Theorem dvrdir 14450

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 1034	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> X e. B )
3		simpr2 1035	. . 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 14352	. . . . 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 1036	. . . . 5 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> Z e. U )
11		eqid 2238	. . . . . 6 |- ( invr ` R ) = ( invr ` R )
12	6, 11	unitinvcl 14430	. . . . 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 14415	. . 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 2238	. . . 4 |- ( .r ` R ) = ( .r ` R )
17	4, 15, 16	ringdir 14324	. . 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 1280	. 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 2239	. . 3 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( .r ` R ) = ( .r ` R ) )
20		eqidd 2239	. . 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 14305	. . . . 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 13819	. . 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 14441	. 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 14441	. . 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 14441	. . 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 6103	. 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 2281	1 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ Z e. U ) ) -> ( ( X .+ Y ) ./ Z ) = ( ( X ./ Z ) .+ ( Y ./ Z ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   .rcmulr 13432   Grpcgrp 13805  SRingcsrg 14267   Ringcrg 14300  Unitcui 14393   invrcinvr 14427  /_rcdvr 14438

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-tpos 6516  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-mulr 13445  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-cmn 14089  df-abl 14090  df-mgp 14218  df-ur 14263  df-srg 14268  df-ring 14302  df-oppr 14373  df-dvdsr 14395  df-unit 14396  df-invr 14428  df-dvr 14439

This theorem is used by:  lringuplu  14503