Step |
Hyp |
Ref |
Expression |
1 |
|
lmod1zr.r |
|- R = { <. ( Base ` ndx ) , { Z } >. , <. ( +g ` ndx ) , { <. <. Z , Z >. , Z >. } >. , <. ( .r ` ndx ) , { <. <. Z , Z >. , Z >. } >. } |
2 |
|
lmod1zr.m |
|- M = ( { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. ( Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , { <. <. Z , I >. , I >. } >. } ) |
3 |
|
elsni |
|- ( p e. { <. Z , I >. } -> p = <. Z , I >. ) |
4 |
|
fveq2 |
|- ( p = <. Z , I >. -> ( 2nd ` p ) = ( 2nd ` <. Z , I >. ) ) |
5 |
4
|
adantl |
|- ( ( ( I e. V /\ Z e. W ) /\ p = <. Z , I >. ) -> ( 2nd ` p ) = ( 2nd ` <. Z , I >. ) ) |
6 |
|
op2ndg |
|- ( ( Z e. W /\ I e. V ) -> ( 2nd ` <. Z , I >. ) = I ) |
7 |
6
|
ancoms |
|- ( ( I e. V /\ Z e. W ) -> ( 2nd ` <. Z , I >. ) = I ) |
8 |
|
snidg |
|- ( I e. V -> I e. { I } ) |
9 |
8
|
adantr |
|- ( ( I e. V /\ Z e. W ) -> I e. { I } ) |
10 |
7 9
|
eqeltrd |
|- ( ( I e. V /\ Z e. W ) -> ( 2nd ` <. Z , I >. ) e. { I } ) |
11 |
10
|
adantr |
|- ( ( ( I e. V /\ Z e. W ) /\ p = <. Z , I >. ) -> ( 2nd ` <. Z , I >. ) e. { I } ) |
12 |
5 11
|
eqeltrd |
|- ( ( ( I e. V /\ Z e. W ) /\ p = <. Z , I >. ) -> ( 2nd ` p ) e. { I } ) |
13 |
3 12
|
sylan2 |
|- ( ( ( I e. V /\ Z e. W ) /\ p e. { <. Z , I >. } ) -> ( 2nd ` p ) e. { I } ) |
14 |
13
|
fmpttd |
|- ( ( I e. V /\ Z e. W ) -> ( p e. { <. Z , I >. } |-> ( 2nd ` p ) ) : { <. Z , I >. } --> { I } ) |
15 |
|
opex |
|- <. Z , I >. e. _V |
16 |
|
simpl |
|- ( ( I e. V /\ Z e. W ) -> I e. V ) |
17 |
|
fsng |
|- ( ( <. Z , I >. e. _V /\ I e. V ) -> ( ( p e. { <. Z , I >. } |-> ( 2nd ` p ) ) : { <. Z , I >. } --> { I } <-> ( p e. { <. Z , I >. } |-> ( 2nd ` p ) ) = { <. <. Z , I >. , I >. } ) ) |
18 |
15 16 17
|
sylancr |
|- ( ( I e. V /\ Z e. W ) -> ( ( p e. { <. Z , I >. } |-> ( 2nd ` p ) ) : { <. Z , I >. } --> { I } <-> ( p e. { <. Z , I >. } |-> ( 2nd ` p ) ) = { <. <. Z , I >. , I >. } ) ) |
19 |
14 18
|
mpbid |
|- ( ( I e. V /\ Z e. W ) -> ( p e. { <. Z , I >. } |-> ( 2nd ` p ) ) = { <. <. Z , I >. , I >. } ) |
20 |
|
xpsng |
|- ( ( Z e. W /\ I e. V ) -> ( { Z } X. { I } ) = { <. Z , I >. } ) |
21 |
20
|
ancoms |
|- ( ( I e. V /\ Z e. W ) -> ( { Z } X. { I } ) = { <. Z , I >. } ) |
22 |
21
|
eqcomd |
|- ( ( I e. V /\ Z e. W ) -> { <. Z , I >. } = ( { Z } X. { I } ) ) |
23 |
22
|
mpteq1d |
|- ( ( I e. V /\ Z e. W ) -> ( p e. { <. Z , I >. } |-> ( 2nd ` p ) ) = ( p e. ( { Z } X. { I } ) |-> ( 2nd ` p ) ) ) |
24 |
19 23
|
eqtr3d |
|- ( ( I e. V /\ Z e. W ) -> { <. <. Z , I >. , I >. } = ( p e. ( { Z } X. { I } ) |-> ( 2nd ` p ) ) ) |
25 |
|
vex |
|- z e. _V |
26 |
|
vex |
|- i e. _V |
27 |
25 26
|
op2ndd |
|- ( p = <. z , i >. -> ( 2nd ` p ) = i ) |
28 |
27
|
mpompt |
|- ( p e. ( { Z } X. { I } ) |-> ( 2nd ` p ) ) = ( z e. { Z } , i e. { I } |-> i ) |
29 |
28
|
a1i |
|- ( ( I e. V /\ Z e. W ) -> ( p e. ( { Z } X. { I } ) |-> ( 2nd ` p ) ) = ( z e. { Z } , i e. { I } |-> i ) ) |
30 |
|
snex |
|- { Z } e. _V |
31 |
1
|
rngbase |
