Step |
Hyp |
Ref |
Expression |
1 |
|
negcl |
|- ( C e. CC -> -u C e. CC ) |
2 |
|
addcn2 |
|- ( ( A e. RR+ /\ B e. CC /\ -u C e. CC ) -> E. y e. RR+ E. z e. RR+ A. u e. CC A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) ) |
3 |
1 2
|
syl3an3 |
|- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> E. y e. RR+ E. z e. RR+ A. u e. CC A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) ) |
4 |
|
negcl |
|- ( v e. CC -> -u v e. CC ) |
5 |
|
fvoveq1 |
|- ( w = -u v -> ( abs ` ( w - -u C ) ) = ( abs ` ( -u v - -u C ) ) ) |
6 |
5
|
breq1d |
|- ( w = -u v -> ( ( abs ` ( w - -u C ) ) < z <-> ( abs ` ( -u v - -u C ) ) < z ) ) |
7 |
6
|
anbi2d |
|- ( w = -u v -> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) <-> ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( -u v - -u C ) ) < z ) ) ) |
8 |
|
oveq2 |
|- ( w = -u v -> ( u + w ) = ( u + -u v ) ) |
9 |
8
|
fvoveq1d |
|- ( w = -u v -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) = ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) ) |
10 |
9
|
breq1d |
|- ( w = -u v -> ( ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A <-> ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) < A ) ) |
11 |
7 10
|
imbi12d |
|- ( w = -u v -> ( ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) <-> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( -u v - -u C ) ) < z ) -> ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) < A ) ) ) |
12 |
11
|
rspcv |
|- ( -u v e. CC -> ( A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) -> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( -u v - -u C ) ) < z ) -> ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) < A ) ) ) |
13 |
4 12
|
syl |
|- ( v e. CC -> ( A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) -> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( -u v - -u C ) ) < z ) -> ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) < A ) ) ) |
14 |
13
|
adantl |
|- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) -> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( -u v - -u C ) ) < z ) -> ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) < A ) ) ) |
15 |
|
simpr |
|- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> v e. CC ) |
16 |
|
simpll3 |
|- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> C e. CC ) |
17 |
15 16
|
neg2subd |
|- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( -u v - -u C ) = ( C - v ) ) |
18 |
17
|
fveq2d |
|- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( abs ` ( -u v - -u C ) ) = ( abs ` ( C - v ) ) ) |
19 |
16 15
|
abssubd |
|- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( abs ` ( C - v ) ) = ( abs ` ( v - C ) ) ) |
20 |
18 19
|
eqtrd |
|- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( abs ` ( -u v - -u C ) ) = ( abs ` ( v - C ) ) ) |
21 |
20
|
breq1d |
|- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( ( abs ` ( -u v - -u C ) ) < z <-> ( abs ` ( v - C ) ) < z ) ) |
22 |
21
|
anbi2d |
|- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( -u v - -u C ) ) < z ) <-> ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) ) ) |
23 |
|
negsub |
|- ( ( u e. CC /\ v e. CC ) -> ( u + -u v ) = ( u - v ) ) |
24 |
23
|
adantll |
|- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( u + -u v ) = ( u - v ) ) |
25 |
|
simpll2 |
|- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> B e. CC ) |
26 |
25 16
|
negsubd |
|- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( B + -u C ) = ( B - C ) ) |
27 |
24 26
|
oveq12d |
|- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( ( u + -u v ) - ( B + -u C ) ) = ( ( u - v ) - ( B - C ) ) ) |
28 |
27
|
fveq2d |
|- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) = ( abs ` ( ( u - v ) - ( B - C ) ) ) ) |
29 |
28
|
breq1d |
|- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) < A <-> ( abs ` ( ( u - v ) - ( B - C ) ) ) < A ) ) |
30 |
22 29
|
imbi12d |
|- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( -u v - -u C ) ) < z ) -> ( abs ` ( ( u + -u v ) - ( B + -u C ) ) ) < A ) <-> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) -> ( abs ` ( ( u - v ) - ( B - C ) ) ) < A ) ) ) |
31 |
14 30
|
sylibd |
|- ( ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) /\ v e. CC ) -> ( A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) -> ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) -> ( abs ` ( ( u - v ) - ( B - C ) ) ) < A ) ) ) |
32 |
31
|
ralrimdva |
|- ( ( ( A e. RR+ /\ B e. CC /\ C e. CC ) /\ u e. CC ) -> ( A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) -> A. v e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) -> ( abs ` ( ( u - v ) - ( B - C ) ) ) < A ) ) ) |
33 |
32
|
ralimdva |
|- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( A. u e. CC A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) -> A. u e. CC A. v e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) -> ( abs ` ( ( u - v ) - ( B - C ) ) ) < A ) ) ) |
34 |
33
|
reximdv |
|- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( E. z e. RR+ A. u e. CC A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) -> E. z e. RR+ A. u e. CC A. v e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) -> ( abs ` ( ( u - v ) - ( B - C ) ) ) < A ) ) ) |
35 |
34
|
reximdv |
|- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( E. y e. RR+ E. z e. RR+ A. u e. CC A. w e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( w - -u C ) ) < z ) -> ( abs ` ( ( u + w ) - ( B + -u C ) ) ) < A ) -> E. y e. RR+ E. z e. RR+ A. u e. CC A. v e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) -> ( abs ` ( ( u - v ) - ( B - C ) ) ) < A ) ) ) |
36 |
3 35
|
mpd |
|- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> E. y e. RR+ E. z e. RR+ A. u e. CC A. v e. CC ( ( ( abs ` ( u - B ) ) < y /\ ( abs ` ( v - C ) ) < z ) -> ( abs ` ( ( u - v ) - ( B - C ) ) ) < A ) ) |