Theorem ringsubdir 14218

Description: Ring multiplication distributes over subtraction. (subdir 8661 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 14162	. . . . . 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 2234	. . . . . 6 |- ( invg ` R ) = ( invg ` R )
8	6, 7	grpinvcl 13778	. . . . 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 2234	. . . . 5 |- ( +g ` R ) = ( +g ` R )
12		ringsubdi.t	. . . . 5 |- .x. = ( .r ` R )
13	6, 11, 12	ringdir 14180	. . . 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 1276	. . 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 14214	. . . 4 |- ( ph -> ( ( ( invg ` R ) ` Y ) .x. Z ) = ( ( invg ` R ) ` ( Y .x. Z ) ) )
16	15	oveq2d 6068	. . 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 2267	. 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 13776	. . . 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 6067	. 2 |- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) .x. Z ) )
22	6, 12	ringcl 14174	. . . 4 |- ( ( R e. Ring /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B )
23	1, 2, 10, 22	syl3anc 1274	. . 3 |- ( ph -> ( X .x. Z ) e. B )
24	6, 12	ringcl 14174	. . . 4 |- ( ( R e. Ring /\ Y e. B /\ Z e. B ) -> ( Y .x. Z ) e. B )
25	1, 5, 10, 24	syl3anc 1274	. . 3 |- ( ph -> ( Y .x. Z ) e. B )
26	6, 11, 7, 18	grpsubval 13776	. . 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 2277	1 |- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X .x. Z ) .- ( Y .x. Z ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   ` cfv 5354  (class class class)co 6052   Basecbs 13229   +g cplusg 13307   .rcmulr 13308   Grpcgrp 13730   invgcminusg 13731   -gcsg 13732   Ringcrg 14157

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-inn 9240  df-2 9298  df-3 9299  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-plusg 13320  df-mulr 13321  df-0g 13488  df-mgm 13586  df-sgrp 13632  df-mnd 13647  df-grp 13733  df-minusg 13734  df-sbg 13735  df-mgp 14082  df-ur 14121  df-ring 14159

This theorem is referenced by: (None)