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Metamath Proof Explorer

Theorem gexdvdsi

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

Ref Expression
Hypotheses gexcl.1 
|- X = ( Base ` G )
gexcl.2 
|- E = ( gEx ` G )
gexid.3 
|- .x. = ( .g ` G )
gexid.4 
|- .0. = ( 0g ` G )
Assertion gexdvdsi 
|- ( ( G e. Grp /\ A e. X /\ E || N ) -> ( N .x. A ) = .0. )

Proof

Step Hyp Ref Expression
1 gexcl.1 
 |-  X = ( Base ` G )
2 gexcl.2 
 |-  E = ( gEx ` G )
3 gexid.3 
 |-  .x. = ( .g ` G )
4 gexid.4 
 |-  .0. = ( 0g ` G )
5 simp3 
 |-  ( ( G e. Grp /\ A e. X /\ E || N ) -> E || N )
6 dvdszrcl 
 |-  ( E || N -> ( E e. ZZ /\ N e. ZZ ) )
7 divides 
 |-  ( ( E e. ZZ /\ N e. ZZ ) -> ( E || N <-> E. x e. ZZ ( x x. E ) = N ) )
8 6 7 biadanii 
 |-  ( E || N <-> ( ( E e. ZZ /\ N e. ZZ ) /\ E. x e. ZZ ( x x. E ) = N ) )
9 5 8 sylib 
 |-  ( ( G e. Grp /\ A e. X /\ E || N ) -> ( ( E e. ZZ /\ N e. ZZ ) /\ E. x e. ZZ ( x x. E ) = N ) )
10 9 simprd 
 |-  ( ( G e. Grp /\ A e. X /\ E || N ) -> E. x e. ZZ ( x x. E ) = N )
11 simpl1 
 |-  ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> G e. Grp )
12 simpr 
 |-  ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> x e. ZZ )
13 9 simplld 
 |-  ( ( G e. Grp /\ A e. X /\ E || N ) -> E e. ZZ )
14 13 adantr 
 |-  ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> E e. ZZ )
15 simpl2 
 |-  ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> A e. X )
16 1 3 mulgass 
 |-  ( ( G e. Grp /\ ( x e. ZZ /\ E e. ZZ /\ A e. X ) ) -> ( ( x x. E ) .x. A ) = ( x .x. ( E .x. A ) ) )
17 11 12 14 15 16 syl13anc 
 |-  ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> ( ( x x. E ) .x. A ) = ( x .x. ( E .x. A ) ) )
18 1 2 3 4 gexid 
 |-  ( A e. X -> ( E .x. A ) = .0. )
19 15 18 syl 
 |-  ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> ( E .x. A ) = .0. )
20 19 oveq2d 
 |-  ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> ( x .x. ( E .x. A ) ) = ( x .x. .0. ) )
21 1 3 4 mulgz 
 |-  ( ( G e. Grp /\ x e. ZZ ) -> ( x .x. .0. ) = .0. )
22 21 3ad2antl1 
 |-  ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> ( x .x. .0. ) = .0. )
23 17 20 22 3eqtrd 
 |-  ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> ( ( x x. E ) .x. A ) = .0. )
24 oveq1 
 |-  ( ( x x. E ) = N -> ( ( x x. E ) .x. A ) = ( N .x. A ) )
25 24 eqeq1d 
 |-  ( ( x x. E ) = N -> ( ( ( x x. E ) .x. A ) = .0. <-> ( N .x. A ) = .0. ) )
26 23 25 syl5ibcom 
 |-  ( ( ( G e. Grp /\ A e. X /\ E || N ) /\ x e. ZZ ) -> ( ( x x. E ) = N -> ( N .x. A ) = .0. ) )
27 26 rexlimdva 
 |-  ( ( G e. Grp /\ A e. X /\ E || N ) -> ( E. x e. ZZ ( x x. E ) = N -> ( N .x. A ) = .0. ) )
28 10 27 mpd 
 |-  ( ( G e. Grp /\ A e. X /\ E || N ) -> ( N .x. A ) = .0. )