Step |
Hyp |
Ref |
Expression |
1 |
|
mendassa.a |
|- A = ( MEndo ` M ) |
2 |
|
mendassa.s |
|- S = ( Scalar ` M ) |
3 |
1
|
mendbas |
|- ( M LMHom M ) = ( Base ` A ) |
4 |
3
|
a1i |
|- ( ( M e. LMod /\ S e. CRing ) -> ( M LMHom M ) = ( Base ` A ) ) |
5 |
|
eqidd |
|- ( ( M e. LMod /\ S e. CRing ) -> ( +g ` A ) = ( +g ` A ) ) |
6 |
1 2
|
mendsca |
|- S = ( Scalar ` A ) |
7 |
6
|
a1i |
|- ( ( M e. LMod /\ S e. CRing ) -> S = ( Scalar ` A ) ) |
8 |
|
eqidd |
|- ( ( M e. LMod /\ S e. CRing ) -> ( .s ` A ) = ( .s ` A ) ) |
9 |
|
eqidd |
|- ( ( M e. LMod /\ S e. CRing ) -> ( Base ` S ) = ( Base ` S ) ) |
10 |
|
eqidd |
|- ( ( M e. LMod /\ S e. CRing ) -> ( +g ` S ) = ( +g ` S ) ) |
11 |
|
eqidd |
|- ( ( M e. LMod /\ S e. CRing ) -> ( .r ` S ) = ( .r ` S ) ) |
12 |
|
eqidd |
|- ( ( M e. LMod /\ S e. CRing ) -> ( 1r ` S ) = ( 1r ` S ) ) |
13 |
|
crngring |
|- ( S e. CRing -> S e. Ring ) |
14 |
13
|
adantl |
|- ( ( M e. LMod /\ S e. CRing ) -> S e. Ring ) |
15 |
1
|
mendring |
|- ( M e. LMod -> A e. Ring ) |
16 |
15
|
adantr |
|- ( ( M e. LMod /\ S e. CRing ) -> A e. Ring ) |
17 |
|
ringgrp |
|- ( A e. Ring -> A e. Grp ) |
18 |
16 17
|
syl |
|- ( ( M e. LMod /\ S e. CRing ) -> A e. Grp ) |
19 |
|
eqid |
|- ( .s ` M ) = ( .s ` M ) |
20 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
21 |
|
eqid |
|- ( Base ` M ) = ( Base ` M ) |
22 |
|
eqid |
|- ( .s ` A ) = ( .s ` A ) |
23 |
1 19 3 2 20 21 22
|
mendvsca |
|- ( ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) ) -> ( x ( .s ` A ) y ) = ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) ) |
24 |
23
|
3adant1 |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ x e. ( Base ` S ) /\ y e. ( M LMHom M ) ) -> ( x ( .s ` A ) y ) = ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) ) |
25 |
21 19 2 20
|
lmhmvsca |
|- ( ( S e. CRing /\ x e. ( Base ` S ) /\ y e. ( M LMHom M ) ) -> ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) e. ( M LMHom M ) ) |
26 |
25
|
3adant1l |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ x e. ( Base ` S ) /\ y e. ( M LMHom M ) ) -> ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) e. ( M LMHom M ) ) |
27 |
24 26
|
eqeltrd |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ x e. ( Base ` S ) /\ y e. ( M LMHom M ) ) -> ( x ( .s ` A ) y ) e. ( M LMHom M ) ) |
28 |
|
simpr2 |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> y e. ( M LMHom M ) ) |
29 |
|
simpr3 |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> z e. ( M LMHom M ) ) |
30 |
|
eqid |
|- ( +g ` M ) = ( +g ` M ) |
31 |
|
eqid |
|- ( +g ` A ) = ( +g ` A ) |
32 |
1 3 30 31
|
mendplusg |
|- ( ( y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) -> ( y ( +g ` A ) z ) = ( y oF ( +g ` M ) z ) ) |
33 |
28 29 32
|
syl2anc |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> ( y ( +g ` A ) z ) = ( y oF ( +g ` M ) z ) ) |
34 |
33
|
