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Metamath Proof Explorer

Theorem rngm2neg

Description: Double negation of a product in a non-unital ring ( mul2neg analog). (Contributed by Mario Carneiro, 4-Dec-2014) Generalization of ringm2neg . (Revised 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 rngm2neg 
|- ( ph -> ( ( N ` X ) .x. ( N ` Y ) ) = ( 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 rnggrp 
 |-  ( R e. Rng -> R e. Grp )
8 4 7 syl 
 |-  ( ph -> R e. Grp )
9 1 3 8 6 grpinvcld 
 |-  ( ph -> ( N ` Y ) e. B )
10 1 2 3 4 5 9 rngmneg1 
 |-  ( ph -> ( ( N ` X ) .x. ( N ` Y ) ) = ( N ` ( X .x. ( N ` Y ) ) ) )
11 1 2 3 4 5 6 rngmneg2 
 |-  ( ph -> ( X .x. ( N ` Y ) ) = ( N ` ( X .x. Y ) ) )
12 11 fveq2d 
 |-  ( ph -> ( N ` ( X .x. ( N ` Y ) ) ) = ( N ` ( N ` ( X .x. Y ) ) ) )
13 1 2 rngcl 
 |-  ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
14 4 5 6 13 syl3anc 
 |-  ( ph -> ( X .x. Y ) e. B )
15 1 3 grpinvinv 
 |-  ( ( R e. Grp /\ ( X .x. Y ) e. B ) -> ( N ` ( N ` ( X .x. Y ) ) ) = ( X .x. Y ) )
16 8 14 15 syl2anc 
 |-  ( ph -> ( N ` ( N ` ( X .x. Y ) ) ) = ( X .x. Y ) )
17 10 12 16 3eqtrd 
 |-  ( ph -> ( ( N ` X ) .x. ( N ` Y ) ) = ( X .x. Y ) )