Theorem rngsubdir 14046

Description: Ring multiplication distributes over subtraction. (subdir 8624 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdir 14151. (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 2231	. . . . 5 |- ( invg ` R ) = ( invg ` R )
5		rnggrp 14032	. . . . . 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 13712	. . . 4 |- ( ph -> ( ( invg ` R ) ` Y ) e. B )
9		rngsubdi.z	. . . 4 |- ( ph -> Z e. B )
10		eqid 2231	. . . . 5 |- ( +g ` R ) = ( +g ` R )
11		rngsubdi.t	. . . . 5 |- .x. = ( .r ` R )
12	3, 10, 11	rngdir 14035	. . . 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 1276	. . 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 14041	. . . 4 |- ( ph -> ( ( ( invg ` R ) ` Y ) .x. Z ) = ( ( invg ` R ) ` ( Y .x. Z ) ) )
15	14	oveq2d 6044	. . 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 2264	. 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 13709	. . . 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 6043	. 2 |- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) .x. Z ) )
21	3, 11	rngcl 14038	. . . 4 |- ( ( R e. Rng /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B )
22	1, 2, 9, 21	syl3anc 1274	. . 3 |- ( ph -> ( X .x. Z ) e. B )
23	3, 11	rngcl 14038	. . . 4 |- ( ( R e. Rng /\ Y e. B /\ Z e. B ) -> ( Y .x. Z ) e. B )
24	1, 7, 9, 23	syl3anc 1274	. . 3 |- ( ph -> ( Y .x. Z ) e. B )
25	3, 10, 4, 17	grpsubval 13709	. . 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 2274	1 |- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X .x. Z ) .- ( Y .x. Z ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   ` cfv 5333  (class class class)co 6028   Basecbs 13162   +g cplusg 13240   .rcmulr 13241   Grpcgrp 13663   invgcminusg 13664   -gcsg 13665  Rngcrng 14026

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-plusg 13253  df-mulr 13254  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-sbg 13668  df-abl 13954  df-mgp 14015  df-rng 14027

This theorem is referenced by:  2idlcpblrng  14619