oveq2d |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) ( y ( +g ` A ) z ) ) = ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) ( y oF ( +g ` M ) z ) ) ) |
35 |
|
simpr1 |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> x e. ( Base ` S ) ) |
36 |
18
|
adantr |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> A e. Grp ) |
37 |
3 31
|
grpcl |
|- ( ( A e. Grp /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) -> ( y ( +g ` A ) z ) e. ( M LMHom M ) ) |
38 |
36 28 29 37
|
syl3anc |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> ( y ( +g ` A ) z ) e. ( M LMHom M ) ) |
39 |
1 19 3 2 20 21 22
|
mendvsca |
|- ( ( x e. ( Base ` S ) /\ ( y ( +g ` A ) z ) e. ( M LMHom M ) ) -> ( x ( .s ` A ) ( y ( +g ` A ) z ) ) = ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) ( y ( +g ` A ) z ) ) ) |
40 |
35 38 39
|
syl2anc |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> ( x ( .s ` A ) ( y ( +g ` A ) z ) ) = ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) ( y ( +g ` A ) z ) ) ) |
41 |
35 28 23
|
syl2anc |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> ( x ( .s ` A ) y ) = ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) ) |
42 |
1 19 3 2 20 21 22
|
mendvsca |
|- ( ( x e. ( Base ` S ) /\ z e. ( M LMHom M ) ) -> ( x ( .s ` A ) z ) = ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) z ) ) |
43 |
35 29 42
|
syl2anc |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> ( x ( .s ` A ) z ) = ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) z ) ) |
44 |
41 43
|
oveq12d |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> ( ( x ( .s ` A ) y ) oF ( +g ` M ) ( x ( .s ` A ) z ) ) = ( ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) oF ( +g ` M ) ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) z ) ) ) |
45 |
27
|
3adant3r3 |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> ( x ( .s ` A ) y ) e. ( M LMHom M ) ) |
46 |
|
eleq1w |
|- ( y = z -> ( y e. ( M LMHom M ) <-> z e. ( M LMHom M ) ) ) |
47 |
46
|
3anbi3d |
|- ( y = z -> ( ( ( M e. LMod /\ S e. CRing ) /\ x e. ( Base ` S ) /\ y e. ( M LMHom M ) ) <-> ( ( M e. LMod /\ S e. CRing ) /\ x e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) ) |
48 |
|
oveq2 |
|- ( y = z -> ( x ( .s ` A ) y ) = ( x ( .s ` A ) z ) ) |
49 |
48
|
eleq1d |
|- ( y = z -> ( ( x ( .s ` A ) y ) e. ( M LMHom M ) <-> ( x ( .s ` A ) z ) e. ( M LMHom M ) ) ) |
50 |
47 49
|
imbi12d |
|- ( y = z -> ( ( ( ( M e. LMod /\ S e. CRing ) /\ x e. ( Base ` S ) /\ y e. ( M LMHom M ) ) -> ( x ( .s ` A ) y ) e. ( M LMHom M ) ) <-> ( ( ( M e. LMod /\ S e. CRing ) /\ x e. ( Base ` S ) /\ z e. ( M LMHom M ) ) -> ( x ( .s ` A ) z ) e. ( M LMHom M ) ) ) ) |
51 |
50 27
|
chvarvv |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ x e. ( Base ` S ) /\ z e. ( M LMHom M ) ) -> ( x ( .s ` A ) z ) e. ( M LMHom M ) ) |
52 |
51
|
3adant3r2 |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> ( x ( .s ` A ) z ) e. ( M LMHom M ) ) |
53 |
1 3 30 31
|
