Description: Linear independence of a set is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015)
Ref | Expression | ||
---|---|---|---|
Hypotheses | lindfmm.b | |
|
lindfmm.c | |
||
Assertion | lindsmm | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lindfmm.b | |
|
2 | lindfmm.c | |
|
3 | ibar | |
|
4 | 3 | 3ad2ant3 | |
5 | f1oi | |
|
6 | f1of | |
|
7 | 5 6 | ax-mp | |
8 | simp3 | |
|
9 | fss | |
|
10 | 7 8 9 | sylancr | |
11 | 1 2 | lindfmm | |
12 | 10 11 | syld3an3 | |
13 | 4 12 | bitr3d | |
14 | lmhmlmod1 | |
|
15 | 14 | 3ad2ant1 | |
16 | 1 | islinds | |
17 | 15 16 | syl | |
18 | lmhmlmod2 | |
|
19 | 18 | 3ad2ant1 | |
20 | 19 | adantr | |
21 | simpr | |
|
22 | f1ores | |
|
23 | f1of1 | |
|
24 | 22 23 | syl | |
25 | 24 | 3adant1 | |
26 | 25 | adantr | |
27 | f1linds | |
|
28 | 20 21 26 27 | syl3anc | |
29 | df-ima | |
|
30 | lindfrn | |
|
31 | 19 30 | sylan | |
32 | 29 31 | eqeltrid | |
33 | 28 32 | impbida | |
34 | coires1 | |
|
35 | 34 | breq1i | |
36 | 33 35 | bitr4di | |
37 | 13 17 36 | 3bitr4d | |