Theorem ringsubdir 13234

Description: Ring multiplication distributes over subtraction. (subdir 8343 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Hypotheses

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

Assertion

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

Proof of Theorem ringsubdir

Step	Hyp	Ref	Expression
1		ringsubdi.r	. . . 4 |- ( ph -> R e. Ring )
2		ringsubdi.x	. . . 4 |- ( ph -> X e. B )
3		ringgrp 13184	. . . . . 6 |- ( R e. Ring -> R e. Grp )
4	1, 3	syl 14	. . . . 5 |- ( ph -> R e. Grp )
5		ringsubdi.y	. . . . 5 |- ( ph -> Y e. B )
6		ringsubdi.b	. . . . . 6 |- B = ( Base ` R )
7		eqid 2177	. . . . . 6 |- ( invg ` R ) = ( invg ` R )
8	6, 7	grpinvcl 12921	. . . . 5 |- ( ( R e. Grp /\ Y e. B ) -> ( ( invg ` R ) ` Y ) e. B )
9	4, 5, 8	syl2anc 411	. . . 4 |- ( ph -> ( ( invg ` R ) ` Y ) e. B )
10		ringsubdi.z	. . . 4 |- ( ph -> Z e. B )
11		eqid 2177	. . . . 5 |- ( +g ` R ) = ( +g ` R )
12		ringsubdi.t	. . . . 5 |- .x. = ( .r ` R )
13	6, 11, 12	ringdir 13202	. . . 4 |- ( ( R e. Ring /\ ( 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 ) ) )
14	1, 2, 9, 10, 13	syl13anc 1240	. . 3 |- ( ph -> ( ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) .x. Z ) = ( ( X .x. Z ) ( +g ` R ) ( ( ( invg ` R ) ` Y ) .x. Z ) ) )
15	6, 12, 7, 1, 5, 10	ringmneg1 13230	. . . 4 |- ( ph -> ( ( ( invg ` R ) ` Y ) .x. Z ) = ( ( invg ` R ) ` ( Y .x. Z ) ) )
16	15	oveq2d 5891	. . 3 |- ( ph -> ( ( X .x. Z ) ( +g ` R ) ( ( ( invg ` R ) ` Y ) .x. Z ) ) = ( ( X .x. Z ) ( +g ` R ) ( ( invg ` R ) ` ( Y .x. Z ) ) ) )
17	14, 16	eqtrd 2210	. 2 |- ( ph -> ( ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) .x. Z ) = ( ( X .x. Z ) ( +g ` R ) ( ( invg ` R ) ` ( Y .x. Z ) ) ) )
18		ringsubdi.m	. . . . 5 |- .- = ( -g ` R )
19	6, 11, 7, 18	grpsubval 12919	. . . 4 |- ( ( X e. B /\ Y e. B ) -> ( X .- Y ) = ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) )
20	2, 5, 19	syl2anc 411	. . 3 |- ( ph -> ( X .- Y ) = ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) )
21	20	oveq1d 5890	. 2 |- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) .x. Z ) )
22	6, 12	ringcl 13196	. . . 4 |- ( ( R e. Ring /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B )
23	1, 2, 10, 22	syl3anc 1238	. . 3 |- ( ph -> ( X .x. Z ) e. B )
24	6, 12	ringcl 13196	. . . 4 |- ( ( R e. Ring /\ Y e. B /\ Z e. B ) -> ( Y .x. Z ) e. B )
25	1, 5, 10, 24	syl3anc 1238	. . 3 |- ( ph -> ( Y .x. Z ) e. B )
26	6, 11, 7, 18	grpsubval 12919	. . 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 ) ) ) )
27	23, 25, 26	syl2anc 411	. 2 |- ( ph -> ( ( X .x. Z ) .- ( Y .x. Z ) ) = ( ( X .x. Z ) ( +g ` R ) ( ( invg ` R ) ` ( Y .x. Z ) ) ) )
28	17, 21, 27	3eqtr4d 2220	1 |- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X .x. Z ) .- ( Y .x. Z ) ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   ` cfv 5217  (class class class)co 5875   Basecbs 12462   +g cplusg 12536   .rcmulr 12537   Grpcgrp 12877   invgcminusg 12878   -gcsg 12879   Ringcrg 13179

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-pre-ltirr 7923  ax-pre-ltadd 7927

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-pnf 7994  df-mnf 7995  df-ltxr 7997  df-inn 8920  df-2 8978  df-3 8979  df-ndx 12465  df-slot 12466  df-base 12468  df-sets 12469  df-plusg 12549  df-mulr 12550  df-0g 12707  df-mgm 12775  df-sgrp 12808  df-mnd 12818  df-grp 12880  df-minusg 12881  df-sbg 12882  df-mgp 13131  df-ur 13143  df-ring 13181

This theorem is referenced by: (None)