Step |
Hyp |
Ref |
Expression |
1 |
|
retop |
|- ( topGen ` ran (,) ) e. Top |
2 |
|
qssre |
|- QQ C_ RR |
3 |
|
uniretop |
|- RR = U. ( topGen ` ran (,) ) |
4 |
3
|
clsss3 |
|- ( ( ( topGen ` ran (,) ) e. Top /\ QQ C_ RR ) -> ( ( cls ` ( topGen ` ran (,) ) ) ` QQ ) C_ RR ) |
5 |
1 2 4
|
mp2an |
|- ( ( cls ` ( topGen ` ran (,) ) ) ` QQ ) C_ RR |
6 |
|
ioof |
|- (,) : ( RR* X. RR* ) --> ~P RR |
7 |
|
ffn |
|- ( (,) : ( RR* X. RR* ) --> ~P RR -> (,) Fn ( RR* X. RR* ) ) |
8 |
|
ovelrn |
|- ( (,) Fn ( RR* X. RR* ) -> ( y e. ran (,) <-> E. z e. RR* E. w e. RR* y = ( z (,) w ) ) ) |
9 |
6 7 8
|
mp2b |
|- ( y e. ran (,) <-> E. z e. RR* E. w e. RR* y = ( z (,) w ) ) |
10 |
|
elioo3g |
|- ( x e. ( z (,) w ) <-> ( ( z e. RR* /\ w e. RR* /\ x e. RR* ) /\ ( z < x /\ x < w ) ) ) |
11 |
10
|
simplbi |
|- ( x e. ( z (,) w ) -> ( z e. RR* /\ w e. RR* /\ x e. RR* ) ) |
12 |
11
|
simp1d |
|- ( x e. ( z (,) w ) -> z e. RR* ) |
13 |
12
|
ad2antrr |
|- ( ( ( x e. ( z (,) w ) /\ y e. QQ ) /\ ( z < y /\ y < w ) ) -> z e. RR* ) |
14 |
11
|
simp2d |
|- ( x e. ( z (,) w ) -> w e. RR* ) |
15 |
14
|
ad2antrr |
|- ( ( ( x e. ( z (,) w ) /\ y e. QQ ) /\ ( z < y /\ y < w ) ) -> w e. RR* ) |
16 |
|
qre |
|- ( y e. QQ -> y e. RR ) |
17 |
16
|
ad2antlr |
|- ( ( ( x e. ( z (,) w ) /\ y e. QQ ) /\ ( z < y /\ y < w ) ) -> y e. RR ) |
18 |
17
|
rexrd |
|- ( ( ( x e. ( z (,) w ) /\ y e. QQ ) /\ ( z < y /\ y < w ) ) -> y e. RR* ) |
19 |
13 15 18
|
3jca |
|- ( ( ( x e. ( z (,) w ) /\ y e. QQ ) /\ ( z < y /\ y < w ) ) -> ( z e. RR* /\ w e. RR* /\ y e. RR* ) ) |
20 |
|
simpr |
|- ( ( ( x e. ( z (,) w ) /\ y e. QQ ) /\ ( z < y /\ y < w ) ) -> ( z < y /\ y < w ) ) |
21 |
|
elioo3g |
|- ( y e. ( z (,) w ) <-> ( ( z e. RR* /\ w e. RR* /\ y e. RR* ) /\ ( z < y /\ y < w ) ) ) |
22 |
19 20 21
|
sylanbrc |
|- ( ( ( x e. ( z (,) w ) /\ y e. QQ ) /\ ( z < y /\ y < w ) ) -> y e. ( z (,) w ) ) |
23 |
|
simplr |
|- ( ( ( x e. ( z (,) w ) /\ y e. QQ ) /\ ( z < y /\ y < w ) ) -> y e. QQ ) |
24 |
|
inelcm |
|- ( ( y e. ( z (,) w ) /\ y e. QQ ) -> ( ( z (,) w ) i^i QQ ) =/= (/) ) |
25 |
22 23 24
|
syl2anc |
