Step |
Hyp |
Ref |
Expression |
1 |
|
ringsubdi.b |
|- B = ( Base ` R ) |
2 |
|
ringsubdi.t |
|- .x. = ( .r ` R ) |
3 |
|
ringsubdi.m |
|- .- = ( -g ` R ) |
4 |
|
ringsubdi.r |
|- ( ph -> R e. Ring ) |
5 |
|
ringsubdi.x |
|- ( ph -> X e. B ) |
6 |
|
ringsubdi.y |
|- ( ph -> Y e. B ) |
7 |
|
ringsubdi.z |
|- ( ph -> Z e. B ) |
8 |
|
ringgrp |
|- ( R e. Ring -> R e. Grp ) |
9 |
4 8
|
syl |
|- ( ph -> R e. Grp ) |
10 |
|
eqid |
|- ( invg ` R ) = ( invg ` R ) |
11 |
1 10
|
grpinvcl |
|- ( ( R e. Grp /\ Y e. B ) -> ( ( invg ` R ) ` Y ) e. B ) |
12 |
9 6 11
|
syl2anc |
|- ( ph -> ( ( invg ` R ) ` Y ) e. B ) |
13 |
|
eqid |
|- ( +g ` R ) = ( +g ` R ) |
14 |
1 13 2
|
ringdir |
|- ( ( R e. Ring /\ ( X e. B /\ ( ( invg ` R ) ` Y ) e. B /\ Z e. B ) ) -> ( ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) .x. Z ) = ( ( X .x. Z ) ( +g ` R ) ( ( ( invg ` R ) ` Y ) .x. Z ) ) ) |
15 |
4 5 12 7 14
|
syl13anc |
|- ( ph -> ( ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) .x. Z ) = ( ( X .x. Z ) ( +g ` R ) ( ( ( invg ` R ) ` Y ) .x. Z ) ) ) |
16 |
1 2 10 4 6 7
|
ringmneg1 |
|- ( ph -> ( ( ( invg ` R ) ` Y ) .x. Z ) = ( ( invg ` R ) ` ( Y .x. Z ) ) ) |
17 |
16
|
oveq2d |
|- ( ph -> ( ( X .x. Z ) ( +g ` R ) ( ( ( invg ` R ) ` Y ) .x. Z ) ) = ( ( X .x. Z ) ( +g ` R ) ( ( invg ` R ) ` ( Y .x. Z ) ) ) ) |
18 |
15 17
|
eqtrd |
|- ( ph -> ( ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) .x. Z ) = ( ( X .x. Z ) ( +g ` R ) ( ( invg ` R ) ` ( Y .x. Z ) ) ) ) |
19 |
1 13 10 3
|
grpsubval |
|- ( ( X e. B /\ Y e. B ) -> ( X .- Y ) = ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) ) |
20 |
5 6 19
|
syl2anc |
|- ( ph -> ( X .- Y ) = ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) ) |
21 |
20
|
oveq1d |
|- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X ( +g ` R ) ( ( invg ` R ) ` Y ) ) .x. Z ) ) |
22 |
1 2
|
ringcl |
|- ( ( R e. Ring /\ X e. B /\ Z e. B ) -> ( X .x. Z ) e. B ) |
23 |
4 5 7 22
|
syl3anc |
|- ( ph -> ( X .x. Z ) e. B ) |
24 |
1 2
|
ringcl |
|- ( ( R e. Ring /\ Y e. B /\ Z e. B ) -> ( Y .x. Z ) e. B ) |
25 |
4 6 7 24
|
syl3anc |
|- ( ph -> ( Y .x. Z ) e. B ) |
26 |
1 13 10 3
|
grpsubval |
|- ( ( ( X .x. Z ) e. B /\ ( Y .x. Z ) e. B ) -> ( ( X .x. Z ) .- ( Y .x. Z ) ) = ( ( X .x. Z ) ( +g ` R ) ( ( invg ` R ) ` ( Y .x. Z ) ) ) ) |
27 |
23 25 26
|
syl2anc |
|- ( ph -> ( ( X .x. Z ) .- ( Y .x. Z ) ) = ( ( X .x. Z ) ( +g ` R ) ( ( invg ` R ) ` ( Y .x. Z ) ) ) ) |
28 |
18 21 27
|
3eqtr4d |
|- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X .x. Z ) .- ( Y .x. Z ) ) ) |