Theorem rngmneg1 14224

Description: Negation of a product in a non-unital ring (mulneg1 8715 analog). In contrast to ringmneg1 14334, 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 2238	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
3		eqid 2238	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
4		rngneglmul.n	. . . . . 6 |- N = ( invg ` R )
5		rngneglmul.r	. . . . . . 7 |- ( ph -> R e. Rng )
6		rnggrp 14215	. . . . . . 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 13841	. . . . 5 |- ( ph -> ( X ( +g ` R ) ( N ` X ) ) = ( 0g ` R ) )
10	9	oveq1d 6093	. . . 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 14222	. . . . 5 |- ( ( R e. Rng /\ Y e. B ) -> ( ( 0g ` R ) .x. Y ) = ( 0g ` R ) )
14	5, 11, 13	syl2anc 415	. . . 4 |- ( ph -> ( ( 0g ` R ) .x. Y ) = ( 0g ` R ) )
15	10, 14	eqtrd 2271	. . 3 |- ( ph -> ( ( X ( +g ` R ) ( N ` X ) ) .x. Y ) = ( 0g ` R ) )
16	1, 12	rngcl 14221	. . . . . 6 |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
17	5, 8, 11, 16	syl3anc 1278	. . . . 5 |- ( ph -> ( X .x. Y ) e. B )
18	1, 4, 7, 8	grpinvcld 13834	. . . . . 6 |- ( ph -> ( N ` X ) e. B )
19	1, 12	rngcl 14221	. . . . . 6 |- ( ( R e. Rng /\ ( N ` X ) e. B /\ Y e. B ) -> ( ( N ` X ) .x. Y ) e. B )
20	5, 18, 11, 19	syl3anc 1278	. . . . 5 |- ( ph -> ( ( N ` X ) .x. Y ) e. B )
21	1, 2, 3, 4	grpinvid1 13837	. . . . 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 1278	. . . 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 14218	. . . . . . 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 2244	. . . . . 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 1280	. . . . 5 |- ( ph -> ( ( X .x. Y ) ( +g ` R ) ( ( N ` X ) .x. Y ) ) = ( ( X ( +g ` R ) ( N ` X ) ) .x. Y ) )
26	25	eqeq1d 2247	. . . 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 2244	1 |- ( ph -> ( ( N ` X ) .x. Y ) = ( N ` ( X .x. Y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5375  (class class class)co 6078   Basecbs 13333   +g cplusg 13411   .rcmulr 13412   0gc0g 13590   Grpcgrp 13785   invgcminusg 13786  Rngcrng 14209

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-addcom 8272  ax-addass 8274  ax-i2m1 8277  ax-0lt1 8278  ax-0id 8280  ax-rnegex 8281  ax-pre-ltirr 8284  ax-pre-ltadd 8288

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-pnf 8355  df-mnf 8356  df-ltxr 8358  df-inn 9287  df-2 9345  df-3 9346  df-ndx 13336  df-slot 13337  df-base 13339  df-sets 13340  df-plusg 13424  df-mulr 13425  df-0g 13592  df-mgm 13656  df-sgrp 13697  df-mnd 13710  df-grp 13788  df-minusg 13789  df-abl 14070  df-mgp 14198  df-rng 14210

This theorem is referenced by:  rngm2neg  14226  rngsubdir  14229