Theorem rngmneg1 14353

Description: Negation of a product in a non-unital ring (mulneg1 8729 analog). In contrast to ringmneg1 14465, 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 14344	. . . . . . 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 13920	. . . . 5 |- ( ph -> ( X ( +g ` R ) ( N ` X ) ) = ( 0g ` R ) )
10	9	oveq1d 6103	. . . 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 14351	. . . . 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 14350	. . . . . 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 13913	. . . . . 6 |- ( ph -> ( N ` X ) e. B )
19	1, 12	rngcl 14350	. . . . . 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 13916	. . . . 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 14347	. . . . . . 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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5380  (class class class)co 6088   Basecbs 13410   +g cplusg 13490   .rcmulr 13491   0gc0g 13669   Grpcgrp 13864   invgcminusg 13865  Rngcrng 14338

This proof depends on 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 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-pre-ltirr 8297  ax-pre-ltadd 8301

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8368  df-mnf 8369  df-ltxr 8371  df-inn 9314  df-2 9372  df-3 9373  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-plusg 13503  df-mulr 13504  df-0g 13671  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-grp 13867  df-minusg 13868  df-abl 14180  df-mgp 14325  df-rng 14339

This theorem is used by:  rngm2neg  14355  rngsubdir  14358