Step |
Hyp |
Ref |
Expression |
1 |
|
stle.1 |
|- A e. CH |
2 |
|
stle.2 |
|- B e. CH |
3 |
|
stle1 |
|- ( S e. States -> ( B e. CH -> ( S ` B ) <_ 1 ) ) |
4 |
2 3
|
mpi |
|- ( S e. States -> ( S ` B ) <_ 1 ) |
5 |
4
|
adantr |
|- ( ( S e. States /\ ( A C_ B /\ ( S ` A ) = 1 ) ) -> ( S ` B ) <_ 1 ) |
6 |
1 2
|
stlei |
|- ( S e. States -> ( A C_ B -> ( S ` A ) <_ ( S ` B ) ) ) |
7 |
6
|
imp |
|- ( ( S e. States /\ A C_ B ) -> ( S ` A ) <_ ( S ` B ) ) |
8 |
7
|
adantrr |
|- ( ( S e. States /\ ( A C_ B /\ ( S ` A ) = 1 ) ) -> ( S ` A ) <_ ( S ` B ) ) |
9 |
|
breq1 |
|- ( ( S ` A ) = 1 -> ( ( S ` A ) <_ ( S ` B ) <-> 1 <_ ( S ` B ) ) ) |
10 |
9
|
ad2antll |
|- ( ( S e. States /\ ( A C_ B /\ ( S ` A ) = 1 ) ) -> ( ( S ` A ) <_ ( S ` B ) <-> 1 <_ ( S ` B ) ) ) |
11 |
8 10
|
mpbid |
|- ( ( S e. States /\ ( A C_ B /\ ( S ` A ) = 1 ) ) -> 1 <_ ( S ` B ) ) |
12 |
|
stcl |
|- ( S e. States -> ( B e. CH -> ( S ` B ) e. RR ) ) |
13 |
2 12
|
mpi |
|- ( S e. States -> ( S ` B ) e. RR ) |
14 |
|
1re |
|- 1 e. RR |
15 |
13 14
|
jctir |
|- ( S e. States -> ( ( S ` B ) e. RR /\ 1 e. RR ) ) |
16 |
15
|
adantr |
|- ( ( S e. States /\ ( A C_ B /\ ( S ` A ) = 1 ) ) -> ( ( S ` B ) e. RR /\ 1 e. RR ) ) |
17 |
|
letri3 |
|- ( ( ( S ` B ) e. RR /\ 1 e. RR ) -> ( ( S ` B ) = 1 <-> ( ( S ` B ) <_ 1 /\ 1 <_ ( S ` B ) ) ) ) |
18 |
16 17
|
syl |
|- ( ( S e. States /\ ( A C_ B /\ ( S ` A ) = 1 ) ) -> ( ( S ` B ) = 1 <-> ( ( S ` B ) <_ 1 /\ 1 <_ ( S ` B ) ) ) ) |
19 |
5 11 18
|
mpbir2and |
|- ( ( S e. States /\ ( A C_ B /\ ( S ` A ) = 1 ) ) -> ( S ` B ) = 1 ) |
20 |
19
|
exp32 |
|- ( S e. States -> ( A C_ B -> ( ( S ` A ) = 1 -> ( S ` B ) = 1 ) ) ) |