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Metamath Proof Explorer

Theorem dvrdir

Description: Distributive law for the division operation of a ring. (Contributed by Thierry Arnoux, 30-Oct-2017)

Ref Expression
Hypotheses dvrdir.b 
|- B = ( Base ` R )
dvrdir.u 
|- U = ( Unit ` R )
dvrdir.p 
|- .+ = ( +g ` R )
dvrdir.t 
|- ./ = ( /r ` R )
Assertion dvrdir 
|- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( X .+ Y ) ./ Z ) = ( ( X ./ Z ) .+ ( Y ./ Z ) ) )

Proof

Step Hyp Ref Expression
1 dvrdir.b 
 |-  B = ( Base ` R )
2 dvrdir.u 
 |-  U = ( Unit ` R )
3 dvrdir.p 
 |-  .+ = ( +g ` R )
4 dvrdir.t 
 |-  ./ = ( /r ` R )
5 simpl 
 |-  ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> R e. Ring )
6 simpr1 
 |-  ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> X e. B )
7 simpr2 
 |-  ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> Y e. B )
8 1 2 unitss 
 |-  U C_ B
9 simpr3 
 |-  ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> Z e. U )
10 eqid 
 |-  ( invr ` R ) = ( invr ` R )
11 2 10 unitinvcl 
 |-  ( ( R e. Ring /\ Z e. U ) -> ( ( invr ` R ) ` Z ) e. U )
12 9 11 syldan 
 |-  ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( invr ` R ) ` Z ) e. U )
13 8 12 sselid 
 |-  ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( invr ` R ) ` Z ) e. B )
14 eqid 
 |-  ( .r ` R ) = ( .r ` R )
15 1 3 14 ringdir 
 |-  ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ ( ( invr ` R ) ` Z ) e. B ) ) -> ( ( X .+ Y ) ( .r ` R ) ( ( invr ` R ) ` Z ) ) = ( ( X ( .r ` R ) ( ( invr ` R ) ` Z ) ) .+ ( Y ( .r ` R ) ( ( invr ` R ) ` Z ) ) ) )
16 5 6 7 13 15 syl13anc 
 |-  ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( X .+ Y ) ( .r ` R ) ( ( invr ` R ) ` Z ) ) = ( ( X ( .r ` R ) ( ( invr ` R ) ` Z ) ) .+ ( Y ( .r ` R ) ( ( invr ` R ) ` Z ) ) ) )
17 ringgrp 
 |-  ( R e. Ring -> R e. Grp )
18 17 adantr 
 |-  ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> R e. Grp )
19 1 3 grpcl 
 |-  ( ( R e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )
20 18 6 7 19 syl3anc 
 |-  ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( X .+ Y ) e. B )
21 1 14 2 10 4 dvrval 
 |-  ( ( ( X .+ Y ) e. B /\ Z e. U ) -> ( ( X .+ Y ) ./ Z ) = ( ( X .+ Y ) ( .r ` R ) ( ( invr ` R ) ` Z ) ) )
22 20 9 21 syl2anc 
 |-  ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( X .+ Y ) ./ Z ) = ( ( X .+ Y ) ( .r ` R ) ( ( invr ` R ) ` Z ) ) )
23 1 14 2 10 4 dvrval 
 |-  ( ( X e. B /\ Z e. U ) -> ( X ./ Z ) = ( X ( .r ` R ) ( ( invr ` R ) ` Z ) ) )
24 6 9 23 syl2anc 
 |-  ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( X ./ Z ) = ( X ( .r ` R ) ( ( invr ` R ) ` Z ) ) )
25 1 14 2 10 4 dvrval 
 |-  ( ( Y e. B /\ Z e. U ) -> ( Y ./ Z ) = ( Y ( .r ` R ) ( ( invr ` R ) ` Z ) ) )
26 7 9 25 syl2anc 
 |-  ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( Y ./ Z ) = ( Y ( .r ` R ) ( ( invr ` R ) ` Z ) ) )
27 24 26 oveq12d 
 |-  ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( X ./ Z ) .+ ( Y ./ Z ) ) = ( ( X ( .r ` R ) ( ( invr ` R ) ` Z ) ) .+ ( Y ( .r ` R ) ( ( invr ` R ) ` Z ) ) ) )
28 16 22 27 3eqtr4d 
 |-  ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( X .+ Y ) ./ Z ) = ( ( X ./ Z ) .+ ( Y ./ Z ) ) )