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Theorem vdcusgra 32197
Description: In a finite complete undirected simple graph with n vertices every vertex has degree (n-1). (Contributed by Alexander van der Vekens, 9-Jul-2018.)
Assertion
Ref Expression
vdcusgra  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin )  ->  A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  ( (
# `  V )  -  1 ) )
Distinct variable groups:    v, E    v, V

Proof of Theorem vdcusgra
StepHypRef Expression
1 nbcusgra 24439 . . . 4  |-  ( ( V ComplUSGrph  E  /\  v  e.  V )  ->  ( <. V ,  E >. Neighbors  v
)  =  ( V 
\  { v } ) )
21adantlr 714 . . 3  |-  ( ( ( V ComplUSGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } ) )
3 cusisusgra 24434 . . . . . . 7  |-  ( V ComplUSGrph  E  ->  V USGrph  E )
43adantr 465 . . . . . 6  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin )  ->  V USGrph  E )
5 hashnbgravdg 24889 . . . . . 6  |-  ( ( V USGrph  E  /\  v  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  v ) )  =  ( ( V VDeg  E
) `  v )
)
64, 5sylan 471 . . . . 5  |-  ( ( ( V ComplUSGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( # `  ( <. V ,  E >. Neighbors  v
) )  =  ( ( V VDeg  E ) `
 v ) )
76adantr 465 . . . 4  |-  ( ( ( ( V ComplUSGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } ) )  ->  ( # `  ( <. V ,  E >. Neighbors  v
) )  =  ( ( V VDeg  E ) `
 v ) )
8 fveq2 5856 . . . . 5  |-  ( (
<. V ,  E >. Neighbors  v
)  =  ( V 
\  { v } )  ->  ( # `  ( <. V ,  E >. Neighbors  v
) )  =  (
# `  ( V  \  { v } ) ) )
9 hashdifsn 12458 . . . . . 6  |-  ( ( V  e.  Fin  /\  v  e.  V )  ->  ( # `  ( V  \  { v } ) )  =  ( ( # `  V
)  -  1 ) )
109adantll 713 . . . . 5  |-  ( ( ( V ComplUSGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( # `  ( V  \  { v } ) )  =  ( ( # `  V
)  -  1 ) )
118, 10sylan9eqr 2506 . . . 4  |-  ( ( ( ( V ComplUSGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } ) )  ->  ( # `  ( <. V ,  E >. Neighbors  v
) )  =  ( ( # `  V
)  -  1 ) )
127, 11eqtr3d 2486 . . 3  |-  ( ( ( ( V ComplUSGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  /\  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } ) )  ->  ( ( V VDeg  E ) `  v
)  =  ( (
# `  V )  -  1 ) )
132, 12mpdan 668 . 2  |-  ( ( ( V ComplUSGrph  E  /\  V  e.  Fin )  /\  v  e.  V
)  ->  ( ( V VDeg  E ) `  v
)  =  ( (
# `  V )  -  1 ) )
1413ralrimiva 2857 1  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin )  ->  A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  ( (
# `  V )  -  1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793    \ cdif 3458   {csn 4014   <.cop 4020   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   Fincfn 7518   1c1 9496    - cmin 9810   #chash 12386   USGrph cusg 24306   Neighbors cnbgra 24393   ComplUSGrph ccusgra 24394   VDeg cvdg 24869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11092  df-xadd 11329  df-fz 11683  df-hash 12387  df-usgra 24309  df-nbgra 24396  df-cusgra 24397  df-vdgr 24870
This theorem is referenced by: (None)
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