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Theorem usgrauvtxvd 30528
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
usgrauvtxvd  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  ( K  e.  ( V UnivVertex  E )  ->  ( ( V VDeg  E ) `  K
)  =  ( (
# `  V )  -  1 ) ) )

Proof of Theorem usgrauvtxvd
StepHypRef Expression
1 uvtxnbgra 23400 . . . 4  |-  ( ( V USGrph  E  /\  K  e.  ( V UnivVertex  E )
)  ->  ( <. V ,  E >. Neighbors  K )  =  ( V  \  { K } ) )
21adantlr 714 . . 3  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  K  e.  ( V UnivVertex  E ) )  -> 
( <. V ,  E >. Neighbors  K )  =  ( V  \  { K } ) )
3 fveq2 5690 . . . . 5  |-  ( (
<. V ,  E >. Neighbors  K
)  =  ( V 
\  { K }
)  ->  ( # `  ( <. V ,  E >. Neighbors  K
) )  =  (
# `  ( V  \  { K } ) ) )
43adantl 466 . . . 4  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  K  e.  ( V UnivVertex  E ) )  /\  ( <. V ,  E >. Neighbors  K )  =  ( V  \  { K } ) )  -> 
( # `  ( <. V ,  E >. Neighbors  K
) )  =  (
# `  ( V  \  { K } ) ) )
5 simpl 457 . . . . . . 7  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  V USGrph  E )
6 uvtxisvtx 23397 . . . . . . 7  |-  ( K  e.  ( V UnivVertex  E )  ->  K  e.  V
)
75, 6anim12i 566 . . . . . 6  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  K  e.  ( V UnivVertex  E ) )  -> 
( V USGrph  E  /\  K  e.  V )
)
87adantr 465 . . . . 5  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  K  e.  ( V UnivVertex  E ) )  /\  ( <. V ,  E >. Neighbors  K )  =  ( V  \  { K } ) )  -> 
( V USGrph  E  /\  K  e.  V )
)
9 hashnbgravdg 23580 . . . . 5  |-  ( ( V USGrph  E  /\  K  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  K ) )  =  ( ( V VDeg  E
) `  K )
)
108, 9syl 16 . . . 4  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  K  e.  ( V UnivVertex  E ) )  /\  ( <. V ,  E >. Neighbors  K )  =  ( V  \  { K } ) )  -> 
( # `  ( <. V ,  E >. Neighbors  K
) )  =  ( ( V VDeg  E ) `
 K ) )
11 hashdifsn 12168 . . . . . . . . 9  |-  ( ( V  e.  Fin  /\  K  e.  V )  ->  ( # `  ( V  \  { K }
) )  =  ( ( # `  V
)  -  1 ) )
1211ex 434 . . . . . . . 8  |-  ( V  e.  Fin  ->  ( K  e.  V  ->  (
# `  ( V  \  { K } ) )  =  ( (
# `  V )  -  1 ) ) )
1312adantl 466 . . . . . . 7  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  ( K  e.  V  ->  (
# `  ( V  \  { K } ) )  =  ( (
# `  V )  -  1 ) ) )
146, 13syl5com 30 . . . . . 6  |-  ( K  e.  ( V UnivVertex  E )  ->  ( ( V USGrph  E  /\  V  e.  Fin )  ->  ( # `  ( V  \  { K }
) )  =  ( ( # `  V
)  -  1 ) ) )
1514impcom 430 . . . . 5  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  K  e.  ( V UnivVertex  E ) )  -> 
( # `  ( V 
\  { K }
) )  =  ( ( # `  V
)  -  1 ) )
1615adantr 465 . . . 4  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  K  e.  ( V UnivVertex  E ) )  /\  ( <. V ,  E >. Neighbors  K )  =  ( V  \  { K } ) )  -> 
( # `  ( V 
\  { K }
) )  =  ( ( # `  V
)  -  1 ) )
174, 10, 163eqtr3d 2482 . . 3  |-  ( ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  K  e.  ( V UnivVertex  E ) )  /\  ( <. V ,  E >. Neighbors  K )  =  ( V  \  { K } ) )  -> 
( ( V VDeg  E
) `  K )  =  ( ( # `  V )  -  1 ) )
182, 17mpdan 668 . 2  |-  ( ( ( V USGrph  E  /\  V  e.  Fin )  /\  K  e.  ( V UnivVertex  E ) )  -> 
( ( V VDeg  E
) `  K )  =  ( ( # `  V )  -  1 ) )
1918ex 434 1  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  ( K  e.  ( V UnivVertex  E )  ->  ( ( V VDeg  E ) `  K
)  =  ( (
# `  V )  -  1 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    \ cdif 3324   {csn 3876   <.cop 3882   class class class wbr 4291   ` cfv 5417  (class class class)co 6090   Fincfn 7309   1c1 9282    - cmin 9594   #chash 12102   USGrph cusg 23263   Neighbors cnbgra 23328   UnivVertex cuvtx 23330   VDeg cvdg 23562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-card 8108  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-n0 10579  df-z 10646  df-uz 10861  df-xadd 11089  df-fz 11437  df-hash 12103  df-usgra 23265  df-nbgra 23331  df-uvtx 23333  df-vdgr 23563
This theorem is referenced by: (None)
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