|- ( ( ( x e. ( z (,) w ) /\ y e. QQ ) /\ ( z < y /\ y < w ) ) -> ( ( z (,) w ) i^i QQ ) =/= (/) ) |
26 |
11
|
simp3d |
|- ( x e. ( z (,) w ) -> x e. RR* ) |
27 |
|
eliooord |
|- ( x e. ( z (,) w ) -> ( z < x /\ x < w ) ) |
28 |
27
|
simpld |
|- ( x e. ( z (,) w ) -> z < x ) |
29 |
27
|
simprd |
|- ( x e. ( z (,) w ) -> x < w ) |
30 |
12 26 14 28 29
|
xrlttrd |
|- ( x e. ( z (,) w ) -> z < w ) |
31 |
|
qbtwnxr |
|- ( ( z e. RR* /\ w e. RR* /\ z < w ) -> E. y e. QQ ( z < y /\ y < w ) ) |
32 |
12 14 30 31
|
syl3anc |
|- ( x e. ( z (,) w ) -> E. y e. QQ ( z < y /\ y < w ) ) |
33 |
25 32
|
r19.29a |
|- ( x e. ( z (,) w ) -> ( ( z (,) w ) i^i QQ ) =/= (/) ) |
34 |
33
|
a1i |
|- ( y = ( z (,) w ) -> ( x e. ( z (,) w ) -> ( ( z (,) w ) i^i QQ ) =/= (/) ) ) |
35 |
|
eleq2 |
|- ( y = ( z (,) w ) -> ( x e. y <-> x e. ( z (,) w ) ) ) |
36 |
|
ineq1 |
|- ( y = ( z (,) w ) -> ( y i^i QQ ) = ( ( z (,) w ) i^i QQ ) ) |
37 |
36
|
neeq1d |
|- ( y = ( z (,) w ) -> ( ( y i^i QQ ) =/= (/) <-> ( ( z (,) w ) i^i QQ ) =/= (/) ) ) |
38 |
34 35 37
|
3imtr4d |
|- ( y = ( z (,) w ) -> ( x e. y -> ( y i^i QQ ) =/= (/) ) ) |
39 |
38
|
rexlimivw |
|- ( E. w e. RR* y = ( z (,) w ) -> ( x e. y -> ( y i^i QQ ) =/= (/) ) ) |
40 |
39
|
rexlimivw |
|- ( E. z e. RR* E. w e. RR* y = ( z (,) w ) -> ( x e. y -> ( y i^i QQ ) =/= (/) ) ) |
41 |
9 40
|
sylbi |
|- ( y e. ran (,) -> ( x e. y -> ( y i^i QQ ) =/= (/) ) ) |
42 |
41
|
rgen |
|- A. y e. ran (,) ( x e. y -> ( y i^i QQ ) =/= (/) ) |
43 |
|
eqidd |
|- ( x e. RR -> ( topGen ` ran (,) ) = ( topGen ` ran (,) ) ) |
44 |
3
|
a1i |
|- ( x e. RR -> RR = U. ( topGen ` ran (,) ) ) |
45 |
|
retopbas |
|- ran (,) e. TopBases |
46 |
45
|
a1i |
|- ( x e. RR -> ran (,) e. TopBases ) |
47 |
2
|
a1i |
|- ( x e. RR -> QQ C_ RR ) |
48 |
|
id |
|- ( x e. RR -> x e. RR ) |
49 |
43 44 46 47 48
|
elcls3 |
|- ( x e. RR -> ( x e. ( ( cls ` ( topGen ` ran (,) ) ) ` QQ ) <-> A. y e. ran (,) ( x e. y -> ( y i^i QQ ) =/= (/) ) ) ) |
50 |
42 49
|
mpbiri |
|- ( x e. RR -> x e. ( ( cls ` ( topGen ` ran (,) ) ) ` QQ ) ) |
51 |
50
|
ssriv |
|- RR C_ ( ( cls ` ( topGen ` ran (,) ) ) ` QQ ) |
52 |
5 51
|
eqssi |
|- ( ( cls ` ( topGen ` ran (,) ) ) ` QQ ) = RR |