Theorem ringmneg1

Description: Negation of a product in a ring. ( mulneg1 analog.) Compared with rngmneg1 , the proof is shorter making use of the existence of a ring unity. (Contributed by Jeff Madsen, 19-Jun-2010) (Revised by Mario Carneiro, 2-Jul-2014)

	Ref	Expression
Hypotheses	ringneglmul.b	|- B = ( Base ` R )
	ringneglmul.t	|- .x. = ( .r ` R )
	ringneglmul.n	|- N = ( invg ` R )
	ringneglmul.r	|- ( ph -> R e. Ring )
	ringneglmul.x	|- ( ph -> X e. B )
	ringneglmul.y	|- ( ph -> Y e. B )
Assertion	ringmneg1	|- ( ph -> ( ( N ` X ) .x. Y ) = ( N ` ( X .x. Y ) ) )

Proof

Step	Hyp	Ref	Expression
1		ringneglmul.b	|- B = ( Base ` R )
2		ringneglmul.t	|- .x. = ( .r ` R )
3		ringneglmul.n	|- N = ( invg ` R )
4		ringneglmul.r	|- ( ph -> R e. Ring )
5		ringneglmul.x	|- ( ph -> X e. B )
6		ringneglmul.y	|- ( ph -> Y e. B )
7		ringgrp	|- ( R e. Ring -> R e. Grp )
8	4 7	syl	|- ( ph -> R e. Grp )
9		eqid	|- ( 1r ` R ) = ( 1r ` R )
10	1 9	ringidcl	|- ( R e. Ring -> ( 1r ` R ) e. B )
11	4 10	syl	|- ( ph -> ( 1r ` R ) e. B )
12	1 3	grpinvcl	|- ( ( R e. Grp /\ ( 1r ` R ) e. B ) -> ( N ` ( 1r ` R ) ) e. B )
13	8 11 12	syl2anc	|- ( ph -> ( N ` ( 1r ` R ) ) e. B )
14	1 2	ringass	|- ( ( R e. Ring /\ ( ( N ` ( 1r ` R ) ) e. B /\ X e. B /\ Y e. B ) ) -> ( ( ( N ` ( 1r ` R ) ) .x. X ) .x. Y ) = ( ( N ` ( 1r ` R ) ) .x. ( X .x. Y ) ) )
15	4 13 5 6 14	syl13anc	|- ( ph -> ( ( ( N ` ( 1r ` R ) ) .x. X ) .x. Y ) = ( ( N ` ( 1r ` R ) ) .x. ( X .x. Y ) ) )
16	1 2 9 3 4 5	ringnegl	|- ( ph -> ( ( N ` ( 1r ` R ) ) .x. X ) = ( N ` X ) )
17	16	oveq1d	|- ( ph -> ( ( ( N ` ( 1r ` R ) ) .x. X ) .x. Y ) = ( ( N ` X ) .x. Y ) )
18	1 2	ringcl	|- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )
19	4 5 6 18	syl3anc	|- ( ph -> ( X .x. Y ) e. B )
20	1 2 9 3 4 19	ringnegl	|- ( ph -> ( ( N ` ( 1r ` R ) ) .x. ( X .x. Y ) ) = ( N ` ( X .x. Y ) ) )
21	15 17 20	3eqtr3d	|- ( ph -> ( ( N ` X ) .x. Y ) = ( N ` ( X .x. Y ) ) )