|- ( { Z } e. _V -> { Z } = ( Base ` R ) ) |
32 |
30 31
|
mp1i |
|- ( ( I e. V /\ Z e. W ) -> { Z } = ( Base ` R ) ) |
33 |
|
eqidd |
|- ( ( I e. V /\ Z e. W ) -> { I } = { I } ) |
34 |
|
mpoeq12 |
|- ( ( { Z } = ( Base ` R ) /\ { I } = { I } ) -> ( z e. { Z } , i e. { I } |-> i ) = ( z e. ( Base ` R ) , i e. { I } |-> i ) ) |
35 |
32 33 34
|
syl2anc |
|- ( ( I e. V /\ Z e. W ) -> ( z e. { Z } , i e. { I } |-> i ) = ( z e. ( Base ` R ) , i e. { I } |-> i ) ) |
36 |
24 29 35
|
3eqtrd |
|- ( ( I e. V /\ Z e. W ) -> { <. <. Z , I >. , I >. } = ( z e. ( Base ` R ) , i e. { I } |-> i ) ) |
37 |
36
|
opeq2d |
|- ( ( I e. V /\ Z e. W ) -> <. ( .s ` ndx ) , { <. <. Z , I >. , I >. } >. = <. ( .s ` ndx ) , ( z e. ( Base ` R ) , i e. { I } |-> i ) >. ) |
38 |
37
|
sneqd |
|- ( ( I e. V /\ Z e. W ) -> { <. ( .s ` ndx ) , { <. <. Z , I >. , I >. } >. } = { <. ( .s ` ndx ) , ( z e. ( Base ` R ) , i e. { I } |-> i ) >. } ) |
39 |
38
|
uneq2d |
|- ( ( I e. V /\ Z e. W ) -> ( { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. ( Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , { <. <. Z , I >. , I >. } >. } ) = ( { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. ( Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , ( z e. ( Base ` R ) , i e. { I } |-> i ) >. } ) ) |
40 |
2 39
|
syl5eq |
|- ( ( I e. V /\ Z e. W ) -> M = ( { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. ( Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , ( z e. ( Base ` R ) , i e. { I } |-> i ) >. } ) ) |
41 |
1
|
ring1 |
|- ( Z e. W -> R e. Ring ) |
42 |
|
eqidd |
|- ( z = a -> i = i ) |
43 |
|
id |
|- ( i = b -> i = b ) |
44 |
42 43
|
cbvmpov |
|- ( z e. ( Base ` R ) , i e. { I } |-> i ) = ( a e. ( Base ` R ) , b e. { I } |-> b ) |
45 |
44
|
opeq2i |
|- <. ( .s ` ndx ) , ( z e. ( Base ` R ) , i e. { I } |-> i ) >. = <. ( .s ` ndx ) , ( a e. ( Base ` R ) , b e. { I } |-> b ) >. |
46 |
45
|
sneqi |
|- { <. ( .s ` ndx ) , ( z e. ( Base ` R ) , i e. { I } |-> i ) >. } = { <. ( .s ` ndx ) , ( a e. ( Base ` R ) , b e. { I } |-> b ) >. } |
47 |
46
|
uneq2i |
|- ( { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. ( Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , ( z e. ( Base ` R ) , i e. { I } |-> i ) >. } ) = ( { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. ( Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , ( a e. ( Base ` R ) , b e. { I } |-> b ) >. } ) |
48 |
47
|
lmod1 |
|- ( ( I e. V /\ R e. Ring ) -> ( { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. ( Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , ( z e. ( Base ` R ) , i e. { I } |-> i ) >. } ) e. LMod ) |
49 |
41 48
|
sylan2 |
|- ( ( I e. V /\ Z e. W ) -> ( { <. ( Base ` ndx ) , { I } >. , <. ( +g ` ndx ) , { <. <. I , I >. , I >. } >. , <. ( Scalar ` ndx ) , R >. } u. { <. ( .s ` ndx ) , ( z e. ( Base ` R ) , i e. { I } |-> i ) >. } ) e. LMod ) |
50 |
40 49
|
eqeltrd |
|- ( ( I e. V /\ Z e. W ) -> M e. LMod ) |