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Metamath Proof Explorer

Theorem isgrpd2e

Description: Deduce a group from its properties. In this version of isgrpd2 , we don't assume there is an expression for the inverse of x . (Contributed by NM, 10-Aug-2013)

Ref Expression
Hypotheses isgrpd2.b 
|- ( ph -> B = ( Base ` G ) )
isgrpd2.p 
|- ( ph -> .+ = ( +g ` G ) )
isgrpd2.z 
|- ( ph -> .0. = ( 0g ` G ) )
isgrpd2.g 
|- ( ph -> G e. Mnd )
isgrpd2e.n 
|- ( ( ph /\ x e. B ) -> E. y e. B ( y .+ x ) = .0. )
Assertion isgrpd2e 
|- ( ph -> G e. Grp )

Proof

Step Hyp Ref Expression
1 isgrpd2.b 
 |-  ( ph -> B = ( Base ` G ) )
2 isgrpd2.p 
 |-  ( ph -> .+ = ( +g ` G ) )
3 isgrpd2.z 
 |-  ( ph -> .0. = ( 0g ` G ) )
4 isgrpd2.g 
 |-  ( ph -> G e. Mnd )
5 isgrpd2e.n 
 |-  ( ( ph /\ x e. B ) -> E. y e. B ( y .+ x ) = .0. )
6 5 ralrimiva 
 |-  ( ph -> A. x e. B E. y e. B ( y .+ x ) = .0. )
7 2 oveqd 
 |-  ( ph -> ( y .+ x ) = ( y ( +g ` G ) x ) )
8 7 3 eqeq12d 
 |-  ( ph -> ( ( y .+ x ) = .0. <-> ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
9 1 8 rexeqbidv 
 |-  ( ph -> ( E. y e. B ( y .+ x ) = .0. <-> E. y e. ( Base ` G ) ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
10 1 9 raleqbidv 
 |-  ( ph -> ( A. x e. B E. y e. B ( y .+ x ) = .0. <-> A. x e. ( Base ` G ) E. y e. ( Base ` G ) ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
11 6 10 mpbid 
 |-  ( ph -> A. x e. ( Base ` G ) E. y e. ( Base ` G ) ( y ( +g ` G ) x ) = ( 0g ` G ) )
12 eqid 
 |-  ( Base ` G ) = ( Base ` G )
13 eqid 
 |-  ( +g ` G ) = ( +g ` G )
14 eqid 
 |-  ( 0g ` G ) = ( 0g ` G )
15 12 13 14 isgrp 
 |-  ( G e. Grp <-> ( G e. Mnd /\ A. x e. ( Base ` G ) E. y e. ( Base ` G ) ( y ( +g ` G ) x ) = ( 0g ` G ) ) )
16 4 11 15 sylanbrc 
 |-  ( ph -> G e. Grp )