mendplusg |
|- ( ( ( x ( .s ` A ) y ) e. ( M LMHom M ) /\ ( x ( .s ` A ) z ) e. ( M LMHom M ) ) -> ( ( x ( .s ` A ) y ) ( +g ` A ) ( x ( .s ` A ) z ) ) = ( ( x ( .s ` A ) y ) oF ( +g ` M ) ( x ( .s ` A ) z ) ) ) |
54 |
45 52 53
|
syl2anc |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> ( ( x ( .s ` A ) y ) ( +g ` A ) ( x ( .s ` A ) z ) ) = ( ( x ( .s ` A ) y ) oF ( +g ` M ) ( x ( .s ` A ) z ) ) ) |
55 |
|
fvexd |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> ( Base ` M ) e. _V ) |
56 |
|
fconst6g |
|- ( x e. ( Base ` S ) -> ( ( Base ` M ) X. { x } ) : ( Base ` M ) --> ( Base ` S ) ) |
57 |
35 56
|
syl |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> ( ( Base ` M ) X. { x } ) : ( Base ` M ) --> ( Base ` S ) ) |
58 |
21 21
|
lmhmf |
|- ( y e. ( M LMHom M ) -> y : ( Base ` M ) --> ( Base ` M ) ) |
59 |
28 58
|
syl |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> y : ( Base ` M ) --> ( Base ` M ) ) |
60 |
21 21
|
lmhmf |
|- ( z e. ( M LMHom M ) -> z : ( Base ` M ) --> ( Base ` M ) ) |
61 |
29 60
|
syl |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> z : ( Base ` M ) --> ( Base ` M ) ) |
62 |
|
simpll |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> M e. LMod ) |
63 |
21 30 2 19 20
|
lmodvsdi |
|- ( ( M e. LMod /\ ( w e. ( Base ` S ) /\ v e. ( Base ` M ) /\ u e. ( Base ` M ) ) ) -> ( w ( .s ` M ) ( v ( +g ` M ) u ) ) = ( ( w ( .s ` M ) v ) ( +g ` M ) ( w ( .s ` M ) u ) ) ) |
64 |
62 63
|
sylan |
|- ( ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) /\ ( w e. ( Base ` S ) /\ v e. ( Base ` M ) /\ u e. ( Base ` M ) ) ) -> ( w ( .s ` M ) ( v ( +g ` M ) u ) ) = ( ( w ( .s ` M ) v ) ( +g ` M ) ( w ( .s ` M ) u ) ) ) |
65 |
55 57 59 61 64
|
caofdi |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) ( y oF ( +g ` M ) z ) ) = ( ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) y ) oF ( +g ` M ) ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) z ) ) ) |
66 |
44 54 65
|
3eqtr4d |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> ( ( x ( .s ` A ) y ) ( +g ` A ) ( x ( .s ` A ) z ) ) = ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) ( y oF ( +g ` M ) z ) ) ) |
67 |
34 40 66
|
3eqtr4d |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( M LMHom M ) /\ z e. ( M LMHom M ) ) ) -> ( x ( .s ` A ) ( y ( +g ` A ) z ) ) = ( ( x ( .s ` A ) y ) ( +g ` A ) ( x ( .s ` A ) z ) ) ) |
68 |
|
fvexd |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( Base ` M ) e. _V ) |
69 |
|
simpr3 |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> z e. ( M LMHom M ) ) |
70 |
69 60
|
syl |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> z : ( Base ` M ) --> ( Base ` M ) ) |
71 |
|
simpr1 |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> x e. ( Base ` S ) ) |
72 |
71 56
|
syl |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( Base ` M ) X. { x } ) : ( Base ` M ) --> ( Base ` S ) ) |
73 |
|
