Theorem rngmneg2 14092

Description: Negation of a product in a non-unital ring (mulneg2 8669 analog). In contrast to ringmneg2 14198, 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
rngmneg2	|- ( ph -> ( X .x. ( N ` Y ) ) = ( N ` ( X .x. Y ) ) )

Proof of Theorem rngmneg2

Step	Hyp	Ref	Expression
1		rngneglmul.b	. . . . . 6 |- B = ( Base ` R )
2		eqid 2232	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
3		eqid 2232	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
4		rngneglmul.n	. . . . . 6 |- N = ( invg ` R )
5		rngneglmul.r	. . . . . . 7 |- ( ph -> R e. Rng )
6		rnggrp 14082	. . . . . . 7 |- ( R e. Rng -> R e. Grp )
7	5, 6	syl 14	. . . . . 6 |- ( ph -> R e. Grp )
8		rngneglmul.y	. . . . . 6 |- ( ph -> Y e. B )
9	1, 2, 3, 4, 7, 8	grplinvd 13768	. . . . 5 |- ( ph -> ( ( N ` Y ) ( +g ` R ) Y ) = ( 0g ` R ) )
10	9	oveq2d 6066	. . . 4 |- ( ph -> ( X .x. ( ( N ` Y ) ( +g ` R ) Y ) ) = ( X .x. ( 0g ` R ) ) )
11		rngneglmul.x	. . . . 5 |- ( ph -> X e. B )
12		rngneglmul.t	. . . . . 6 |- .x. = ( .r ` R )
13	1, 12, 3	rngrz 14090	. . . . 5 |- ( ( R e. Rng /\ X e. B ) -> ( X .x. ( 0g ` R ) ) = ( 0g ` R ) )
14	5, 11, 13	syl2anc 411	. . . 4 |- ( ph -> ( X .x. ( 0g ` R ) ) = ( 0g ` R ) )
15	10, 14	eqtrd 2265	. . 3 |- ( ph -> ( X .x. ( ( N ` Y ) ( +g ` R ) Y ) ) = ( 0g ` R ) )
16	1, 12	rngcl 14088	. . . . . 6 |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
17	5, 11, 8, 16	syl3anc 1274	. . . . 5 |- ( ph -> ( X .x. Y ) e. B )
18	1, 4, 7, 8	grpinvcld 13762	. . . . . 6 |- ( ph -> ( N ` Y ) e. B )
19	1, 12	rngcl 14088	. . . . . 6 |- ( ( R e. Rng /\ X e. B /\ ( N ` Y ) e. B ) -> ( X .x. ( N ` Y ) ) e. B )
20	5, 11, 18, 19	syl3anc 1274	. . . . 5 |- ( ph -> ( X .x. ( N ` Y ) ) e. B )
21	1, 2, 3, 4	grpinvid2 13766	. . . . 5 |- ( ( R e. Grp /\ ( X .x. Y ) e. B /\ ( X .x. ( N ` Y ) ) e. B ) -> ( ( N ` ( X .x. Y ) ) = ( X .x. ( N ` Y ) ) <-> ( ( X .x. ( N ` Y ) ) ( +g ` R ) ( X .x. Y ) ) = ( 0g ` R ) ) )
22	7, 17, 20, 21	syl3anc 1274	. . . 4 |- ( ph -> ( ( N ` ( X .x. Y ) ) = ( X .x. ( N ` Y ) ) <-> ( ( X .x. ( N ` Y ) ) ( +g ` R ) ( X .x. Y ) ) = ( 0g ` R ) ) )
23	1, 2, 12	rngdi 14084	. . . . . . 7 |- ( ( R e. Rng /\ ( X e. B /\ ( N ` Y ) e. B /\ Y e. B ) ) -> ( X .x. ( ( N ` Y ) ( +g ` R ) Y ) ) = ( ( X .x. ( N ` Y ) ) ( +g ` R ) ( X .x. Y ) ) )
24	23	eqcomd 2238	. . . . . 6 |- ( ( R e. Rng /\ ( X e. B /\ ( N ` Y ) e. B /\ Y e. B ) ) -> ( ( X .x. ( N ` Y ) ) ( +g ` R ) ( X .x. Y ) ) = ( X .x. ( ( N ` Y ) ( +g ` R ) Y ) ) )
25	5, 11, 18, 8, 24	syl13anc 1276	. . . . 5 |- ( ph -> ( ( X .x. ( N ` Y ) ) ( +g ` R ) ( X .x. Y ) ) = ( X .x. ( ( N ` Y ) ( +g ` R ) Y ) ) )
26	25	eqeq1d 2241	. . . 4 |- ( ph -> ( ( ( X .x. ( N ` Y ) ) ( +g ` R ) ( X .x. Y ) ) = ( 0g ` R ) <-> ( X .x. ( ( N ` Y ) ( +g ` R ) Y ) ) = ( 0g ` R ) ) )
27	22, 26	bitrd 188	. . 3 |- ( ph -> ( ( N ` ( X .x. Y ) ) = ( X .x. ( N ` Y ) ) <-> ( X .x. ( ( N ` Y ) ( +g ` R ) Y ) ) = ( 0g ` R ) ) )
28	15, 27	mpbird 167	. 2 |- ( ph -> ( N ` ( X .x. Y ) ) = ( X .x. ( N ` Y ) ) )
29	28	eqcomd 2238	1 |- ( ph -> ( X .x. ( N ` Y ) ) = ( N ` ( X .x. Y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   .rcmulr 13291   0gc0g 13469   Grpcgrp 13713   invgcminusg 13714  Rngcrng 14076

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717  df-abl 14004  df-mgp 14065  df-rng 14077

This theorem is referenced by:  rngm2neg  14093  rngsubdi  14095