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Metamath Proof Explorer

Theorem odcong

Description: If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. (Contributed by Stefan O'Rear, 5-Sep-2015)

Ref Expression
Hypotheses odcl.1 
|- X = ( Base ` G )
odcl.2 
|- O = ( od ` G )
odid.3 
|- .x. = ( .g ` G )
odid.4 
|- .0. = ( 0g ` G )
Assertion odcong 
|- ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( O ` A ) || ( M - N ) <-> ( M .x. A ) = ( N .x. A ) ) )

Proof

Step Hyp Ref Expression
1 odcl.1 
 |-  X = ( Base ` G )
2 odcl.2 
 |-  O = ( od ` G )
3 odid.3 
 |-  .x. = ( .g ` G )
4 odid.4 
 |-  .0. = ( 0g ` G )
5 zsubcl 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
6 1 2 3 4 oddvds 
 |-  ( ( G e. Grp /\ A e. X /\ ( M - N ) e. ZZ ) -> ( ( O ` A ) || ( M - N ) <-> ( ( M - N ) .x. A ) = .0. ) )
7 5 6 syl3an3 
 |-  ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( O ` A ) || ( M - N ) <-> ( ( M - N ) .x. A ) = .0. ) )
8 simp1 
 |-  ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> G e. Grp )
9 simp3l 
 |-  ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> M e. ZZ )
10 simp3r 
 |-  ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> N e. ZZ )
11 simp2 
 |-  ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> A e. X )
12 eqid 
 |-  ( -g ` G ) = ( -g ` G )
13 1 3 12 mulgsubdir 
 |-  ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ A e. X ) ) -> ( ( M - N ) .x. A ) = ( ( M .x. A ) ( -g ` G ) ( N .x. A ) ) )
14 8 9 10 11 13 syl13anc 
 |-  ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( M - N ) .x. A ) = ( ( M .x. A ) ( -g ` G ) ( N .x. A ) ) )
15 14 eqeq1d 
 |-  ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( ( M - N ) .x. A ) = .0. <-> ( ( M .x. A ) ( -g ` G ) ( N .x. A ) ) = .0. ) )
16 1 3 mulgcl 
 |-  ( ( G e. Grp /\ M e. ZZ /\ A e. X ) -> ( M .x. A ) e. X )
17 8 9 11 16 syl3anc 
 |-  ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M .x. A ) e. X )
18 1 3 mulgcl 
 |-  ( ( G e. Grp /\ N e. ZZ /\ A e. X ) -> ( N .x. A ) e. X )
19 8 10 11 18 syl3anc 
 |-  ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( N .x. A ) e. X )
20 1 4 12 grpsubeq0 
 |-  ( ( G e. Grp /\ ( M .x. A ) e. X /\ ( N .x. A ) e. X ) -> ( ( ( M .x. A ) ( -g ` G ) ( N .x. A ) ) = .0. <-> ( M .x. A ) = ( N .x. A ) ) )
21 8 17 19 20 syl3anc 
 |-  ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( ( M .x. A ) ( -g ` G ) ( N .x. A ) ) = .0. <-> ( M .x. A ) = ( N .x. A ) ) )
22 7 15 21 3bitrd 
 |-  ( ( G e. Grp /\ A e. X /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( O ` A ) || ( M - N ) <-> ( M .x. A ) = ( N .x. A ) ) )