simpr2 |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> y e. ( Base ` S ) ) |
74 |
|
fconst6g |
|- ( y e. ( Base ` S ) -> ( ( Base ` M ) X. { y } ) : ( Base ` M ) --> ( Base ` S ) ) |
75 |
73 74
|
syl |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( Base ` M ) X. { y } ) : ( Base ` M ) --> ( Base ` S ) ) |
76 |
|
simpll |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> M e. LMod ) |
77 |
|
eqid |
|- ( +g ` S ) = ( +g ` S ) |
78 |
21 30 2 19 20 77
|
lmodvsdir |
|- ( ( M e. LMod /\ ( w e. ( Base ` S ) /\ v e. ( Base ` S ) /\ u e. ( Base ` M ) ) ) -> ( ( w ( +g ` S ) v ) ( .s ` M ) u ) = ( ( w ( .s ` M ) u ) ( +g ` M ) ( v ( .s ` M ) u ) ) ) |
79 |
76 78
|
sylan |
|- ( ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) /\ ( w e. ( Base ` S ) /\ v e. ( Base ` S ) /\ u e. ( Base ` M ) ) ) -> ( ( w ( +g ` S ) v ) ( .s ` M ) u ) = ( ( w ( .s ` M ) u ) ( +g ` M ) ( v ( .s ` M ) u ) ) ) |
80 |
68 70 72 75 79
|
caofdir |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( ( ( Base ` M ) X. { x } ) oF ( +g ` S ) ( ( Base ` M ) X. { y } ) ) oF ( .s ` M ) z ) = ( ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) z ) oF ( +g ` M ) ( ( ( Base ` M ) X. { y } ) oF ( .s ` M ) z ) ) ) |
81 |
14
|
adantr |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> S e. Ring ) |
82 |
20 77
|
ringacl |
|- ( ( S e. Ring /\ x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( x ( +g ` S ) y ) e. ( Base ` S ) ) |
83 |
81 71 73 82
|
syl3anc |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( x ( +g ` S ) y ) e. ( Base ` S ) ) |
84 |
1 19 3 2 20 21 22
|
mendvsca |
|- ( ( ( x ( +g ` S ) y ) e. ( Base ` S ) /\ z e. ( M LMHom M ) ) -> ( ( x ( +g ` S ) y ) ( .s ` A ) z ) = ( ( ( Base ` M ) X. { ( x ( +g ` S ) y ) } ) oF ( .s ` M ) z ) ) |
85 |
83 69 84
|
syl2anc |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( x ( +g ` S ) y ) ( .s ` A ) z ) = ( ( ( Base ` M ) X. { ( x ( +g ` S ) y ) } ) oF ( .s ` M ) z ) ) |
86 |
68 71 73
|
ofc12 |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( ( Base ` M ) X. { x } ) oF ( +g ` S ) ( ( Base ` M ) X. { y } ) ) = ( ( Base ` M ) X. { ( x ( +g ` S ) y ) } ) ) |
87 |
86
|
oveq1d |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( ( ( Base ` M ) X. { x } ) oF ( +g ` S ) ( ( Base ` M ) X. { y } ) ) oF ( .s ` M ) z ) = ( ( ( Base ` M ) X. { ( x ( +g ` S ) y ) } ) oF ( .s ` M ) z ) ) |
88 |
85 87
|
eqtr4d |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( x ( +g ` S ) y ) ( .s ` A ) z ) = ( ( ( ( Base ` M ) X. { x } ) oF ( +g ` S ) ( ( Base ` M ) X. { y } ) ) oF ( .s ` M ) z ) ) |
89 |
51
|
3adant3r2 |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( x ( .s ` A ) z ) e. ( M LMHom M ) ) |
90 |
|
eleq1w |
|- ( x = y -> ( x e. ( Base ` S ) <-> y e. ( Base ` S ) ) ) |
91 |
90
|
3anbi2d |
|- ( x = y -> ( ( ( M e. LMod /\ S e. CRing ) /\ x e. ( Base ` S ) /\ z e. ( M LMHom M ) ) <-> ( ( M e. LMod /\ S e. CRing ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) ) |
