Theorem rngsubdir 13910

Description: Ring multiplication distributes over subtraction. (subdir 8528 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdir 14015. (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 2229	. . . . 5 |- ( invg ` R ) = ( invg ` R )
5		rnggrp 13896	. . . . . 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 13577	. . . 4 |- ( ph -> ( ( invg ` R ) ` Y ) e. B )
9		rngsubdi.z	. . . 4 |- ( ph -> Z e. B )
10		eqid 2229	. . . . 5 |- ( +g ` R ) = ( +g ` R )
11		rngsubdi.t	. . . . 5 |- .x. = ( .r ` R )
12	3, 10, 11	rngdir 13899	. . . 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 1273	. . 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 13905	. . . 4 |- ( ph -> ( ( ( invg ` R ) ` Y ) .x. Z ) = ( ( invg ` R ) ` ( Y .x. Z ) ) )
15	14	oveq2d 6016	. . 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 2262	. 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 13574	. . . 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 6015	. 2 |- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) .x. Z ) )
21	3, 11	rngcl 13902	. . . 4 |- ( ( R e. Rng /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B )
22	1, 2, 9, 21	syl3anc 1271	. . 3 |- ( ph -> ( X .x. Z ) e. B )
23	3, 11	rngcl 13902	. . . 4 |- ( ( R e. Rng /\ Y e. B /\ Z e. B ) -> ( Y .x. Z ) e. B )
24	1, 7, 9, 23	syl3anc 1271	. . 3 |- ( ph -> ( Y .x. Z ) e. B )
25	3, 10, 4, 17	grpsubval 13574	. . 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 2272	1 |- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X .x. Z ) .- ( Y .x. Z ) ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   ` cfv 5317  (class class class)co 6000   Basecbs 13027   +g cplusg 13105   .rcmulr 13106   Grpcgrp 13528   invgcminusg 13529   -gcsg 13530  Rngcrng 13890

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-inn 9107  df-2 9165  df-3 9166  df-ndx 13030  df-slot 13031  df-base 13033  df-sets 13034  df-plusg 13118  df-mulr 13119  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-grp 13531  df-minusg 13532  df-sbg 13533  df-abl 13819  df-mgp 13879  df-rng 13891

This theorem is referenced by:  2idlcpblrng  14481