ( K  -  1 ) ) )  ->  ( # `
 V )  =  ( ( K  x.  ( K  -  1
) )  +  1 ) ) )
414, 40mpd 15 . 2  |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  A. v  e.  V  (
( V VDeg  E ) `  v )  =  K )  ->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) )
4241ex 434 1  |-  ( ( V FriendGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  K  ->  ( # `  V
)  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   (/)c0 3793   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Fincfn 7535   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    - cmin 9824   NN0cn0 10816   #chash 12407   USGrph cusg 24456   2SPathOnOt c2spthot 24982   VDeg cvdg 25019   FriendGrph cfrgra 25114
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-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 11821  df-seq 12110  df-exp 12169  df-hash 12408  df-word 12545  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322  df-sum 13520  df-usgra 24459  df-nbgra 24546  df-wlk 24634  df-trail 24635  df-pth 24636  df-spth 24637  df-wlkon 24640  df-spthon 24643  df-2wlkonot 24984  df-2spthonot 24986  df-2spthsot 24987  df-vdgr 25020  df-frgra 25115
This theorem is referenced by:  frrusgraord  25197
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