Theorem rngsubdi 13963

Description: Ring multiplication distributes over subtraction. (subdi 8563 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdi 14068. (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
rngsubdi	|- ( ph -> ( X .x. ( Y .- Z ) ) = ( ( X .x. Y ) .- ( X .x. Z ) ) )

Proof of Theorem rngsubdi

Step	Hyp	Ref	Expression
1		rngsubdi.r	. . . 4 |- ( ph -> R e. Rng )
2		rngsubdi.x	. . . 4 |- ( ph -> X e. B )
3		rngsubdi.y	. . . 4 |- ( ph -> Y e. B )
4		rngsubdi.b	. . . . 5 |- B = ( Base ` R )
5		eqid 2231	. . . . 5 |- ( invg ` R ) = ( invg ` R )
6		rnggrp 13950	. . . . . 6 |- ( R e. Rng -> R e. Grp )
7	1, 6	syl 14	. . . . 5 |- ( ph -> R e. Grp )
8		rngsubdi.z	. . . . 5 |- ( ph -> Z e. B )
9	4, 5, 7, 8	grpinvcld 13631	. . . 4 |- ( ph -> ( ( invg ` R ) ` Z ) e. B )
10		eqid 2231	. . . . 5 |- ( +g ` R ) = ( +g ` R )
11		rngsubdi.t	. . . . 5 |- .x. = ( .r ` R )
12	4, 10, 11	rngdi 13952	. . . 4 |- ( ( R e. Rng /\ ( 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 ) ) ) )
13	1, 2, 3, 9, 12	syl13anc 1275	. . 3 |- ( ph -> ( X .x. ( Y ( +g ` R ) ( ( invg ` R ) ` Z ) ) ) = ( ( X .x. Y ) ( +g ` R ) ( X .x. ( ( invg ` R ) ` Z ) ) ) )
14	4, 11, 5, 1, 2, 8	rngmneg2 13960	. . . 4 |- ( ph -> ( X .x. ( ( invg ` R ) ` Z ) ) = ( ( invg ` R ) ` ( X .x. Z ) ) )
15	14	oveq2d 6033	. . 3 |- ( ph -> ( ( X .x. Y ) ( +g ` R ) ( X .x. ( ( invg ` R ) ` Z ) ) ) = ( ( X .x. Y ) ( +g ` R ) ( ( invg ` R ) ` ( X .x. Z ) ) ) )
16	13, 15	eqtrd 2264	. 2 |- ( ph -> ( X .x. ( Y ( +g ` R ) ( ( invg ` R ) ` Z ) ) ) = ( ( X .x. Y ) ( +g ` R ) ( ( invg ` R ) ` ( X .x. Z ) ) ) )
17		rngsubdi.m	. . . . 5 |- .- = ( -g ` R )
18	4, 10, 5, 17	grpsubval 13628	. . . 4 |- ( ( Y e. B /\ Z e. B ) -> ( Y .- Z ) = ( Y ( +g ` R ) ( ( invg ` R ) ` Z ) ) )
19	3, 8, 18	syl2anc 411	. . 3 |- ( ph -> ( Y .- Z ) = ( Y ( +g ` R ) ( ( invg ` R ) ` Z ) ) )
20	19	oveq2d 6033	. 2 |- ( ph -> ( X .x. ( Y .- Z ) ) = ( X .x. ( Y ( +g ` R ) ( ( invg ` R ) ` Z ) ) ) )
21	4, 11	rngcl 13956	. . . 4 |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
22	1, 2, 3, 21	syl3anc 1273	. . 3 |- ( ph -> ( X .x. Y ) e. B )
23	4, 11	rngcl 13956	. . . 4 |- ( ( R e. Rng /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B )
24	1, 2, 8, 23	syl3anc 1273	. . 3 |- ( ph -> ( X .x. Z ) e. B )
25	4, 10, 5, 17	grpsubval 13628	. . 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 ) ) ) )
26	22, 24, 25	syl2anc 411	. 2 |- ( ph -> ( ( X .x. Y ) .- ( X .x. Z ) ) = ( ( X .x. Y ) ( +g ` R ) ( ( invg ` R ) ` ( X .x. Z ) ) ) )
27	16, 20, 26	3eqtr4d 2274	1 |- ( ph -> ( X .x. ( Y .- Z ) ) = ( ( X .x. Y ) .- ( X .x. Z ) ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   .rcmulr 13160   Grpcgrp 13582   invgcminusg 13583   -gcsg 13584  Rngcrng 13944

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 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

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 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-sbg 13587  df-abl 13873  df-mgp 13933  df-rng 13945

This theorem is referenced by:  2idlcpblrng  14536