Metamath Proof Explorer

Theorem frrusgrord

Description: If a nonempty finite friendship graph is k-regular, its order is k(k-1)+1. This corresponds to claim 3 in Huneke p. 2: "Next we claim that the number n of vertices in G is exactly k(k-1)+1.". Variant of frrusgrord0 , using the definition RegUSGraph ( df-rusgr ). (Contributed by Alexander van der Vekens, 25-Aug-2018) (Revised by AV, 26-May-2021) (Proof shortened by AV, 12-Jan-2022)

Ref Expression
Hypothesis frrusgrord0.v 
|- V = ( Vtx ` G )
Assertion frrusgrord 
|- ( ( V e. Fin /\ V =/= (/) ) -> ( ( G e. FriendGraph /\ G RegUSGraph K ) -> ( # ` V ) = ( ( K x. ( K - 1 ) ) + 1 ) ) )

Proof

Step Hyp Ref Expression
1 frrusgrord0.v 
 |-  V = ( Vtx ` G )
2 rusgrrgr 
 |-  ( G RegUSGraph K -> G RegGraph K )
3 eqid 
 |-  ( VtxDeg ` G ) = ( VtxDeg ` G )
4 1 3 rgrprop 
 |-  ( G RegGraph K -> ( K e. NN0* /\ A. v e. V ( ( VtxDeg ` G ) ` v ) = K ) )
5 2 4 syl 
 |-  ( G RegUSGraph K -> ( K e. NN0* /\ A. v e. V ( ( VtxDeg ` G ) ` v ) = K ) )
6 5 simprd 
 |-  ( G RegUSGraph K -> A. v e. V ( ( VtxDeg ` G ) ` v ) = K )
7 1 frrusgrord0 
 |-  ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) -> ( A. v e. V ( ( VtxDeg ` G ) ` v ) = K -> ( # ` V ) = ( ( K x. ( K - 1 ) ) + 1 ) ) )
8 6 7 syl5 
 |-  ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) -> ( G RegUSGraph K -> ( # ` V ) = ( ( K x. ( K - 1 ) ) + 1 ) ) )
9 8 3expb 
 |-  ( ( G e. FriendGraph /\ ( V e. Fin /\ V =/= (/) ) ) -> ( G RegUSGraph K -> ( # ` V ) = ( ( K x. ( K - 1 ) ) + 1 ) ) )
10 9 expcom 
 |-  ( ( V e. Fin /\ V =/= (/) ) -> ( G e. FriendGraph -> ( G RegUSGraph K -> ( # ` V ) = ( ( K x. ( K - 1 ) ) + 1 ) ) ) )
11 10 impd 
 |-  ( ( V e. Fin /\ V =/= (/) ) -> ( ( G e. FriendGraph /\ G RegUSGraph K ) -> ( # ` V ) = ( ( K x. ( K - 1 ) ) + 1 ) ) )