Metamath Proof Explorer

Theorem hstrbi

Description: Strong CH-state theorem (bidirectional version). Theorem in Mayet3 p. 10 and its converse. (Contributed by NM, 30-Jun-2006) (New usage is discouraged.)

Ref Expression
Hypotheses hstr.1 
|- A e. CH
hstr.2 
|- B e. CH
Assertion hstrbi 
|- ( A. f e. CHStates ( ( normh ` ( f ` A ) ) = 1 -> ( normh ` ( f ` B ) ) = 1 ) <-> A C_ B )

Proof

Step Hyp Ref Expression
1 hstr.1 
 |-  A e. CH
2 hstr.2 
 |-  B e. CH
3 1 2 hstri 
 |-  ( A. f e. CHStates ( ( normh ` ( f ` A ) ) = 1 -> ( normh ` ( f ` B ) ) = 1 ) -> A C_ B )
4 hstles 
 |-  ( ( ( f e. CHStates /\ A e. CH ) /\ ( B e. CH /\ A C_ B ) ) -> ( ( normh ` ( f ` A ) ) = 1 -> ( normh ` ( f ` B ) ) = 1 ) )
5 2 4 mpanr1 
 |-  ( ( ( f e. CHStates /\ A e. CH ) /\ A C_ B ) -> ( ( normh ` ( f ` A ) ) = 1 -> ( normh ` ( f ` B ) ) = 1 ) )
6 1 5 mpanl2 
 |-  ( ( f e. CHStates /\ A C_ B ) -> ( ( normh ` ( f ` A ) ) = 1 -> ( normh ` ( f ` B ) ) = 1 ) )
7 6 expcom 
 |-  ( A C_ B -> ( f e. CHStates -> ( ( normh ` ( f ` A ) ) = 1 -> ( normh ` ( f ` B ) ) = 1 ) ) )
8 7 ralrimiv 
 |-  ( A C_ B -> A. f e. CHStates ( ( normh ` ( f ` A ) ) = 1 -> ( normh ` ( f ` B ) ) = 1 ) )
9 3 8 impbii 
 |-  ( A. f e. CHStates ( ( normh ` ( f ` A ) ) = 1 -> ( normh ` ( f ` B ) ) = 1 ) <-> A C_ B )