Metamath Proof Explorer

Theorem stlesi

Description: Ordering law for states. (Contributed by NM, 24-Oct-1999) (New usage is discouraged.)

Ref Expression
Hypotheses stle.1 
|- A e. CH
stle.2 
|- B e. CH
Assertion stlesi 
|- ( S e. States -> ( A C_ B -> ( ( S ` A ) = 1 -> ( S ` B ) = 1 ) ) )

Proof

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 ) ) )