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Theorem cusgraisrusgra 25511
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 25036 . . . 4  |-  ( ( V ComplUSGrph  E  /\  v  e.  V )  ->  ( <. V ,  E >. Neighbors  v
)  =  ( V 
\  { v } ) )
21ralrimiva 2846 . . 3  |-  ( V ComplUSGrph  E  ->  A. v  e.  V  ( <. V ,  E >. Neighbors 
v )  =  ( V  \  { v } ) )
3 cusisusgra 25031 . . . . . . . 8  |-  ( V ComplUSGrph  E  ->  V USGrph  E )
43adantr 466 . . . . . . 7  |-  ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  ->  V USGrph  E )
54adantr 466 . . . . . 6  |-  ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  A. v  e.  V  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } ) )  ->  V USGrph  E )
6 hashnncl 12544 . . . . . . . . 9  |-  ( V  e.  Fin  ->  (
( # `  V )  e.  NN  <->  V  =/=  (/) ) )
7 nnm1nn0 10911 . . . . . . . . 9  |-  ( (
# `  V )  e.  NN  ->  ( ( # `
 V )  - 
1 )  e.  NN0 )
86, 7syl6bir 232 . . . . . . . 8  |-  ( V  e.  Fin  ->  ( V  =/=  (/)  ->  ( ( # `
 V )  - 
1 )  e.  NN0 ) )
98imp 430 . . . . . . 7  |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  (
( # `  V )  -  1 )  e. 
NN0 )
109ad2antlr 731 . . . . . 6  |-  ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  A. v  e.  V  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } ) )  -> 
( ( # `  V
)  -  1 )  e.  NN0 )
114anim1i 570 . . . . . . . . . . . 12  |-  ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  v  e.  V
)  ->  ( V USGrph  E  /\  v  e.  V
) )
1211adantr 466 . . . . . . . . . . 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 25486 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  v  e.  V )  ->  ( # `
 ( <. V ,  E >. Neighbors  v ) )  =  ( ( V VDeg  E
) `  v )
)
1412, 13syl 17 . . . . . . . . . 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 5881 . . . . . . . . . . 11  |-  ( (
<. V ,  E >. Neighbors  v
)  =  ( V 
\  { v } )  ->  ( # `  ( <. V ,  E >. Neighbors  v
) )  =  (
# `  ( V  \  { v } ) ) )
16 simprl 762 . . . . . . . . . . . 12  |-  ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  ->  V  e.  Fin )
17 hashdifsn 12586 . . . . . . . . . . . 12  |-  ( ( V  e.  Fin  /\  v  e.  V )  ->  ( # `  ( V  \  { v } ) )  =  ( ( # `  V
)  -  1 ) )
1816, 17sylan 473 . . . . . . . . . . 11  |-  ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  v  e.  V
)  ->  ( # `  ( V  \  { v } ) )  =  ( ( # `  V
)  -  1 ) )
1915, 18sylan9eqr 2492 . . . . . . . . . 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 2472 . . . . . . . . 9  |-  ( ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  v  e.  V
)  /\  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } ) )  ->  ( ( V VDeg  E ) `  v
)  =  ( (
# `  V )  -  1 ) )
2120ex 435 . . . . . . . 8  |-  ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  v  e.  V
)  ->  ( ( <. V ,  E >. Neighbors  v
)  =  ( V 
\  { v } )  ->  ( ( V VDeg  E ) `  v
)  =  ( (
# `  V )  -  1 ) ) )
2221ralimdva 2840 . . . . . . 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 430 . . . . . 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 24911 . . . . . . . . . . 11  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
253, 24syl 17 . . . . . . . . . 10  |-  ( V ComplUSGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2625adantr 466 . . . . . . . . 9  |-  ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  -> 
( V  e.  _V  /\  E  e.  _V )
)
27 ovex 6333 . . . . . . . . . 10  |-  ( (
# `  V )  -  1 )  e. 
_V
2827a1i 11 . . . . . . . . 9  |-  ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  -> 
( ( # `  V
)  -  1 )  e.  _V )
29 df-3an 984 . . . . . . . . 9  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  (
( # `  V )  -  1 )  e. 
_V )  <->  ( ( V  e.  _V  /\  E  e.  _V )  /\  (
( # `  V )  -  1 )  e. 
_V ) )
3026, 28, 29sylanbrc 668 . . . . . . . 8  |-  ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  -> 
( V  e.  _V  /\  E  e.  _V  /\  ( ( # `  V
)  -  1 )  e.  _V ) )
3130adantr 466 . . . . . . 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 25500 . . . . . . 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 17 . . . . . 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 1188 . . . . 5  |-  ( ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  /\  A. v  e.  V  ( <. V ,  E >. Neighbors  v )  =  ( V  \  { v } ) )  ->  <. V ,  E >. RegUSGrph  (
( # `  V )  -  1 ) )
3534expcom 436 . . . 4  |-  ( A. v  e.  V  ( <. V ,  E >. Neighbors  v
)  =  ( V 
\  { v } )  ->  ( ( V ComplUSGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) )  ->  <. V ,  E >. RegUSGrph  (
( # `  V )  -  1 ) ) )
3635expd 437 . . 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 37 . 2  |-  ( V ComplUSGrph  E  ->  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  <. V ,  E >. RegUSGrph  ( ( # `  V
)  -  1 ) ) )
38373impib 1203 1  |-  ( ( V ComplUSGrph  E  /\  V  e. 
Fin  /\  V  =/=  (/) )  ->  <. V ,  E >. RegUSGrph  ( ( # `  V
)  -  1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   _Vcvv 3087    \ cdif 3439   (/)c0 3767   {csn 4002   <.cop 4008   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Fincfn 7577   1c1 9539    - cmin 9859   NNcn 10609   NN0cn0 10869   #chash 12512   USGrph cusg 24903   Neighbors cnbgra 24990   ComplUSGrph ccusgra 24991   VDeg cvdg 25466   RegUSGrph crusgra 25496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-xadd 11410  df-fz 11783  df-hash 12513  df-usgra 24906  df-nbgra 24993  df-cusgra 24994  df-vdgr 25467  df-rgra 25497  df-rusgra 25498
This theorem is referenced by:  cusgraiffrusgra  25513
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