Theorem rngmneg2 13825

Description: Negation of a product in a non-unital ring (mulneg2 8503 analog). In contrast to ringmneg2 13931, 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 2207	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
3		eqid 2207	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
4		rngneglmul.n	. . . . . 6 |- N = ( invg ` R )
5		rngneglmul.r	. . . . . . 7 |- ( ph -> R e. Rng )
6		rnggrp 13815	. . . . . . 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 13502	. . . . 5 |- ( ph -> ( ( N ` Y ) ( +g ` R ) Y ) = ( 0g ` R ) )
10	9	oveq2d 5983	. . . 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 13823	. . . . 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 2240	. . 3 |- ( ph -> ( X .x. ( ( N ` Y ) ( +g ` R ) Y ) ) = ( 0g ` R ) )
16	1, 12	rngcl 13821	. . . . . 6 |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
17	5, 11, 8, 16	syl3anc 1250	. . . . 5 |- ( ph -> ( X .x. Y ) e. B )
18	1, 4, 7, 8	grpinvcld 13496	. . . . . 6 |- ( ph -> ( N ` Y ) e. B )
19	1, 12	rngcl 13821	. . . . . 6 |- ( ( R e. Rng /\ X e. B /\ ( N ` Y ) e. B ) -> ( X .x. ( N ` Y ) ) e. B )
20	5, 11, 18, 19	syl3anc 1250	. . . . 5 |- ( ph -> ( X .x. ( N ` Y ) ) e. B )
21	1, 2, 3, 4	grpinvid2 13500	. . . . 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 1250	. . . 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 13817	. . . . . . 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 2213	. . . . . 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 1252	. . . . 5 |- ( ph -> ( ( X .x. ( N ` Y ) ) ( +g ` R ) ( X .x. Y ) ) = ( X .x. ( ( N ` Y ) ( +g ` R ) Y ) ) )
26	25	eqeq1d 2216	. . . 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 2213	1 |- ( ph -> ( X .x. ( N ` Y ) ) = ( N ` ( X .x. Y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2178   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   .rcmulr 13025   0gc0g 13203   Grpcgrp 13447   invgcminusg 13448  Rngcrng 13809

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-plusg 13037  df-mulr 13038  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-abl 13738  df-mgp 13798  df-rng 13810

This theorem is referenced by:  rngm2neg  13826  rngsubdi  13828