Step |
Hyp |
Ref |
Expression |
1 |
|
ioonct.b |
|- ( ph -> A e. RR* ) |
2 |
|
ioonct.c |
|- ( ph -> B e. RR* ) |
3 |
|
ioonct.l |
|- ( ph -> A < B ) |
4 |
|
ioonct.a |
|- C = ( A (,) B ) |
5 |
|
ioontr |
|- ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) B ) ) = ( A (,) B ) |
6 |
5
|
a1i |
|- ( ( ph /\ C ~<_ _om ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) B ) ) = ( A (,) B ) ) |
7 |
|
ioossre |
|- ( A (,) B ) C_ RR |
8 |
7
|
a1i |
|- ( ( ph /\ C ~<_ _om ) -> ( A (,) B ) C_ RR ) |
9 |
4
|
breq1i |
|- ( C ~<_ _om <-> ( A (,) B ) ~<_ _om ) |
10 |
9
|
biimpi |
|- ( C ~<_ _om -> ( A (,) B ) ~<_ _om ) |
11 |
|
nnenom |
|- NN ~~ _om |
12 |
11
|
ensymi |
|- _om ~~ NN |
13 |
12
|
a1i |
|- ( C ~<_ _om -> _om ~~ NN ) |
14 |
|
domentr |
|- ( ( ( A (,) B ) ~<_ _om /\ _om ~~ NN ) -> ( A (,) B ) ~<_ NN ) |
15 |
10 13 14
|
syl2anc |
|- ( C ~<_ _om -> ( A (,) B ) ~<_ NN ) |
16 |
15
|
adantl |
|- ( ( ph /\ C ~<_ _om ) -> ( A (,) B ) ~<_ NN ) |
17 |
|
rectbntr0 |
|- ( ( ( A (,) B ) C_ RR /\ ( A (,) B ) ~<_ NN ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) B ) ) = (/) ) |
18 |
8 16 17
|
syl2anc |
|- ( ( ph /\ C ~<_ _om ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) B ) ) = (/) ) |
19 |
6 18
|
eqtr3d |
|- ( ( ph /\ C ~<_ _om ) -> ( A (,) B ) = (/) ) |
20 |
|
ioon0 |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A (,) B ) =/= (/) <-> A < B ) ) |
21 |
1 2 20
|
syl2anc |
|- ( ph -> ( ( A (,) B ) =/= (/) <-> A < B ) ) |
22 |
3 21
|
mpbird |
|- ( ph -> ( A (,) B ) =/= (/) ) |
23 |
22
|
neneqd |
|- ( ph -> -. ( A (,) B ) = (/) ) |
24 |
23
|
adantr |
|- ( ( ph /\ C ~<_ _om ) -> -. ( A (,) B ) = (/) ) |
25 |
19 24
|
pm2.65da |
|- ( ph -> -. C ~<_ _om ) |