92 |
|
oveq1 |
|- ( x = y -> ( x ( .s ` A ) z ) = ( y ( .s ` A ) z ) ) |
93 |
92
|
eleq1d |
|- ( x = y -> ( ( x ( .s ` A ) z ) e. ( M LMHom M ) <-> ( y ( .s ` A ) z ) e. ( M LMHom M ) ) ) |
94 |
91 93
|
imbi12d |
|- ( x = y -> ( ( ( ( M e. LMod /\ S e. CRing ) /\ x e. ( Base ` S ) /\ z e. ( M LMHom M ) ) -> ( x ( .s ` A ) z ) e. ( M LMHom M ) ) <-> ( ( ( M e. LMod /\ S e. CRing ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) -> ( y ( .s ` A ) z ) e. ( M LMHom M ) ) ) ) |
95 |
94 51
|
chvarvv |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) -> ( y ( .s ` A ) z ) e. ( M LMHom M ) ) |
96 |
95
|
3adant3r1 |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( y ( .s ` A ) z ) e. ( M LMHom M ) ) |
97 |
1 3 30 31
|
mendplusg |
|- ( ( ( x ( .s ` A ) z ) e. ( M LMHom M ) /\ ( y ( .s ` A ) z ) e. ( M LMHom M ) ) -> ( ( x ( .s ` A ) z ) ( +g ` A ) ( y ( .s ` A ) z ) ) = ( ( x ( .s ` A ) z ) oF ( +g ` M ) ( y ( .s ` A ) z ) ) ) |
98 |
89 96 97
|
syl2anc |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( x ( .s ` A ) z ) ( +g ` A ) ( y ( .s ` A ) z ) ) = ( ( x ( .s ` A ) z ) oF ( +g ` M ) ( y ( .s ` A ) z ) ) ) |
99 |
71 69 42
|
syl2anc |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( x ( .s ` A ) z ) = ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) z ) ) |
100 |
1 19 3 2 20 21 22
|
mendvsca |
|- ( ( y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) -> ( y ( .s ` A ) z ) = ( ( ( Base ` M ) X. { y } ) oF ( .s ` M ) z ) ) |
101 |
73 69 100
|
syl2anc |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( y ( .s ` A ) z ) = ( ( ( Base ` M ) X. { y } ) oF ( .s ` M ) z ) ) |
102 |
99 101
|
oveq12d |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( x ( .s ` A ) z ) oF ( +g ` M ) ( y ( .s ` A ) z ) ) = ( ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) z ) oF ( +g ` M ) ( ( ( Base ` M ) X. { y } ) oF ( .s ` M ) z ) ) ) |
103 |
98 102
|
eqtrd |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( x ( .s ` A ) z ) ( +g ` A ) ( y ( .s ` A ) z ) ) = ( ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) z ) oF ( +g ` M ) ( ( ( Base ` M ) X. { y } ) oF ( .s ` M ) z ) ) ) |
104 |
80 88 103
|
3eqtr4d |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( x ( +g ` S ) y ) ( .s ` A ) z ) = ( ( x ( .s ` A ) z ) ( +g ` A ) ( y ( .s ` A ) z ) ) ) |
105 |
|
ovexd |
|- ( ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) /\ k e. ( Base ` M ) ) -> ( x ( .r ` S ) y ) e. _V ) |
106 |
70
|
ffvelrnda |
|- ( ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) /\ k e. ( Base ` M ) ) -> ( z ` k ) e. ( Base ` M ) ) |
107 |
|
fconstmpt |
|- ( ( Base ` M ) X. { ( x ( .r ` S ) y ) } ) = ( k e. ( Base ` M ) |-> ( x ( .r ` S ) y ) ) |
108 |
107
|
a1i |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( Base ` M ) X. { ( x ( .r ` S ) y ) } ) = ( k e. ( Base ` M ) |-> ( x ( .r ` S ) y ) ) ) |
109 |
70
