Theorem rngsubdir 13448

Description: Ring multiplication distributes over subtraction. (subdir 8405 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdir 13553. (Revised by AV, 23-Feb-2025.)

Hypotheses

Ref	Expression
rngsubdi.b	|- B = ( Base ` R )
rngsubdi.t	|- .x. = ( .r ` R )
rngsubdi.m	|- .- = ( -g ` R )
rngsubdi.r	|- ( ph -> R e. Rng )
rngsubdi.x	|- ( ph -> X e. B )
rngsubdi.y	|- ( ph -> Y e. B )
rngsubdi.z	|- ( ph -> Z e. B )

Assertion

Ref	Expression
rngsubdir	|- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X .x. Z ) .- ( Y .x. Z ) ) )

Proof of Theorem rngsubdir

Step	Hyp	Ref	Expression
1		rngsubdi.r	. . . 4 |- ( ph -> R e. Rng )
2		rngsubdi.x	. . . 4 |- ( ph -> X e. B )
3		rngsubdi.b	. . . . 5 |- B = ( Base ` R )
4		eqid 2193	. . . . 5 |- ( invg ` R ) = ( invg ` R )
5		rnggrp 13434	. . . . . 6 |- ( R e. Rng -> R e. Grp )
6	1, 5	syl 14	. . . . 5 |- ( ph -> R e. Grp )
7		rngsubdi.y	. . . . 5 |- ( ph -> Y e. B )
8	3, 4, 6, 7	grpinvcld 13121	. . . 4 |- ( ph -> ( ( invg ` R ) ` Y ) e. B )
9		rngsubdi.z	. . . 4 |- ( ph -> Z e. B )
10		eqid 2193	. . . . 5 |- ( +g ` R ) = ( +g ` R )
11		rngsubdi.t	. . . . 5 |- .x. = ( .r ` R )
12	3, 10, 11	rngdir 13437	. . . 4 |- ( ( R e. Rng /\ ( X e. B /\ ( ( invg ` R ) ` Y ) e. B /\ Z e. B ) ) -> ( ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) .x. Z ) = ( ( X .x. Z ) ( +g ` R ) ( ( ( invg ` R ) ` Y ) .x. Z ) ) )
13	1, 2, 8, 9, 12	syl13anc 1251	. . 3 |- ( ph -> ( ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) .x. Z ) = ( ( X .x. Z ) ( +g ` R ) ( ( ( invg ` R ) ` Y ) .x. Z ) ) )
14	3, 11, 4, 1, 7, 9	rngmneg1 13443	. . . 4 |- ( ph -> ( ( ( invg ` R ) ` Y ) .x. Z ) = ( ( invg ` R ) ` ( Y .x. Z ) ) )
15	14	oveq2d 5934	. . 3 |- ( ph -> ( ( X .x. Z ) ( +g ` R ) ( ( ( invg ` R ) ` Y ) .x. Z ) ) = ( ( X .x. Z ) ( +g ` R ) ( ( invg ` R ) ` ( Y .x. Z ) ) ) )
16	13, 15	eqtrd 2226	. 2 |- ( ph -> ( ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) .x. Z ) = ( ( X .x. Z ) ( +g ` R ) ( ( invg ` R ) ` ( Y .x. Z ) ) ) )
17		rngsubdi.m	. . . . 5 |- .- = ( -g ` R )
18	3, 10, 4, 17	grpsubval 13118	. . . 4 |- ( ( X e. B /\ Y e. B ) -> ( X .- Y ) = ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) )
19	2, 7, 18	syl2anc 411	. . 3 |- ( ph -> ( X .- Y ) = ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) )
20	19	oveq1d 5933	. 2 |- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) .x. Z ) )
21	3, 11	rngcl 13440	. . . 4 |- ( ( R e. Rng /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B )
22	1, 2, 9, 21	syl3anc 1249	. . 3 |- ( ph -> ( X .x. Z ) e. B )
23	3, 11	rngcl 13440	. . . 4 |- ( ( R e. Rng /\ Y e. B /\ Z e. B ) -> ( Y .x. Z ) e. B )
24	1, 7, 9, 23	syl3anc 1249	. . 3 |- ( ph -> ( Y .x. Z ) e. B )
25	3, 10, 4, 17	grpsubval 13118	. . 3 |- ( ( ( X .x. Z ) e. B /\ ( Y .x. Z ) e. B ) -> ( ( X .x. Z ) .- ( Y .x. Z ) ) = ( ( X .x. Z ) ( +g ` R ) ( ( invg ` R ) ` ( Y .x. Z ) ) ) )
26	22, 24, 25	syl2anc 411	. 2 |- ( ph -> ( ( X .x. Z ) .- ( Y .x. Z ) ) = ( ( X .x. Z ) ( +g ` R ) ( ( invg ` R ) ` ( Y .x. Z ) ) ) )
27	16, 20, 26	3eqtr4d 2236	1 |- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X .x. Z ) .- ( Y .x. Z ) ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   .rcmulr 12696   Grpcgrp 13072   invgcminusg 13073   -gcsg 13074  Rngcrng 13428

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-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-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-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-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-sbg 13077  df-abl 13357  df-mgp 13417  df-rng 13429

This theorem is referenced by:  2idlcpblrng  14019