Theorem dvrdir 13775

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 13679	. . . . 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 2196	. . . . . 6 |- ( invr ` R ) = ( invr ` R )
12	6, 11	unitinvcl 13755	. . . . 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 13740	. . 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 2196	. . . 4 |- ( .r ` R ) = ( .r ` R )
17	4, 15, 16	ringdir 13651	. . 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 2197	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( .r ` R ) = ( .r ` R ) )
20		eqidd 2197	. . 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 13633	. . . . 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 13216	. . 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 13766	. 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 13766	. . 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 13766	. . 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 5943	. 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 2239	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 2167   ` cfv 5259  (class class class)co 5925   Basecbs 12703   +g cplusg 12780   .rcmulr 12781   Grpcgrp 13202  SRingcsrg 13595   Ringcrg 13628  Unitcui 13719   invrcinvr 13752  /_rcdvr 13763

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-lttrn 8010  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-tpos 6312  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-mulr 12794  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206  df-cmn 13492  df-abl 13493  df-mgp 13553  df-ur 13592  df-srg 13596  df-ring 13630  df-oppr 13700  df-dvdsr 13721  df-unit 13722  df-invr 13753  df-dvr 13764

This theorem is referenced by:  lringuplu  13828