|
feqmptd |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> z = ( k e. ( Base ` M ) |-> ( z ` k ) ) ) |
110 |
68 105 106 108 109
|
offval2 |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( ( Base ` M ) X. { ( x ( .r ` S ) y ) } ) oF ( .s ` M ) z ) = ( k e. ( Base ` M ) |-> ( ( x ( .r ` S ) y ) ( .s ` M ) ( z ` k ) ) ) ) |
111 |
|
eqid |
|- ( .r ` S ) = ( .r ` S ) |
112 |
20 111
|
ringcl |
|- ( ( S e. Ring /\ x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( x ( .r ` S ) y ) e. ( Base ` S ) ) |
113 |
81 71 73 112
|
syl3anc |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( x ( .r ` S ) y ) e. ( Base ` S ) ) |
114 |
1 19 3 2 20 21 22
|
mendvsca |
|- ( ( ( x ( .r ` S ) y ) e. ( Base ` S ) /\ z e. ( M LMHom M ) ) -> ( ( x ( .r ` S ) y ) ( .s ` A ) z ) = ( ( ( Base ` M ) X. { ( x ( .r ` S ) y ) } ) oF ( .s ` M ) z ) ) |
115 |
113 69 114
|
syl2anc |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( x ( .r ` S ) y ) ( .s ` A ) z ) = ( ( ( Base ` M ) X. { ( x ( .r ` S ) y ) } ) oF ( .s ` M ) z ) ) |
116 |
71
|
adantr |
|- ( ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) /\ k e. ( Base ` M ) ) -> x e. ( Base ` S ) ) |
117 |
|
ovexd |
|- ( ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) /\ k e. ( Base ` M ) ) -> ( y ( .s ` M ) ( z ` k ) ) e. _V ) |
118 |
|
fconstmpt |
|- ( ( Base ` M ) X. { x } ) = ( k e. ( Base ` M ) |-> x ) |
119 |
118
|
a1i |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( Base ` M ) X. { x } ) = ( k e. ( Base ` M ) |-> x ) ) |
120 |
|
simplr2 |
|- ( ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) /\ k e. ( Base ` M ) ) -> y e. ( Base ` S ) ) |
121 |
|
fconstmpt |
|- ( ( Base ` M ) X. { y } ) = ( k e. ( Base ` M ) |-> y ) |
122 |
121
|
a1i |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( Base ` M ) X. { y } ) = ( k e. ( Base ` M ) |-> y ) ) |
123 |
68 120 106 122 109
|
offval2 |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( ( Base ` M ) X. { y } ) oF ( .s ` M ) z ) = ( k e. ( Base ` M ) |-> ( y ( .s ` M ) ( z ` k ) ) ) ) |
124 |
101 123
|
eqtrd |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( y ( .s ` A ) z ) = ( k e. ( Base ` M ) |-> ( y ( .s ` M ) ( z ` k ) ) ) ) |
125 |
68 116 117 119 124
|
offval2 |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) ( y ( .s ` A ) z ) ) = ( k e. ( Base ` M ) |-> ( x ( .s ` M ) ( y ( .s ` M ) ( z ` k ) ) ) ) ) |
126 |
1 19 3 2 20 21 22
|
mendvsca |
|- ( ( x e. ( Base ` S ) /\ ( y ( .s ` A ) z ) e. ( M LMHom M ) ) -> ( x ( .s ` A ) ( y ( .s ` A ) z ) ) = ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) ( y ( .s ` A ) z ) ) ) |
127 |
71 96 126
|
syl2anc |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( x ( .s ` A ) ( y ( .s ` A ) z ) ) = ( ( ( Base ` M ) X. { x } ) oF ( .s ` M ) ( y ( .s ` A ) z ) ) ) |
128 |
76
|
adantr |
|- ( ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) /\ k e. ( Base ` M ) ) -> M e. LMod ) |
