Theorem rngmneg1 13959

Description: Negation of a product in a non-unital ring (mulneg1 8573 analog). In contrast to ringmneg1 14065, 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 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 13950	. . . . . . 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 13638	. . . . 5 |- ( ph -> ( X ( +g ` R ) ( N ` X ) ) = ( 0g ` R ) )
10	9	oveq1d 6032	. . . 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 13957	. . . . 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 2264	. . 3 |- ( ph -> ( ( X ( +g ` R ) ( N ` X ) ) .x. Y ) = ( 0g ` R ) )
16	1, 12	rngcl 13956	. . . . . 6 |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
17	5, 8, 11, 16	syl3anc 1273	. . . . 5 |- ( ph -> ( X .x. Y ) e. B )
18	1, 4, 7, 8	grpinvcld 13631	. . . . . 6 |- ( ph -> ( N ` X ) e. B )
19	1, 12	rngcl 13956	. . . . . 6 |- ( ( R e. Rng /\ ( N ` X ) e. B /\ Y e. B ) -> ( ( N ` X ) .x. Y ) e. B )
20	5, 18, 11, 19	syl3anc 1273	. . . . 5 |- ( ph -> ( ( N ` X ) .x. Y ) e. B )
21	1, 2, 3, 4	grpinvid1 13634	. . . . 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 1273	. . . 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 13953	. . . . . . 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 2237	. . . . . 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 1275	. . . . 5 |- ( ph -> ( ( X .x. Y ) ( +g ` R ) ( ( N ` X ) .x. Y ) ) = ( ( X ( +g ` R ) ( N ` X ) ) .x. Y ) )
26	25	eqeq1d 2240	. . . 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 2237	1 |- ( ph -> ( ( N ` X ) .x. Y ) = ( N ` ( X .x. Y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   .rcmulr 13160   0gc0g 13338   Grpcgrp 13582   invgcminusg 13583  Rngcrng 13944

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-abl 13873  df-mgp 13933  df-rng 13945

This theorem is referenced by:  rngm2neg  13961  rngsubdir  13964