Step |
Hyp |
Ref |
Expression |
1 |
|
df-rab |
|- { x e. CC | ( x - A ) e. ( ZZ>= ` B ) } = { x | ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) } |
2 |
|
simp2 |
|- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> x e. CC ) |
3 |
|
zcn |
|- ( A e. ZZ -> A e. CC ) |
4 |
3
|
3ad2ant1 |
|- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> A e. CC ) |
5 |
2 4
|
npcand |
|- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> ( ( x - A ) + A ) = x ) |
6 |
|
eluzadd |
|- ( ( ( x - A ) e. ( ZZ>= ` B ) /\ A e. ZZ ) -> ( ( x - A ) + A ) e. ( ZZ>= ` ( B + A ) ) ) |
7 |
6
|
ancoms |
|- ( ( A e. ZZ /\ ( x - A ) e. ( ZZ>= ` B ) ) -> ( ( x - A ) + A ) e. ( ZZ>= ` ( B + A ) ) ) |
8 |
7
|
3adant2 |
|- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> ( ( x - A ) + A ) e. ( ZZ>= ` ( B + A ) ) ) |
9 |
5 8
|
eqeltrrd |
|- ( ( A e. ZZ /\ x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> x e. ( ZZ>= ` ( B + A ) ) ) |
10 |
9
|
3expib |
|- ( A e. ZZ -> ( ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> x e. ( ZZ>= ` ( B + A ) ) ) ) |
11 |
10
|
adantr |
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) -> x e. ( ZZ>= ` ( B + A ) ) ) ) |
12 |
|
eluzelcn |
|- ( x e. ( ZZ>= ` ( B + A ) ) -> x e. CC ) |
13 |
12
|
a1i |
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( x e. ( ZZ>= ` ( B + A ) ) -> x e. CC ) ) |
14 |
|
eluzsub |
|- ( ( B e. ZZ /\ A e. ZZ /\ x e. ( ZZ>= ` ( B + A ) ) ) -> ( x - A ) e. ( ZZ>= ` B ) ) |
15 |
14
|
3expia |
|- ( ( B e. ZZ /\ A e. ZZ ) -> ( x e. ( ZZ>= ` ( B + A ) ) -> ( x - A ) e. ( ZZ>= ` B ) ) ) |
16 |
15
|
ancoms |
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( x e. ( ZZ>= ` ( B + A ) ) -> ( x - A ) e. ( ZZ>= ` B ) ) ) |
17 |
13 16
|
jcad |
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( x e. ( ZZ>= ` ( B + A ) ) -> ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) ) ) |
18 |
11 17
|
impbid |
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) <-> x e. ( ZZ>= ` ( B + A ) ) ) ) |
19 |
18
|
abbi1dv |
|- ( ( A e. ZZ /\ B e. ZZ ) -> { x | ( x e. CC /\ ( x - A ) e. ( ZZ>= ` B ) ) } = ( ZZ>= ` ( B + A ) ) ) |
20 |
1 19
|
eqtrid |
|- ( ( A e. ZZ /\ B e. ZZ ) -> { x e. CC | ( x - A ) e. ( ZZ>= ` B ) } = ( ZZ>= ` ( B + A ) ) ) |