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Theorem cusgraisrusgra 24761
Description: A complete undirected simple graph with n vertices (at least one) is (n-1)-regular. (Contributed by Alexander van der Vekens, 10-Jul-2018.)
Assertion
Ref Expression
cusgraisrusgra  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  <. V ,  E >. RegUSGrph  ( ( # `  V
)  -  1 ) )

Proof of Theorem cusgraisrusgra
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 nbcusgra 24286 . . . 4  |-  ( ( V ComplUSGrph  E  /\  v  e.  V )  ->  ( <. V ,  E >. Neighbors  v
)  =  ( V 
\  { v } ) )
21ralrimiva 2881 . . 3  |-  ( V ComplUSGrph  E  ->  A. v  e.  V  ( <. V ,  E >. Neighbors 
v )  =  ( V  \  { v } ) )
3 cusisusgra 24281 . . . . . . . 8  |-  ( V ComplUSGrph  E  ->  V USGrph  E )
43adantr 465 . . . . . . 7  |-  ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  ->  V USGrph  E )
54adantr 465 . . . . . 6  |-  ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  A. v  e.  V  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } ) )  ->  V USGrph  E )
6 hashnncl 12416 . . . . . . . . 9  |-  ( V  e.  Fin  ->  (
( # `  V )  e.  NN  <->  V  =/=  (/) ) )
7 nnm1nn0 10849 . . . . . . . . 9  |-  ( (
# `  V )  e.  NN  ->  ( ( # `
 V )  - 
1 )  e.  NN0 )
86, 7syl6bir 229 . . . . . . . 8  |-  ( V  e.  Fin  ->  ( V  =/=  (/)  ->  ( ( # `
 V )  - 
1 )  e.  NN0 ) )
98imp 429 . . . . . . 7  |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  (
( # `  V )  -  1 )  e. 
NN0 )
109ad2antlr 726 . . . . . 6  |-  ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  A. v  e.  V  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } ) )  -> 
( ( # `  V
)  -  1 )  e.  NN0 )
114anim1i 568 . . . . . . . . . . . 12  |-  ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  v  e.  V
)  ->  ( V USGrph  E  /\  v  e.  V
) )
1211adantr 465 . . . . . . . . . . 11  |-  ( ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  v  e.  V
)  /\  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } ) )  ->  ( V USGrph  E  /\  v  e.  V
) )
13 hashnbgravdg 24736 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  v  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  v ) )  =  ( ( V VDeg  E
) `  v )
)
1412, 13syl 16 . . . . . . . . . 10  |-  ( ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  v  e.  V
)  /\  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } ) )  ->  ( # `  ( <. V ,  E >. Neighbors  v
) )  =  ( ( V VDeg  E ) `
 v ) )
15 fveq2 5872 . . . . . . . . . . 11  |-  ( (
<. V ,  E >. Neighbors  v
)  =  ( V 
\  { v } )  ->  ( # `  ( <. V ,  E >. Neighbors  v
) )  =  (
# `  ( V  \  { v } ) ) )
16 simprl 755 . . . . . . . . . . . 12  |-  ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  ->  V  e.  Fin )
17 hashdifsn 12457 . . . . . . . . . . . 12  |-  ( ( V  e.  Fin  /\  v  e.  V )  ->  ( # `  ( V  \  { v } ) )  =  ( ( # `  V
)  -  1 ) )
1816, 17sylan 471 . . . . . . . . . . 11  |-  ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  v  e.  V
)  ->  ( # `  ( V  \  { v } ) )  =  ( ( # `  V
)  -  1 ) )
1915, 18sylan9eqr 2530 . . . . . . . . . 10  |-  ( ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  v  e.  V
)  /\  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } ) )  ->  ( # `  ( <. V ,  E >. Neighbors  v
) )  =  ( ( # `  V
)  -  1 ) )
2014, 19eqtr3d 2510 . . . . . . . . 9  |-  ( ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  v  e.  V
)  /\  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } ) )  ->  ( ( V VDeg  E ) `  v
)  =  ( (
# `  V )  -  1 ) )
2120ex 434 . . . . . . . 8  |-  ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  v  e.  V
)  ->  ( ( <. V ,  E >. Neighbors  v
)  =  ( V 
\  { v } )  ->  ( ( V VDeg  E ) `  v
)  =  ( (
# `  V )  -  1 ) ) )
2221ralimdva 2875 . . . . . . 7  |-  ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  -> 
( A. v  e.  V  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } )  ->  A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  ( (
# `  V )  -  1 ) ) )
2322imp 429 . . . . . 6  |-  ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  A. v  e.  V  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } ) )  ->  A. v  e.  V  ( ( V VDeg  E
) `  v )  =  ( ( # `  V )  -  1 ) )
24 usgrav 24161 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
253, 24syl 16 . . . . . . . . . 10  |-  ( V ComplUSGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2625adantr 465 . . . . . . . . 9  |-  ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  -> 
( V  e.  _V  /\  E  e.  _V )
)
