Step |
Hyp |
Ref |
Expression |
1 |
|
o1add2.1 |
|- ( ( ph /\ x e. A ) -> B e. V ) |
2 |
|
o1add2.2 |
|- ( ( ph /\ x e. A ) -> C e. V ) |
3 |
|
o1add2.3 |
|- ( ph -> ( x e. A |-> B ) e. O(1) ) |
4 |
|
o1add2.4 |
|- ( ph -> ( x e. A |-> C ) e. O(1) ) |
5 |
1
|
ralrimiva |
|- ( ph -> A. x e. A B e. V ) |
6 |
|
dmmptg |
|- ( A. x e. A B e. V -> dom ( x e. A |-> B ) = A ) |
7 |
5 6
|
syl |
|- ( ph -> dom ( x e. A |-> B ) = A ) |
8 |
|
o1dm |
|- ( ( x e. A |-> B ) e. O(1) -> dom ( x e. A |-> B ) C_ RR ) |
9 |
3 8
|
syl |
|- ( ph -> dom ( x e. A |-> B ) C_ RR ) |
10 |
7 9
|
eqsstrrd |
|- ( ph -> A C_ RR ) |
11 |
|
reex |
|- RR e. _V |
12 |
11
|
ssex |
|- ( A C_ RR -> A e. _V ) |
13 |
10 12
|
syl |
|- ( ph -> A e. _V ) |
14 |
|
eqidd |
|- ( ph -> ( x e. A |-> B ) = ( x e. A |-> B ) ) |
15 |
|
eqidd |
|- ( ph -> ( x e. A |-> C ) = ( x e. A |-> C ) ) |
16 |
13 1 2 14 15
|
offval2 |
|- ( ph -> ( ( x e. A |-> B ) oF - ( x e. A |-> C ) ) = ( x e. A |-> ( B - C ) ) ) |
17 |
|
o1sub |
|- ( ( ( x e. A |-> B ) e. O(1) /\ ( x e. A |-> C ) e. O(1) ) -> ( ( x e. A |-> B ) oF - ( x e. A |-> C ) ) e. O(1) ) |
18 |
3 4 17
|
syl2anc |
|- ( ph -> ( ( x e. A |-> B ) oF - ( x e. A |-> C ) ) e. O(1) ) |
19 |
16 18
|
eqeltrrd |
|- ( ph -> ( x e. A |-> ( B - C ) ) e. O(1) ) |