Description: Lemma for minvec . The infimum of the distances to A is a real number. (Contributed by Mario Carneiro, 16-Jun-2014) (Revised by Mario Carneiro, 15-Oct-2015) (Revised by AV, 3-Oct-2020)
Ref | Expression | ||
---|---|---|---|
Hypotheses | minvec.x | |
|
minvec.m | |
||
minvec.n | |
||
minvec.u | |
||
minvec.y | |
||
minvec.w | |
||
minvec.a | |
||
minvec.j | |
||
minvec.r | |
||
minvec.s | |
||
Assertion | minveclem4c | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | minvec.x | |
|
2 | minvec.m | |
|
3 | minvec.n | |
|
4 | minvec.u | |
|
5 | minvec.y | |
|
6 | minvec.w | |
|
7 | minvec.a | |
|
8 | minvec.j | |
|
9 | minvec.r | |
|
10 | minvec.s | |
|
11 | 1 2 3 4 5 6 7 8 9 | minveclem1 | |
12 | 11 | simp1d | |
13 | 11 | simp2d | |
14 | 0re | |
|
15 | 11 | simp3d | |
16 | breq1 | |
|
17 | 16 | ralbidv | |
18 | 17 | rspcev | |
19 | 14 15 18 | sylancr | |
20 | infrecl | |
|
21 | 12 13 19 20 | syl3anc | |
22 | 10 21 | eqeltrid | |