27 ovex 6320 . . . . . . . . . 10  |-  ( (
# `  V )  -  1 )  e. 
_V
2827a1i 11 . . . . . . . . 9  |-  ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  -> 
( ( # `  V
)  -  1 )  e.  _V )
29 df-3an 975 . . . . . . . . 9  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  (
( # `  V )  -  1 )  e. 
_V )  <->  ( ( V  e.  _V  /\  E  e.  _V )  /\  (
( # `  V )  -  1 )  e. 
_V ) )
3026, 28, 29sylanbrc 664 . . . . . . . 8  |-  ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  -> 
( V  e.  _V  /\  E  e.  _V  /\  ( ( # `  V
)  -  1 )  e.  _V ) )
3130adantr 465 . . . . . . 7  |-  ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  A. v  e.  V  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } ) )  -> 
( V  e.  _V  /\  E  e.  _V  /\  ( ( # `  V
)  -  1 )  e.  _V ) )
32 isrusgra 24750 . . . . . . 7  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  (
( # `  V )  -  1 )  e. 
_V )  ->  ( <. V ,  E >. RegUSGrph  (
( # `  V )  -  1 )  <->  ( V USGrph  E  /\  ( ( # `  V )  -  1 )  e.  NN0  /\  A. v  e.  V  ( ( V VDeg  E ) `
 v )  =  ( ( # `  V
)  -  1 ) ) ) )
3331, 32syl 16 . . . . . 6  |-  ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  A. v  e.  V  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } ) )  -> 
( <. V ,  E >. RegUSGrph 
( ( # `  V
)  -  1 )  <-> 
( V USGrph  E  /\  ( ( # `  V
)  -  1 )  e.  NN0  /\  A. v  e.  V  ( ( V VDeg  E ) `  v
)  =  ( (
# `  V )  -  1 ) ) ) )
345, 10, 23, 33mpbir3and 1179 . . . . 5  |-  ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  A. v  e.  V  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } ) )  ->  <. V ,  E >. RegUSGrph  (
( # `  V )  -  1 ) )
3534expcom 435 . . . 4  |-  ( A. v  e.  V  ( <. V ,  E >. Neighbors  v
)  =  ( V 
\  { v } )  ->  ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  ->  <. V ,  E >. RegUSGrph  (
( # `  V )  -  1 ) ) )
3635expd 436 . . 3  |-  ( A. v  e.  V  ( <. V ,  E >. Neighbors  v
)  =  ( V 
\  { v } )  ->  ( V ComplUSGrph  E  ->  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  <. V ,  E >. RegUSGrph  ( ( # `  V
)  -  1 ) ) ) )
372, 36mpcom 36 . 2  |-  ( V ComplUSGrph  E  ->  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  <. V ,  E >. RegUSGrph  ( ( # `  V
)  -  1 ) ) )
38373impib 1194 1  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  <. V ,  E >. RegUSGrph  ( ( # `  V
)  -  1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   _Vcvv 3118    \ cdif 3478   (/)c0 3790   {csn 4033   <.cop 4039   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Fincfn 7528   1c1 9505    - cmin 9817   NNcn 10548   NN0cn0 10807   #chash 12385   USGrph cusg 24153   Neighbors cnbgra 24240   ComplUSGrph ccusgra 24241   VDeg cvdg 24716   RegUSGrph crusgra 24746
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-xadd 11331  df-fz 11685  df-hash 12386  df-usgra 24156  df-nbgra 24243  df-cusgra 24244  df-vdgr 24717  df-rgra 24747  df-rusgra 24748
This theorem is referenced by:  cusgraiffrusgra  24763
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