Metamath Proof Explorer

Theorem dprdgrp

Description: Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016)

Ref Expression
Assertion dprdgrp 
|- ( G dom DProd S -> G e. Grp )

Proof

Step Hyp Ref Expression
1 reldmdprd 
 |-  Rel dom DProd
2 1 brrelex2i 
 |-  ( G dom DProd S -> S e. _V )
3 2 dmexd 
 |-  ( G dom DProd S -> dom S e. _V )
4 eqid 
 |-  dom S = dom S
5 eqid 
 |-  ( Cntz ` G ) = ( Cntz ` G )
6 eqid 
 |-  ( 0g ` G ) = ( 0g ` G )
7 eqid 
 |-  ( mrCls ` ( SubGrp ` G ) ) = ( mrCls ` ( SubGrp ` G ) )
8 5 6 7 dmdprd 
 |-  ( ( dom S e. _V /\ dom S = dom S ) -> ( G dom DProd S <-> ( G e. Grp /\ S : dom S --> ( SubGrp ` G ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( ( Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( ( mrCls ` ( SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) ) )
9 3 4 8 sylancl 
 |-  ( G dom DProd S -> ( G dom DProd S <-> ( G e. Grp /\ S : dom S --> ( SubGrp ` G ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( ( Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( ( mrCls ` ( SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) ) )
10 9 ibi 
 |-  ( G dom DProd S -> ( G e. Grp /\ S : dom S --> ( SubGrp ` G ) /\ A. x e. dom S ( A. y e. ( dom S \ { x } ) ( S ` x ) C_ ( ( Cntz ` G ) ` ( S ` y ) ) /\ ( ( S ` x ) i^i ( ( mrCls ` ( SubGrp ` G ) ) ` U. ( S " ( dom S \ { x } ) ) ) ) = { ( 0g ` G ) } ) ) )
11 10 simp1d 
 |-  ( G dom DProd S -> G e. Grp )