Step |
Hyp |
Ref |
Expression |
1 |
|
uzfbas.1 |
|- Z = ( ZZ>= ` M ) |
2 |
1
|
uzrest |
|- ( M e. ZZ -> ( ran ZZ>= |`t Z ) = ( ZZ>= " Z ) ) |
3 |
|
zfbas |
|- ran ZZ>= e. ( fBas ` ZZ ) |
4 |
|
0nelfb |
|- ( ran ZZ>= e. ( fBas ` ZZ ) -> -. (/) e. ran ZZ>= ) |
5 |
3 4
|
ax-mp |
|- -. (/) e. ran ZZ>= |
6 |
|
imassrn |
|- ( ZZ>= " Z ) C_ ran ZZ>= |
7 |
2 6
|
eqsstrdi |
|- ( M e. ZZ -> ( ran ZZ>= |`t Z ) C_ ran ZZ>= ) |
8 |
7
|
sseld |
|- ( M e. ZZ -> ( (/) e. ( ran ZZ>= |`t Z ) -> (/) e. ran ZZ>= ) ) |
9 |
5 8
|
mtoi |
|- ( M e. ZZ -> -. (/) e. ( ran ZZ>= |`t Z ) ) |
10 |
|
uzssz |
|- ( ZZ>= ` M ) C_ ZZ |
11 |
1 10
|
eqsstri |
|- Z C_ ZZ |
12 |
|
trfbas2 |
|- ( ( ran ZZ>= e. ( fBas ` ZZ ) /\ Z C_ ZZ ) -> ( ( ran ZZ>= |`t Z ) e. ( fBas ` Z ) <-> -. (/) e. ( ran ZZ>= |`t Z ) ) ) |
13 |
3 11 12
|
mp2an |
|- ( ( ran ZZ>= |`t Z ) e. ( fBas ` Z ) <-> -. (/) e. ( ran ZZ>= |`t Z ) ) |
14 |
9 13
|
sylibr |
|- ( M e. ZZ -> ( ran ZZ>= |`t Z ) e. ( fBas ` Z ) ) |
15 |
2 14
|
eqeltrrd |
|- ( M e. ZZ -> ( ZZ>= " Z ) e. ( fBas ` Z ) ) |