MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cusgraiffrusgra Structured version   Unicode version

Theorem cusgraiffrusgra 25344
Description: A finite undirected simple graph with n vertices is complete iff it is (n-1)-regular. Hint: If the definition of RegGrph allowed for  k  e.  ZZ, then the assumption  V  =/=  (/) could be removed. Furthermore, if the definition of RegGrph also allowed for  k  e.  ( ZZ  u.  { +oo } ), then the theorem would also hold for inifinite graphs. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
Assertion
Ref Expression
cusgraiffrusgra  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( V ComplUSGrph  E  <->  <. V ,  E >. RegUSGrph  (
( # `  V )  -  1 ) ) )

Proof of Theorem cusgraiffrusgra
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 cusgraisrusgra 25342 . . . . . 6  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  <. V ,  E >. RegUSGrph  ( ( # `  V
)  -  1 ) )
21a1d 25 . . . . 5  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( V USGrph  E  ->  <. V ,  E >. RegUSGrph 
( ( # `  V
)  -  1 ) ) )
323exp 1196 . . . 4  |-  ( V ComplUSGrph  E  ->  ( V  e. 
Fin  ->  ( V  =/=  (/)  ->  ( V USGrph  E  -> 
<. V ,  E >. RegUSGrph  (
( # `  V )  -  1 ) ) ) ) )
43com14 88 . . 3  |-  ( V USGrph  E  ->  ( V  e. 
Fin  ->  ( V  =/=  (/)  ->  ( V ComplUSGrph  E  ->  <. V ,  E >. RegUSGrph  (
( # `  V )  -  1 ) ) ) ) )
543imp 1191 . 2  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( V ComplUSGrph  E  ->  <. V ,  E >. RegUSGrph 
( ( # `  V
)  -  1 ) ) )
6 rusgraprop 25333 . . . 4  |-  ( <. V ,  E >. RegUSGrph  (
( # `  V )  -  1 )  -> 
( V USGrph  E  /\  ( ( # `  V
)  -  1 )  e.  NN0  /\  A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  ( (
# `  V )  -  1 ) ) )
76simp3d 1011 . . 3  |-  ( <. V ,  E >. RegUSGrph  (
( # `  V )  -  1 )  ->  A. v  e.  V  ( ( V VDeg  E
) `  v )  =  ( ( # `  V )  -  1 ) )
8 vdiscusgra 25325 . . . 4  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  ( A. v  e.  V  ( ( V VDeg  E
) `  v )  =  ( ( # `  V )  -  1 )  ->  V ComplUSGrph  E ) )
983adant3 1017 . . 3  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  (
( V VDeg  E ) `  v )  =  ( ( # `  V
)  -  1 )  ->  V ComplUSGrph  E ) )
107, 9syl5 30 . 2  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( <. V ,  E >. RegUSGrph  ( (
# `  V )  -  1 )  ->  V ComplUSGrph  E ) )
115, 10impbid 191 1  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  ( V ComplUSGrph  E  <->  <. V ,  E >. RegUSGrph  (
( # `  V )  -  1 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   (/)c0 3737   <.cop 3977   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   Fincfn 7553   1c1 9522    - cmin 9840   NN0cn0 10835   #chash 12450   USGrph cusg 24734   ComplUSGrph ccusgra 24822   VDeg cvdg 25297   RegUSGrph crusgra 25327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-xadd 11371  df-fz 11725  df-hash 12451  df-usgra 24737  df-nbgra 24824  df-cusgra 24825  df-uvtx 24826  df-vdgr 25298  df-rgra 25328  df-rusgra 25329
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator