Theorem ringsubdi 14299

Description: Ring multiplication distributes over subtraction. (subdi 8675 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
ringsubdi	|- ( ph -> ( X .x. ( Y .- Z ) ) = ( ( X .x. Y ) .- ( X .x. Z ) ) )

Proof of Theorem ringsubdi

Step	Hyp	Ref	Expression
1		ringsubdi.r	. . . 4 |- ( ph -> R e. Ring )
2		ringsubdi.x	. . . 4 |- ( ph -> X e. B )
3		ringsubdi.y	. . . 4 |- ( ph -> Y e. B )
4		ringgrp 14244	. . . . . 6 |- ( R e. Ring -> R e. Grp )
5	1, 4	syl 14	. . . . 5 |- ( ph -> R e. Grp )
6		ringsubdi.z	. . . . 5 |- ( ph -> Z e. B )
7		ringsubdi.b	. . . . . 6 |- B = ( Base ` R )
8		eqid 2234	. . . . . 6 |- ( invg ` R ) = ( invg ` R )
9	7, 8	grpinvcl 13803	. . . . 5 |- ( ( R e. Grp /\ Z e. B ) -> ( ( invg ` R ) ` Z ) e. B )
10	5, 6, 9	syl2anc 411	. . . 4 |- ( ph -> ( ( invg ` R ) ` Z ) e. B )
11		eqid 2234	. . . . 5 |- ( +g ` R ) = ( +g ` R )
12		ringsubdi.t	. . . . 5 |- .x. = ( .r ` R )
13	7, 11, 12	ringdi 14261	. . . 4 |- ( ( R e. Ring /\ ( X e. B /\ Y e. B /\ ( ( invg ` R ) ` Z ) e. B ) ) -> ( X .x. ( Y ( +g ` R ) ( ( invg ` R ) ` Z ) ) ) = ( ( X .x. Y ) ( +g ` R ) ( X .x. ( ( invg ` R ) ` Z ) ) ) )
14	1, 2, 3, 10, 13	syl13anc 1276	. . 3 |- ( ph -> ( X .x. ( Y ( +g ` R ) ( ( invg ` R ) ` Z ) ) ) = ( ( X .x. Y ) ( +g ` R ) ( X .x. ( ( invg ` R ) ` Z ) ) ) )
15	7, 12, 8, 1, 2, 6	ringmneg2 14297	. . . 4 |- ( ph -> ( X .x. ( ( invg ` R ) ` Z ) ) = ( ( invg ` R ) ` ( X .x. Z ) ) )
16	15	oveq2d 6074	. . 3 |- ( ph -> ( ( X .x. Y ) ( +g ` R ) ( X .x. ( ( invg ` R ) ` Z ) ) ) = ( ( X .x. Y ) ( +g ` R ) ( ( invg ` R ) ` ( X .x. Z ) ) ) )
17	14, 16	eqtrd 2267	. 2 |- ( ph -> ( X .x. ( Y ( +g ` R ) ( ( invg ` R ) ` Z ) ) ) = ( ( X .x. Y ) ( +g ` R ) ( ( invg ` R ) ` ( X .x. Z ) ) ) )
18		ringsubdi.m	. . . . 5 |- .- = ( -g ` R )
19	7, 11, 8, 18	grpsubval 13801	. . . 4 |- ( ( Y e. B /\ Z e. B ) -> ( Y .- Z ) = ( Y ( +g ` R ) ( ( invg ` R ) ` Z ) ) )
20	3, 6, 19	syl2anc 411	. . 3 |- ( ph -> ( Y .- Z ) = ( Y ( +g ` R ) ( ( invg ` R ) ` Z ) ) )
21	20	oveq2d 6074	. 2 |- ( ph -> ( X .x. ( Y .- Z ) ) = ( X .x. ( Y ( +g ` R ) ( ( invg ` R ) ` Z ) ) ) )
22	7, 12	ringcl 14256	. . . 4 |- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
23	1, 2, 3, 22	syl3anc 1274	. . 3 |- ( ph -> ( X .x. Y ) e. B )
24	7, 12	ringcl 14256	. . . 4 |- ( ( R e. Ring /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B )
25	1, 2, 6, 24	syl3anc 1274	. . 3 |- ( ph -> ( X .x. Z ) e. B )
26	7, 11, 8, 18	grpsubval 13801	. . 3 |- ( ( ( X .x. Y ) e. B /\ ( X .x. Z ) e. B ) -> ( ( X .x. Y ) .- ( X .x. Z ) ) = ( ( X .x. Y ) ( +g ` R ) ( ( invg ` R ) ` ( X .x. Z ) ) ) )
27	23, 25, 26	syl2anc 411	. 2 |- ( ph -> ( ( X .x. Y ) .- ( X .x. Z ) ) = ( ( X .x. Y ) ( +g ` R ) ( ( invg ` R ) ` ( X .x. Z ) ) ) )
28	17, 21, 27	3eqtr4d 2277	1 |- ( ph -> ( X .x. ( Y .- Z ) ) = ( ( X .x. Y ) .- ( X .x. Z ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   .rcmulr 13375   Grpcgrp 13755   invgcminusg 13756   -gcsg 13757   Ringcrg 14239

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 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

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 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-sbg 13760  df-mgp 14160  df-ur 14203  df-ring 14241

This theorem is referenced by: (None)