Theorem rngmneg2 14042

Description: Negation of a product in a non-unital ring (mulneg2 8634 analog). In contrast to ringmneg2 14148, 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 2231	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
3		eqid 2231	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
4		rngneglmul.n	. . . . . 6 |- N = ( invg ` R )
5		rngneglmul.r	. . . . . . 7 |- ( ph -> R e. Rng )
6		rnggrp 14032	. . . . . . 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 13718	. . . . 5 |- ( ph -> ( ( N ` Y ) ( +g ` R ) Y ) = ( 0g ` R ) )
10	9	oveq2d 6044	. . . 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 14040	. . . . 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 2264	. . 3 |- ( ph -> ( X .x. ( ( N ` Y ) ( +g ` R ) Y ) ) = ( 0g ` R ) )
16	1, 12	rngcl 14038	. . . . . 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 13712	. . . . . 6 |- ( ph -> ( N ` Y ) e. B )
19	1, 12	rngcl 14038	. . . . . 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 13716	. . . . 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 14034	. . . . . . 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 2237	. . . . . 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 2240	. . . 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 2237	1 |- ( ph -> ( X .x. ( N ` Y ) ) = ( N ` ( X .x. Y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   ` cfv 5333  (class class class)co 6028   Basecbs 13162   +g cplusg 13240   .rcmulr 13241   0gc0g 13419   Grpcgrp 13663   invgcminusg 13664  Rngcrng 14026

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-plusg 13253  df-mulr 13254  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-abl 13954  df-mgp 14015  df-rng 14027

This theorem is referenced by:  rngm2neg  14043  rngsubdi  14045