Theorem rusgrprop0 26463

Description: The properties of a k-regular simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.)

Hypotheses

Ref	Expression
isrusgr0.v	|- V = (Vtx ` G )
isrusgr0.d	|- D = (VtxDeg ` G )

Assertion

Ref	Expression
rusgrprop0	|- ( G RegUSGraph K -> ( G e. USGraph /\ K e. NN0* /\ A. v e. V ( D ` v ) = K ) )

Distinct variable groups:   v, G   v, K

Allowed substitution hints:   D( v)   V( v)

Proof of Theorem rusgrprop0

Step	Hyp	Ref	Expression
1		rusgrprop 26458	. 2 |- ( G RegUSGraph K -> ( G e. USGraph /\ G RegGraph K ) )
2		isrusgr0.v	. . . . 5 |- V = (Vtx ` G )
3		isrusgr0.d	. . . . 5 |- D = (VtxDeg ` G )
4	2, 3	rgrprop 26456	. . . 4 |- ( G RegGraph K -> ( K e. NN0* /\ A. v e. V ( D ` v ) = K ) )
5	4	anim2i 593	. . 3 |- ( ( G e. USGraph /\ G RegGraph K ) -> ( G e. USGraph /\ ( K e. NN0* /\ A. v e. V ( D ` v ) = K ) ) )
6		3anass 1042	. . 3 |- ( ( G e. USGraph /\ K e. NN0* /\ A. v e. V ( D ` v ) = K ) <-> ( G e. USGraph /\ ( K e. NN0* /\ A. v e. V ( D ` v ) = K ) ) )
7	5, 6	sylibr 224	. 2 |- ( ( G e. USGraph /\ G RegGraph K ) -> ( G e. USGraph /\ K e. NN0* /\ A. v e. V ( D ` v ) = K ) )
8	1, 7	syl 17	1 |- ( G RegUSGraph K -> ( G e. USGraph /\ K e. NN0* /\ A. v e. V ( D ` v ) = K ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   ` cfv 5888  NN0*cxnn0 11363  Vtxcvtx 25874   USGraph cusgr 26044  VtxDegcvtxdg 26361   RegGraph crgr 26451   RegUSGraph crusgr 26452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896  df-rgr 26453  df-rusgr 26454

This theorem is referenced by:  frusgrnn0  26467  cusgrm1rusgr  26478  rusgrpropnb  26479  frgrreg  27252  frgrregord013  27253