Metamath Proof Explorer

Theorem rngmneg1

Description: Negation of a product in a non-unital ring ( mulneg1 analog). In contrast to ringmneg1 , the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025)

Ref Expression
Hypotheses 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 rngmneg1 
|- ( ph -> ( ( N ` X ) .x. Y ) = ( N ` ( X .x. Y ) ) )

Proof

Step Hyp Ref Expression
1 rngneglmul.b 
 |-  B = ( Base ` R )
2 rngneglmul.t 
 |-  .x. = ( .r ` R )
3 rngneglmul.n 
 |-  N = ( invg ` R )
4 rngneglmul.r 
 |-  ( ph -> R e. Rng )
5 rngneglmul.x 
 |-  ( ph -> X e. B )
6 rngneglmul.y 
 |-  ( ph -> Y e. B )
7 eqid 
 |-  ( +g ` R ) = ( +g ` R )
8 eqid 
 |-  ( 0g ` R ) = ( 0g ` R )
9 rnggrp 
 |-  ( R e. Rng -> R e. Grp )
10 4 9 syl 
 |-  ( ph -> R e. Grp )
11 1 7 8 3 10 5 grprinvd 
 |-  ( ph -> ( X ( +g ` R ) ( N ` X ) ) = ( 0g ` R ) )
12 11 oveq1d 
 |-  ( ph -> ( ( X ( +g ` R ) ( N ` X ) ) .x. Y ) = ( ( 0g ` R ) .x. Y ) )
13 1 2 8 rnglz 
 |-  ( ( R e. Rng /\ Y e. B ) -> ( ( 0g ` R ) .x. Y ) = ( 0g ` R ) )
14 4 6 13 syl2anc 
 |-  ( ph -> ( ( 0g ` R ) .x. Y ) = ( 0g ` R ) )
15 12 14 eqtrd 
 |-  ( ph -> ( ( X ( +g ` R ) ( N ` X ) ) .x. Y ) = ( 0g ` R ) )
16 1 2 rngcl 
 |-  ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
17 4 5 6 16 syl3anc 
 |-  ( ph -> ( X .x. Y ) e. B )
18 1 3 10 5 grpinvcld 
 |-  ( ph -> ( N ` X ) e. B )
19 1 2 rngcl 
 |-  ( ( R e. Rng /\ ( N ` X ) e. B /\ Y e. B ) -> ( ( N ` X ) .x. Y ) e. B )
20 4 18 6 19 syl3anc 
 |-  ( ph -> ( ( N ` X ) .x. Y ) e. B )
21 1 7 8 3 grpinvid1 
 |-  ( ( 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 10 17 20 21 syl3anc 
 |-  ( ph -> ( ( N ` ( X .x. Y ) ) = ( ( N ` X ) .x. Y ) <-> ( ( X .x. Y ) ( +g ` R ) ( ( N ` X ) .x. Y ) ) = ( 0g ` R ) ) )
23 1 7 2 rngdir 
 |-  ( ( 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 
 |-  ( ( 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 4 5 18 6 24 syl13anc 
 |-  ( ph -> ( ( X .x. Y ) ( +g ` R ) ( ( N ` X ) .x. Y ) ) = ( ( X ( +g ` R ) ( N ` X ) ) .x. Y ) )
26 25 eqeq1d 
 |-  ( 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 
 |-  ( ph -> ( ( N ` ( X .x. Y ) ) = ( ( N ` X ) .x. Y ) <-> ( ( X ( +g ` R ) ( N ` X ) ) .x. Y ) = ( 0g ` R ) ) )
28 15 27 mpbird 
 |-  ( ph -> ( N ` ( X .x. Y ) ) = ( ( N ` X ) .x. Y ) )
29 28 eqcomd 
 |-  ( ph -> ( ( N ` X ) .x. Y ) = ( N ` ( X .x. Y ) ) )