Description: A family is independent iff the linear combinations homomorphism is injective. (Contributed by Stefan O'Rear, 28-Feb-2015)
Ref | Expression | ||
---|---|---|---|
Hypotheses | islindf5.f | |
|
islindf5.b | |
||
islindf5.c | |
||
islindf5.v | |
||
islindf5.e | |
||
islindf5.t | |
||
islindf5.i | |
||
islindf5.r | |
||
islindf5.a | |
||
Assertion | islindf5 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islindf5.f | |
|
2 | islindf5.b | |
|
3 | islindf5.c | |
|
4 | islindf5.v | |
|
5 | islindf5.e | |
|
6 | islindf5.t | |
|
7 | islindf5.i | |
|
8 | islindf5.r | |
|
9 | islindf5.a | |
|
10 | eqid | |
|
11 | eqid | |
|
12 | eqid | |
|
13 | eqid | |
|
14 | 3 10 4 11 12 13 | islindf4 | |
15 | 6 7 9 14 | syl3anc | |
16 | oveq1 | |
|
17 | 16 | oveq2d | |
18 | ovex | |
|
19 | 17 5 18 | fvmpt | |
20 | 19 | adantl | |
21 | 20 | eqeq1d | |
22 | 10 | lmodring | |
23 | 6 22 | syl | |
24 | 8 23 | eqeltrd | |
25 | eqid | |
|
26 | 1 25 | frlm0 | |
27 | 24 7 26 | syl2anc | |
28 | 8 | fveq2d | |
29 | 28 | sneqd | |
30 | 29 | xpeq2d | |
31 | 27 30 | eqtr3d | |
32 | 31 | adantr | |
33 | 32 | eqeq2d | |
34 | 21 33 | imbi12d | |
35 | 34 | ralbidva | |
36 | 8 | eqcomd | |
37 | 36 | oveq1d | |
38 | 37 1 | eqtr4di | |
39 | 38 | fveq2d | |
40 | 39 2 | eqtr4di | |
41 | 40 | raleqdv | |
42 | 35 41 | bitr4d | |
43 | 15 42 | bitr4d | |
44 | 1 2 3 4 5 6 7 8 9 | frlmup1 | |
45 | lmghm | |
|
46 | eqid | |
|
47 | 2 3 46 11 | ghmf1 | |
48 | 44 45 47 | 3syl | |
49 | 43 48 | bitr4d | |