Step |
Hyp |
Ref |
Expression |
1 |
|
srgz.b |
|- B = ( Base ` R ) |
2 |
|
srgz.t |
|- .x. = ( .r ` R ) |
3 |
|
srgz.z |
|- .0. = ( 0g ` R ) |
4 |
|
eqid |
|- ( mulGrp ` R ) = ( mulGrp ` R ) |
5 |
|
eqid |
|- ( +g ` R ) = ( +g ` R ) |
6 |
1 4 5 2 3
|
issrg |
|- ( R e. SRing <-> ( R e. CMnd /\ ( mulGrp ` R ) e. Mnd /\ A. x e. B ( A. y e. B A. z e. B ( ( x .x. ( y ( +g ` R ) z ) ) = ( ( x .x. y ) ( +g ` R ) ( x .x. z ) ) /\ ( ( x ( +g ` R ) y ) .x. z ) = ( ( x .x. z ) ( +g ` R ) ( y .x. z ) ) ) /\ ( ( .0. .x. x ) = .0. /\ ( x .x. .0. ) = .0. ) ) ) ) |
7 |
6
|
simp3bi |
|- ( R e. SRing -> A. x e. B ( A. y e. B A. z e. B ( ( x .x. ( y ( +g ` R ) z ) ) = ( ( x .x. y ) ( +g ` R ) ( x .x. z ) ) /\ ( ( x ( +g ` R ) y ) .x. z ) = ( ( x .x. z ) ( +g ` R ) ( y .x. z ) ) ) /\ ( ( .0. .x. x ) = .0. /\ ( x .x. .0. ) = .0. ) ) ) |
8 |
7
|
r19.21bi |
|- ( ( R e. SRing /\ x e. B ) -> ( A. y e. B A. z e. B ( ( x .x. ( y ( +g ` R ) z ) ) = ( ( x .x. y ) ( +g ` R ) ( x .x. z ) ) /\ ( ( x ( +g ` R ) y ) .x. z ) = ( ( x .x. z ) ( +g ` R ) ( y .x. z ) ) ) /\ ( ( .0. .x. x ) = .0. /\ ( x .x. .0. ) = .0. ) ) ) |
9 |
8
|
simprrd |
|- ( ( R e. SRing /\ x e. B ) -> ( x .x. .0. ) = .0. ) |
10 |
9
|
ralrimiva |
|- ( R e. SRing -> A. x e. B ( x .x. .0. ) = .0. ) |
11 |
|
oveq1 |
|- ( x = X -> ( x .x. .0. ) = ( X .x. .0. ) ) |
12 |
11
|
eqeq1d |
|- ( x = X -> ( ( x .x. .0. ) = .0. <-> ( X .x. .0. ) = .0. ) ) |
13 |
12
|
rspcv |
|- ( X e. B -> ( A. x e. B ( x .x. .0. ) = .0. -> ( X .x. .0. ) = .0. ) ) |
14 |
10 13
|
mpan9 |
|- ( ( R e. SRing /\ X e. B ) -> ( X .x. .0. ) = .0. ) |