Metamath Proof Explorer

Theorem rngosubdir

Description: Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010)

Ref Expression
Hypotheses ringsubdi.1 
|- G = ( 1st ` R )
ringsubdi.2 
|- H = ( 2nd ` R )
ringsubdi.3 
|- X = ran G
ringsubdi.4 
|- D = ( /g ` G )
Assertion rngosubdir 
|- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A D B ) H C ) = ( ( A H C ) D ( B H C ) ) )

Proof

Step Hyp Ref Expression
1 ringsubdi.1 
 |-  G = ( 1st ` R )
2 ringsubdi.2 
 |-  H = ( 2nd ` R )
3 ringsubdi.3 
 |-  X = ran G
4 ringsubdi.4 
 |-  D = ( /g ` G )
5 eqid 
 |-  ( inv ` G ) = ( inv ` G )
6 1 3 5 4 rngosub 
 |-  ( ( R e. RingOps /\ A e. X /\ B e. X ) -> ( A D B ) = ( A G ( ( inv ` G ) ` B ) ) )
7 6 3adant3r3 
 |-  ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A D B ) = ( A G ( ( inv ` G ) ` B ) ) )
8 7 oveq1d 
 |-  ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A D B ) H C ) = ( ( A G ( ( inv ` G ) ` B ) ) H C ) )
9 1 2 3 rngocl 
 |-  ( ( R e. RingOps /\ A e. X /\ C e. X ) -> ( A H C ) e. X )
10 9 3adant3r2 
 |-  ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H C ) e. X )
11 1 2 3 rngocl 
 |-  ( ( R e. RingOps /\ B e. X /\ C e. X ) -> ( B H C ) e. X )
12 11 3adant3r1 
 |-  ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( B H C ) e. X )
13 10 12 jca 
 |-  ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A H C ) e. X /\ ( B H C ) e. X ) )
14 1 3 5 4 rngosub 
 |-  ( ( R e. RingOps /\ ( A H C ) e. X /\ ( B H C ) e. X ) -> ( ( A H C ) D ( B H C ) ) = ( ( A H C ) G ( ( inv ` G ) ` ( B H C ) ) ) )
15 14 3expb 
 |-  ( ( R e. RingOps /\ ( ( A H C ) e. X /\ ( B H C ) e. X ) ) -> ( ( A H C ) D ( B H C ) ) = ( ( A H C ) G ( ( inv ` G ) ` ( B H C ) ) ) )
16 13 15 syldan 
 |-  ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A H C ) D ( B H C ) ) = ( ( A H C ) G ( ( inv ` G ) ` ( B H C ) ) ) )
17 idd 
 |-  ( R e. RingOps -> ( A e. X -> A e. X ) )
18 1 3 5 rngonegcl 
 |-  ( ( R e. RingOps /\ B e. X ) -> ( ( inv ` G ) ` B ) e. X )
19 18 ex 
 |-  ( R e. RingOps -> ( B e. X -> ( ( inv ` G ) ` B ) e. X ) )
20 idd 
 |-  ( R e. RingOps -> ( C e. X -> C e. X ) )
21 17 19 20 3anim123d 
 |-  ( R e. RingOps -> ( ( A e. X /\ B e. X /\ C e. X ) -> ( A e. X /\ ( ( inv ` G ) ` B ) e. X /\ C e. X ) ) )
22 21 imp 
 |-  ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A e. X /\ ( ( inv ` G ) ` B ) e. X /\ C e. X ) )
23 1 2 3 rngodir 
 |-  ( ( R e. RingOps /\ ( A e. X /\ ( ( inv ` G ) ` B ) e. X /\ C e. X ) ) -> ( ( A G ( ( inv ` G ) ` B ) ) H C ) = ( ( A H C ) G ( ( ( inv ` G ) ` B ) H C ) ) )
24 22 23 syldan 
 |-  ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G ( ( inv ` G ) ` B ) ) H C ) = ( ( A H C ) G ( ( ( inv ` G ) ` B ) H C ) ) )
25 1 2 3 5 rngoneglmul 
 |-  ( ( R e. RingOps /\ B e. X /\ C e. X ) -> ( ( inv ` G ) ` ( B H C ) ) = ( ( ( inv ` G ) ` B ) H C ) )
26 25 3adant3r1 
 |-  ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( inv ` G ) ` ( B H C ) ) = ( ( ( inv ` G ) ` B ) H C ) )
27 26 oveq2d 
 |-  ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A H C ) G ( ( inv ` G ) ` ( B H C ) ) ) = ( ( A H C ) G ( ( ( inv ` G ) ` B ) H C ) ) )
28 24 27 eqtr4d 
 |-  ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G ( ( inv ` G ) ` B ) ) H C ) = ( ( A H C ) G ( ( inv ` G ) ` ( B H C ) ) ) )
29 16 28 eqtr4d 
 |-  ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A H C ) D ( B H C ) ) = ( ( A G ( ( inv ` G ) ` B ) ) H C ) )
30 8 29 eqtr4d 
 |-  ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A D B ) H C ) = ( ( A H C ) D ( B H C ) ) )