Theorem dvrdir 13639

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

Hypotheses

Ref	Expression
dvrdir.b	|- B = ( Base ` R )
dvrdir.u	|- U = (Unit ` R )
dvrdir.p	|- .+ = ( +g ` R )
dvrdir.t	|- ./ = (/r ` R )

Assertion

Ref	Expression
dvrdir	|- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( X .+ Y ) ./ Z ) = ( ( X ./ Z ) .+ ( Y ./ Z ) ) )

Proof of Theorem dvrdir

Step	Hyp	Ref	Expression
1		simpl 109	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> R e. Ring )
2		simpr1 1005	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> X e. B )
3		simpr2 1006	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> Y e. B )
4		dvrdir.b	. . . . 5 |- B = ( Base ` R )
5	4	a1i 9	. . . 4 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> B = ( Base ` R ) )
6		dvrdir.u	. . . . 5 |- U = (Unit ` R )
7	6	a1i 9	. . . 4 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> U = (Unit ` R ) )
8		ringsrg 13543	. . . . 5 |- ( R e. Ring -> R e. SRing )
9	8	adantr 276	. . . 4 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> R e. SRing )
10		simpr3 1007	. . . . 5 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> Z e. U )
11		eqid 2193	. . . . . 6 |- ( invr ` R ) = ( invr ` R )
12	6, 11	unitinvcl 13619	. . . . 5 |- ( ( R e. Ring /\ Z e. U ) -> ( ( invr ` R ) ` Z ) e. U )
13	10, 12	syldan 282	. . . 4 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( invr ` R ) ` Z ) e. U )
14	5, 7, 9, 13	unitcld 13604	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( invr ` R ) ` Z ) e. B )
15		dvrdir.p	. . . 4 |- .+ = ( +g ` R )
16		eqid 2193	. . . 4 |- ( .r ` R ) = ( .r ` R )
17	4, 15, 16	ringdir 13515	. . 3 |- ( ( 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 ) ) ) )
18	1, 2, 3, 14, 17	syl13anc 1251	. 2 |- ( ( 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 ) ) ) )
19		eqidd 2194	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( .r ` R ) = ( .r ` R ) )
20		eqidd 2194	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( invr ` R ) = ( invr ` R ) )
21		dvrdir.t	. . . 4 |- ./ = (/r ` R )
22	21	a1i 9	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ./ = (/r ` R ) )
23		ringgrp 13497	. . . . 5 |- ( R e. Ring -> R e. Grp )
24	23	adantr 276	. . . 4 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> R e. Grp )
25	4, 15, 24, 2, 3	grpcld 13086	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( X .+ Y ) e. B )
26	5, 19, 7, 20, 22, 1, 25, 10	dvrvald 13630	. 2 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( X .+ Y ) ./ Z ) = ( ( X .+ Y ) ( .r ` R ) ( ( invr ` R ) ` Z ) ) )
27	5, 19, 7, 20, 22, 1, 2, 10	dvrvald 13630	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( X ./ Z ) = ( X ( .r ` R ) ( ( invr ` R ) ` Z ) ) )
28	5, 19, 7, 20, 22, 1, 3, 10	dvrvald 13630	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( Y ./ Z ) = ( Y ( .r ` R ) ( ( invr ` R ) ` Z ) ) )
29	27, 28	oveq12d 5936	. 2 |- ( ( 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 ) ) ) )
30	18, 26, 29	3eqtr4d 2236	1 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( X .+ Y ) ./ Z ) = ( ( X ./ Z ) .+ ( Y ./ Z ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   .rcmulr 12696   Grpcgrp 13072  SRingcsrg 13459   Ringcrg 13492  Unitcui 13583   invrcinvr 13616  /_rcdvr 13627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-tpos 6298  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-cmn 13356  df-abl 13357  df-mgp 13417  df-ur 13456  df-srg 13460  df-ring 13494  df-oppr 13564  df-dvdsr 13585  df-unit 13586  df-invr 13617  df-dvr 13628

This theorem is referenced by:  lringuplu  13692