Step |
Hyp |
Ref |
Expression |
1 |
|
o1dif.1 |
|- ( ( ph /\ x e. A ) -> B e. CC ) |
2 |
|
o1dif.2 |
|- ( ( ph /\ x e. A ) -> C e. CC ) |
3 |
|
o1dif.3 |
|- ( ph -> ( x e. A |-> ( B - C ) ) e. O(1) ) |
4 |
|
o1sub |
|- ( ( ( x e. A |-> B ) e. O(1) /\ ( x e. A |-> ( B - C ) ) e. O(1) ) -> ( ( x e. A |-> B ) oF - ( x e. A |-> ( B - C ) ) ) e. O(1) ) |
5 |
4
|
expcom |
|- ( ( x e. A |-> ( B - C ) ) e. O(1) -> ( ( x e. A |-> B ) e. O(1) -> ( ( x e. A |-> B ) oF - ( x e. A |-> ( B - C ) ) ) e. O(1) ) ) |
6 |
3 5
|
syl |
|- ( ph -> ( ( x e. A |-> B ) e. O(1) -> ( ( x e. A |-> B ) oF - ( x e. A |-> ( B - C ) ) ) e. O(1) ) ) |
7 |
1 2
|
subcld |
|- ( ( ph /\ x e. A ) -> ( B - C ) e. CC ) |
8 |
7
|
ralrimiva |
|- ( ph -> A. x e. A ( B - C ) e. CC ) |
9 |
|
dmmptg |
|- ( A. x e. A ( B - C ) e. CC -> dom ( x e. A |-> ( B - C ) ) = A ) |
10 |
8 9
|
syl |
|- ( ph -> dom ( x e. A |-> ( B - C ) ) = A ) |
11 |
|
o1dm |
|- ( ( x e. A |-> ( B - C ) ) e. O(1) -> dom ( x e. A |-> ( B - C ) ) C_ RR ) |
12 |
3 11
|
syl |
|- ( ph -> dom ( x e. A |-> ( B - C ) ) C_ RR ) |
13 |
10 12
|
eqsstrrd |
|- ( ph -> A C_ RR ) |
14 |
|
reex |
|- RR e. _V |
15 |
14
|
ssex |
|- ( A C_ RR -> A e. _V ) |
16 |
13 15
|
syl |
|- ( ph -> A e. _V ) |
17 |
|
eqidd |
|- ( ph -> ( x e. A |-> B ) = ( x e. A |-> B ) ) |
18 |
|
eqidd |
|- ( ph -> ( x e. A |-> ( B - C ) ) = ( x e. A |-> ( B - C ) ) ) |
19 |
16 1 7 17 18
|
offval2 |
|- ( ph -> ( ( x e. A |-> B ) oF - ( x e. A |-> ( B - C ) ) ) = ( x e. A |-> ( B - ( B - C ) ) ) ) |
20 |
1 2
|
nncand |
|- ( ( ph /\ x e. A ) -> ( B - ( B - C ) ) = C ) |
21 |
20
|
mpteq2dva |
|- ( ph -> ( x e. A |-> ( B - ( B - C ) ) ) = ( x e. A |-> C ) ) |
22 |
19 21
|
eqtrd |
|- ( ph -> ( ( x e. A |-> B ) oF - ( x e. A |-> ( B - C ) ) ) = ( x e. A |-> C ) ) |
23 |
22
|
eleq1d |
|- ( ph -> ( ( ( x e. A |-> B ) oF - ( x e. A |-> ( B - C ) ) ) e. O(1) <-> ( x e. A |-> C ) e. O(1) ) ) |
24 |
6 23
|
sylibd |
|- ( ph -> ( ( x e. A |-> B ) e. O(1) -> ( x e. A |-> C ) e. O(1) ) ) |
25 |
|
o1add |
|- ( ( ( x e. A |-> ( B - C ) ) e. O(1) /\ ( x e. A |-> C ) e. O(1) ) -> ( ( x e. A |-> ( B - C ) ) oF + ( x e. A |-> C ) ) e. O(1) ) |
26 |
25
|
ex |
|- ( ( x e. A |-> ( B - C ) ) e. O(1) -> ( ( x e. A |-> C ) e. O(1) -> ( ( x e. A |-> ( B - C ) ) oF + ( x e. A |-> C ) ) e. O(1) ) ) |
27 |
3 26
|
syl |
|- ( ph -> ( ( x e. A |-> C ) e. O(1) -> ( ( x e. A |-> ( B - C ) ) oF + ( x e. A |-> C ) ) e. O(1) ) ) |
28 |
|
eqidd |
|- ( ph -> ( x e. A |-> C ) = ( x e. A |-> C ) ) |
29 |
16 7 2 18 28
|
offval2 |
|- ( ph -> ( ( x e. A |-> ( B - C ) ) oF + ( x e. A |-> C ) ) = ( x e. A |-> ( ( B - C ) + C ) ) ) |
30 |
1 2
|
npcand |
|- ( ( ph /\ x e. A ) -> ( ( B - C ) + C ) = B ) |
31 |
30
|
mpteq2dva |
|- ( ph -> ( x e. A |-> ( ( B - C ) + C ) ) = ( x e. A |-> B ) ) |
32 |
29 31
|
eqtrd |
|- ( ph -> ( ( x e. A |-> ( B - C ) ) oF + ( x e. A |-> C ) ) = ( x e. A |-> B ) ) |
33 |
32
|
eleq1d |
|- ( ph -> ( ( ( x e. A |-> ( B - C ) ) oF + ( x e. A |-> C ) ) e. O(1) <-> ( x e. A |-> B ) e. O(1) ) ) |
34 |
27 33
|
sylibd |
|- ( ph -> ( ( x e. A |-> C ) e. O(1) -> ( x e. A |-> B ) e. O(1) ) ) |
35 |
24 34
|
impbid |
|- ( ph -> ( ( x e. A |-> B ) e. O(1) <-> ( x e. A |-> C ) e. O(1) ) ) |