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 ) ) )