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Metamath Proof Explorer

Theorem gexod

Description: Any group element is annihilated by any multiple of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016)

Ref Expression
Hypotheses gexod.1 
|- X = ( Base ` G )
gexod.2 
|- E = ( gEx ` G )
gexod.3 
|- O = ( od ` G )
Assertion gexod 
|- ( ( G e. Grp /\ A e. X ) -> ( O ` A ) || E )

Proof

Step Hyp Ref Expression
1 gexod.1 
 |-  X = ( Base ` G )
2 gexod.2 
 |-  E = ( gEx ` G )
3 gexod.3 
 |-  O = ( od ` G )
4 eqid 
 |-  ( .g ` G ) = ( .g ` G )
5 eqid 
 |-  ( 0g ` G ) = ( 0g ` G )
6 1 2 4 5 gexid 
 |-  ( A e. X -> ( E ( .g ` G ) A ) = ( 0g ` G ) )
7 6 adantl 
 |-  ( ( G e. Grp /\ A e. X ) -> ( E ( .g ` G ) A ) = ( 0g ` G ) )
8 1 2 gexcl 
 |-  ( G e. Grp -> E e. NN0 )
9 8 adantr 
 |-  ( ( G e. Grp /\ A e. X ) -> E e. NN0 )
10 9 nn0zd 
 |-  ( ( G e. Grp /\ A e. X ) -> E e. ZZ )
11 1 3 4 5 oddvds 
 |-  ( ( G e. Grp /\ A e. X /\ E e. ZZ ) -> ( ( O ` A ) || E <-> ( E ( .g ` G ) A ) = ( 0g ` G ) ) )
12 10 11 mpd3an3 
 |-  ( ( G e. Grp /\ A e. X ) -> ( ( O ` A ) || E <-> ( E ( .g ` G ) A ) = ( 0g ` G ) ) )
13 7 12 mpbird 
 |-  ( ( G e. Grp /\ A e. X ) -> ( O ` A ) || E )