Step |
Hyp |
Ref |
Expression |
1 |
|
qusmul2.h |
|- Q = ( R /s ( R ~QG I ) ) |
2 |
|
qusmul2.v |
|- B = ( Base ` R ) |
3 |
|
qusmul2.p |
|- .x. = ( .r ` R ) |
4 |
|
qusmul2.a |
|- .X. = ( .r ` Q ) |
5 |
|
qusmul2.1 |
|- ( ph -> R e. Ring ) |
6 |
|
qusmul2.2 |
|- ( ph -> I e. ( 2Ideal ` R ) ) |
7 |
|
qusmul2.3 |
|- ( ph -> X e. B ) |
8 |
|
qusmul2.4 |
|- ( ph -> Y e. B ) |
9 |
1
|
a1i |
|- ( ph -> Q = ( R /s ( R ~QG I ) ) ) |
10 |
2
|
a1i |
|- ( ph -> B = ( Base ` R ) ) |
11 |
6
|
2idllidld |
|- ( ph -> I e. ( LIdeal ` R ) ) |
12 |
|
eqid |
|- ( LIdeal ` R ) = ( LIdeal ` R ) |
13 |
12
|
lidlsubg |
|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) ) -> I e. ( SubGrp ` R ) ) |
14 |
5 11 13
|
syl2anc |
|- ( ph -> I e. ( SubGrp ` R ) ) |
15 |
|
eqid |
|- ( R ~QG I ) = ( R ~QG I ) |
16 |
2 15
|
eqger |
|- ( I e. ( SubGrp ` R ) -> ( R ~QG I ) Er B ) |
17 |
14 16
|
syl |
|- ( ph -> ( R ~QG I ) Er B ) |
18 |
|
eqid |
|- ( 2Ideal ` R ) = ( 2Ideal ` R ) |
19 |
2 15 18 3
|
2idlcpbl |
|- ( ( R e. Ring /\ I e. ( 2Ideal ` R ) ) -> ( ( x ( R ~QG I ) y /\ z ( R ~QG I ) t ) -> ( x .x. z ) ( R ~QG I ) ( y .x. t ) ) ) |
20 |
5 6 19
|
syl2anc |
|- ( ph -> ( ( x ( R ~QG I ) y /\ z ( R ~QG I ) t ) -> ( x .x. z ) ( R ~QG I ) ( y .x. t ) ) ) |
21 |
2 3
|
ringcl |
|- ( ( R e. Ring /\ p e. B /\ q e. B ) -> ( p .x. q ) e. B ) |
22 |
21
|
3expb |
|- ( ( R e. Ring /\ ( p e. B /\ q e. B ) ) -> ( p .x. q ) e. B ) |
23 |
5 22
|
sylan |
|- ( ( ph /\ ( p e. B /\ q e. B ) ) -> ( p .x. q ) e. B ) |
24 |
23
|
caovclg |
|- ( ( ph /\ ( y e. B /\ t e. B ) ) -> ( y .x. t ) e. B ) |
25 |
9 10 17 5 20 24 3 4
|
qusmulval |
|- ( ( ph /\ X e. B /\ Y e. B ) -> ( [ X ] ( R ~QG I ) .X. [ Y ] ( R ~QG I ) ) = [ ( X .x. Y ) ] ( R ~QG I ) ) |
26 |
7 8 25
|
mpd3an23 |
|- ( ph -> ( [ X ] ( R ~QG I ) .X. [ Y ] ( R ~QG I ) ) = [ ( X .x. Y ) ] ( R ~QG I ) ) |