Theorem ringsubdir 14072

Description: Ring multiplication distributes over subtraction. (subdir 8565 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 14016	. . . . . 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 2231	. . . . . 6 |- ( invg ` R ) = ( invg ` R )
8	6, 7	grpinvcl 13632	. . . . 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 2231	. . . . 5 |- ( +g ` R ) = ( +g ` R )
12		ringsubdi.t	. . . . 5 |- .x. = ( .r ` R )
13	6, 11, 12	ringdir 14034	. . . 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 1275	. . 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 14068	. . . 4 |- ( ph -> ( ( ( invg ` R ) ` Y ) .x. Z ) = ( ( invg ` R ) ` ( Y .x. Z ) ) )
16	15	oveq2d 6034	. . 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 2264	. 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 13630	. . . 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 6033	. 2 |- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) .x. Z ) )
22	6, 12	ringcl 14028	. . . 4 |- ( ( R e. Ring /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B )
23	1, 2, 10, 22	syl3anc 1273	. . 3 |- ( ph -> ( X .x. Z ) e. B )
24	6, 12	ringcl 14028	. . . 4 |- ( ( R e. Ring /\ Y e. B /\ Z e. B ) -> ( Y .x. Z ) e. B )
25	1, 5, 10, 24	syl3anc 1273	. . 3 |- ( ph -> ( Y .x. Z ) e. B )
26	6, 11, 7, 18	grpsubval 13630	. . 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 2274	1 |- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X .x. Z ) .- ( Y .x. Z ) ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6018   Basecbs 13083   +g cplusg 13161   .rcmulr 13162   Grpcgrp 13584   invgcminusg 13585   -gcsg 13586   Ringcrg 14011

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13086  df-slot 13087  df-base 13089  df-sets 13090  df-plusg 13174  df-mulr 13175  df-0g 13342  df-mgm 13440  df-sgrp 13486  df-mnd 13501  df-grp 13587  df-minusg 13588  df-sbg 13589  df-mgp 13936  df-ur 13975  df-ring 14013

This theorem is referenced by: (None)