Theorem rngmneg1 13298

Description: Negation of a product in a non-unital ring (mulneg1 8381 analog). In contrast to ringmneg1 13402, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.)

Hypotheses

Ref	Expression
rngneglmul.b	|- B = ( Base ` R )
rngneglmul.t	|- .x. = ( .r ` R )
rngneglmul.n	|- N = ( invg ` R )
rngneglmul.r	|- ( ph -> R e. Rng )
rngneglmul.x	|- ( ph -> X e. B )
rngneglmul.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
rngmneg1	|- ( ph -> ( ( N ` X ) .x. Y ) = ( N ` ( X .x. Y ) ) )

Proof of Theorem rngmneg1

Step	Hyp	Ref	Expression
1		rngneglmul.b	. . . . . 6 |- B = ( Base ` R )
2		eqid 2189	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
3		eqid 2189	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
4		rngneglmul.n	. . . . . 6 |- N = ( invg ` R )
5		rngneglmul.r	. . . . . . 7 |- ( ph -> R e. Rng )
6		rnggrp 13289	. . . . . . 7 |- ( R e. Rng -> R e. Grp )
7	5, 6	syl 14	. . . . . 6 |- ( ph -> R e. Grp )
8		rngneglmul.x	. . . . . 6 |- ( ph -> X e. B )
9	1, 2, 3, 4, 7, 8	grprinvd 12997	. . . . 5 |- ( ph -> ( X ( +g ` R ) ( N ` X ) ) = ( 0g ` R ) )
10	9	oveq1d 5910	. . . 4 |- ( ph -> ( ( X ( +g ` R ) ( N ` X ) ) .x. Y ) = ( ( 0g ` R ) .x. Y ) )
11		rngneglmul.y	. . . . 5 |- ( ph -> Y e. B )
12		rngneglmul.t	. . . . . 6 |- .x. = ( .r ` R )
13	1, 12, 3	rnglz 13296	. . . . 5 |- ( ( R e. Rng /\ Y e. B ) -> ( ( 0g ` R ) .x. Y ) = ( 0g ` R ) )
14	5, 11, 13	syl2anc 411	. . . 4 |- ( ph -> ( ( 0g ` R ) .x. Y ) = ( 0g ` R ) )
15	10, 14	eqtrd 2222	. . 3 |- ( ph -> ( ( X ( +g ` R ) ( N ` X ) ) .x. Y ) = ( 0g ` R ) )
16	1, 12	rngcl 13295	. . . . . 6 |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
17	5, 8, 11, 16	syl3anc 1249	. . . . 5 |- ( ph -> ( X .x. Y ) e. B )
18	1, 4, 7, 8	grpinvcld 12990	. . . . . 6 |- ( ph -> ( N ` X ) e. B )
19	1, 12	rngcl 13295	. . . . . 6 |- ( ( R e. Rng /\ ( N ` X ) e. B /\ Y e. B ) -> ( ( N ` X ) .x. Y ) e. B )
20	5, 18, 11, 19	syl3anc 1249	. . . . 5 |- ( ph -> ( ( N ` X ) .x. Y ) e. B )
21	1, 2, 3, 4	grpinvid1 12993	. . . . 5 |- ( ( R e. Grp /\ ( X .x. Y ) e. B /\ ( ( N ` X ) .x. Y ) e. B ) -> ( ( N ` ( X .x. Y ) ) = ( ( N ` X ) .x. Y ) <-> ( ( X .x. Y ) ( +g ` R ) ( ( N ` X ) .x. Y ) ) = ( 0g ` R ) ) )
22	7, 17, 20, 21	syl3anc 1249	. . . 4 |- ( ph -> ( ( N ` ( X .x. Y ) ) = ( ( N ` X ) .x. Y ) <-> ( ( X .x. Y ) ( +g ` R ) ( ( N ` X ) .x. Y ) ) = ( 0g ` R ) ) )
23	1, 2, 12	rngdir 13292	. . . . . . 7 |- ( ( R e. Rng /\ ( X e. B /\ ( N ` X ) e. B /\ Y e. B ) ) -> ( ( X ( +g ` R ) ( N ` X ) ) .x. Y ) = ( ( X .x. Y ) ( +g ` R ) ( ( N ` X ) .x. Y ) ) )
24	23	eqcomd 2195	. . . . . 6 |- ( ( R e. Rng /\ ( X e. B /\ ( N ` X ) e. B /\ Y e. B ) ) -> ( ( X .x. Y ) ( +g ` R ) ( ( N ` X ) .x. Y ) ) = ( ( X ( +g ` R ) ( N ` X ) ) .x. Y ) )
25	5, 8, 18, 11, 24	syl13anc 1251	. . . . 5 |- ( ph -> ( ( X .x. Y ) ( +g ` R ) ( ( N ` X ) .x. Y ) ) = ( ( X ( +g ` R ) ( N ` X ) ) .x. Y ) )
26	25	eqeq1d 2198	. . . 4 |- ( ph -> ( ( ( X .x. Y ) ( +g ` R ) ( ( N ` X ) .x. Y ) ) = ( 0g ` R ) <-> ( ( X ( +g ` R ) ( N ` X ) ) .x. Y ) = ( 0g ` R ) ) )
27	22, 26	bitrd 188	. . 3 |- ( ph -> ( ( N ` ( X .x. Y ) ) = ( ( N ` X ) .x. Y ) <-> ( ( X ( +g ` R ) ( N ` X ) ) .x. Y ) = ( 0g ` R ) ) )
28	15, 27	mpbird 167	. 2 |- ( ph -> ( N ` ( X .x. Y ) ) = ( ( N ` X ) .x. Y ) )
29	28	eqcomd 2195	1 |- ( ph -> ( ( N ` X ) .x. Y ) = ( N ` ( X .x. Y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2160   ` cfv 5235  (class class class)co 5895   Basecbs 12511   +g cplusg 12586   .rcmulr 12587   0gc0g 12758   Grpcgrp 12942   invgcminusg 12943  Rngcrng 13283

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-addcom 7940  ax-addass 7942  ax-i2m1 7945  ax-0lt1 7946  ax-0id 7948  ax-rnegex 7949  ax-pre-ltirr 7952  ax-pre-ltadd 7956

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-pnf 8023  df-mnf 8024  df-ltxr 8026  df-inn 8949  df-2 9007  df-3 9008  df-ndx 12514  df-slot 12515  df-base 12517  df-sets 12518  df-plusg 12599  df-mulr 12600  df-0g 12760  df-mgm 12829  df-sgrp 12862  df-mnd 12875  df-grp 12945  df-minusg 12946  df-abl 13223  df-mgp 13272  df-rng 13284

This theorem is referenced by:  rngm2neg  13300  rngsubdir  13303