Metamath Proof Explorer

Theorem frrusgrord0lem

Description: Lemma for frrusgrord0 . (Contributed by AV, 12-Jan-2022)

Ref Expression
Hypothesis frrusgrord0.v 
|- V = ( Vtx ` G )
Assertion frrusgrord0lem 
|- ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ A. v e. V ( ( VtxDeg ` G ) ` v ) = K ) -> ( K e. CC /\ ( # ` V ) e. CC /\ ( # ` V ) =/= 0 ) )

Proof

Step Hyp Ref Expression
1 frrusgrord0.v 
 |-  V = ( Vtx ` G )
2 frgrusgr 
 |-  ( G e. FriendGraph -> G e. USGraph )
3 2 anim1i 
 |-  ( ( G e. FriendGraph /\ V e. Fin ) -> ( G e. USGraph /\ V e. Fin ) )
4 1 isfusgr 
 |-  ( G e. FinUSGraph <-> ( G e. USGraph /\ V e. Fin ) )
5 3 4 sylibr 
 |-  ( ( G e. FriendGraph /\ V e. Fin ) -> G e. FinUSGraph )
6 eqid 
 |-  ( VtxDeg ` G ) = ( VtxDeg ` G )
7 1 6 fusgrregdegfi 
 |-  ( ( G e. FinUSGraph /\ V =/= (/) ) -> ( A. v e. V ( ( VtxDeg ` G ) ` v ) = K -> K e. NN0 ) )
8 5 7 stoic3 
 |-  ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) -> ( A. v e. V ( ( VtxDeg ` G ) ` v ) = K -> K e. NN0 ) )
9 8 imp 
 |-  ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ A. v e. V ( ( VtxDeg ` G ) ` v ) = K ) -> K e. NN0 )
10 9 nn0cnd 
 |-  ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ A. v e. V ( ( VtxDeg ` G ) ` v ) = K ) -> K e. CC )
11 hashcl 
 |-  ( V e. Fin -> ( # ` V ) e. NN0 )
12 11 nn0cnd 
 |-  ( V e. Fin -> ( # ` V ) e. CC )
13 12 3ad2ant2 
 |-  ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) -> ( # ` V ) e. CC )
14 13 adantr 
 |-  ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ A. v e. V ( ( VtxDeg ` G ) ` v ) = K ) -> ( # ` V ) e. CC )
15 hasheq0 
 |-  ( V e. Fin -> ( ( # ` V ) = 0 <-> V = (/) ) )
16 15 biimpd 
 |-  ( V e. Fin -> ( ( # ` V ) = 0 -> V = (/) ) )
17 16 necon3d 
 |-  ( V e. Fin -> ( V =/= (/) -> ( # ` V ) =/= 0 ) )
18 17 imp 
 |-  ( ( V e. Fin /\ V =/= (/) ) -> ( # ` V ) =/= 0 )
19 18 3adant1 
 |-  ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) -> ( # ` V ) =/= 0 )
20 19 adantr 
 |-  ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ A. v e. V ( ( VtxDeg ` G ) ` v ) = K ) -> ( # ` V ) =/= 0 )
21 10 14 20 3jca 
 |-  ( ( ( G e. FriendGraph /\ V e. Fin /\ V =/= (/) ) /\ A. v e. V ( ( VtxDeg ` G ) ` v ) = K ) -> ( K e. CC /\ ( # ` V ) e. CC /\ ( # ` V ) =/= 0 ) )