Metamath Proof Explorer

Theorem mulgsubdir

Description: Subtraction of a group element from itself. (Contributed by Mario Carneiro, 13-Dec-2014)

Ref Expression
Hypotheses mulgsubdir.b 
|- B = ( Base ` G )
mulgsubdir.t 
|- .x. = ( .g ` G )
mulgsubdir.d 
|- .- = ( -g ` G )
Assertion mulgsubdir 
|- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M - N ) .x. X ) = ( ( M .x. X ) .- ( N .x. X ) ) )

Proof

Step Hyp Ref Expression
1 mulgsubdir.b 
 |-  B = ( Base ` G )
2 mulgsubdir.t 
 |-  .x. = ( .g ` G )
3 mulgsubdir.d 
 |-  .- = ( -g ` G )
4 znegcl 
 |-  ( N e. ZZ -> -u N e. ZZ )
5 eqid 
 |-  ( +g ` G ) = ( +g ` G )
6 1 2 5 mulgdir 
 |-  ( ( G e. Grp /\ ( M e. ZZ /\ -u N e. ZZ /\ X e. B ) ) -> ( ( M + -u N ) .x. X ) = ( ( M .x. X ) ( +g ` G ) ( -u N .x. X ) ) )
7 4 6 syl3anr2 
 |-  ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M + -u N ) .x. X ) = ( ( M .x. X ) ( +g ` G ) ( -u N .x. X ) ) )
8 simpr1 
 |-  ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> M e. ZZ )
9 8 zcnd 
 |-  ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> M e. CC )
10 simpr2 
 |-  ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> N e. ZZ )
11 10 zcnd 
 |-  ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> N e. CC )
12 9 11 negsubd 
 |-  ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( M + -u N ) = ( M - N ) )
13 12 oveq1d 
 |-  ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M + -u N ) .x. X ) = ( ( M - N ) .x. X ) )
14 eqid 
 |-  ( invg ` G ) = ( invg ` G )
15 1 2 14 mulgneg 
 |-  ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( -u N .x. X ) = ( ( invg ` G ) ` ( N .x. X ) ) )
16 15 3adant3r1 
 |-  ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( -u N .x. X ) = ( ( invg ` G ) ` ( N .x. X ) ) )
17 16 oveq2d 
 |-  ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M .x. X ) ( +g ` G ) ( -u N .x. X ) ) = ( ( M .x. X ) ( +g ` G ) ( ( invg ` G ) ` ( N .x. X ) ) ) )
18 1 2 mulgcl 
 |-  ( ( G e. Grp /\ M e. ZZ /\ X e. B ) -> ( M .x. X ) e. B )
19 18 3adant3r2 
 |-  ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( M .x. X ) e. B )
20 1 2 mulgcl 
 |-  ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( N .x. X ) e. B )
21 20 3adant3r1 
 |-  ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( N .x. X ) e. B )
22 1 5 14 3 grpsubval 
 |-  ( ( ( M .x. X ) e. B /\ ( N .x. X ) e. B ) -> ( ( M .x. X ) .- ( N .x. X ) ) = ( ( M .x. X ) ( +g ` G ) ( ( invg ` G ) ` ( N .x. X ) ) ) )
23 19 21 22 syl2anc 
 |-  ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M .x. X ) .- ( N .x. X ) ) = ( ( M .x. X ) ( +g ` G ) ( ( invg ` G ) ` ( N .x. X ) ) ) )
24 17 23 eqtr4d 
 |-  ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M .x. X ) ( +g ` G ) ( -u N .x. X ) ) = ( ( M .x. X ) .- ( N .x. X ) ) )
25 7 13 24 3eqtr3d 
 |-  ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M - N ) .x. X ) = ( ( M .x. X ) .- ( N .x. X ) ) )