129 |
21 2 19 20 111
|
lmodvsass |
|- ( ( M e. LMod /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ ( z ` k ) e. ( Base ` M ) ) ) -> ( ( x ( .r ` S ) y ) ( .s ` M ) ( z ` k ) ) = ( x ( .s ` M ) ( y ( .s ` M ) ( z ` k ) ) ) ) |
130 |
128 116 120 106 129
|
syl13anc |
|- ( ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) /\ k e. ( Base ` M ) ) -> ( ( x ( .r ` S ) y ) ( .s ` M ) ( z ` k ) ) = ( x ( .s ` M ) ( y ( .s ` M ) ( z ` k ) ) ) ) |
131 |
130
|
mpteq2dva |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( k e. ( Base ` M ) |-> ( ( x ( .r ` S ) y ) ( .s ` M ) ( z ` k ) ) ) = ( k e. ( Base ` M ) |-> ( x ( .s ` M ) ( y ( .s ` M ) ( z ` k ) ) ) ) ) |
132 |
125 127 131
|
3eqtr4d |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( x ( .s ` A ) ( y ( .s ` A ) z ) ) = ( k e. ( Base ` M ) |-> ( ( x ( .r ` S ) y ) ( .s ` M ) ( z ` k ) ) ) ) |
133 |
110 115 132
|
3eqtr4d |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) /\ z e. ( M LMHom M ) ) ) -> ( ( x ( .r ` S ) y ) ( .s ` A ) z ) = ( x ( .s ` A ) ( y ( .s ` A ) z ) ) ) |
134 |
14
|
adantr |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ x e. ( M LMHom M ) ) -> S e. Ring ) |
135 |
|
eqid |
|- ( 1r ` S ) = ( 1r ` S ) |
136 |
20 135
|
ringidcl |
|- ( S e. Ring -> ( 1r ` S ) e. ( Base ` S ) ) |
137 |
134 136
|
syl |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ x e. ( M LMHom M ) ) -> ( 1r ` S ) e. ( Base ` S ) ) |
138 |
1 19 3 2 20 21 22
|
mendvsca |
|- ( ( ( 1r ` S ) e. ( Base ` S ) /\ x e. ( M LMHom M ) ) -> ( ( 1r ` S ) ( .s ` A ) x ) = ( ( ( Base ` M ) X. { ( 1r ` S ) } ) oF ( .s ` M ) x ) ) |
139 |
137 138
|
sylancom |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ x e. ( M LMHom M ) ) -> ( ( 1r ` S ) ( .s ` A ) x ) = ( ( ( Base ` M ) X. { ( 1r ` S ) } ) oF ( .s ` M ) x ) ) |
140 |
|
fvexd |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ x e. ( M LMHom M ) ) -> ( Base ` M ) e. _V ) |
141 |
21 21
|
lmhmf |
|- ( x e. ( M LMHom M ) -> x : ( Base ` M ) --> ( Base ` M ) ) |
142 |
141
|
adantl |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ x e. ( M LMHom M ) ) -> x : ( Base ` M ) --> ( Base ` M ) ) |
143 |
|
simpll |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ x e. ( M LMHom M ) ) -> M e. LMod ) |
144 |
21 2 19 135
|
lmodvs1 |
|- ( ( M e. LMod /\ y e. ( Base ` M ) ) -> ( ( 1r ` S ) ( .s ` M ) y ) = y ) |
145 |
143 144
|
sylan |
|- ( ( ( ( M e. LMod /\ S e. CRing ) /\ x e. ( M LMHom M ) ) /\ y e. ( Base ` M ) ) -> ( ( 1r ` S ) ( .s ` M ) y ) = y ) |
146 |
140 142 137 145
|
caofid0l |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ x e. ( M LMHom M ) ) -> ( ( ( Base ` M ) X. { ( 1r ` S ) } ) oF ( .s ` M ) x ) = x ) |
147 |
139 146
|
eqtrd |
|- ( ( ( M e. LMod /\ S e. CRing ) /\ x e. ( M LMHom M ) ) -> ( ( 1r ` S ) ( .s ` A ) x ) = x ) |
148 |
4 5 7 8 9 10 11 12 14 18 27 67 104 133 147
|
islmodd |
|- ( ( M e. LMod /\ S e. CRing ) -> A e. LMod ) |