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Metamath Proof Explorer

Theorem pmap11

Description: The projective map of a Hilbert lattice is one-to-one. Part of Theorem 15.5 of MaedaMaeda p. 62. (Contributed by NM, 22-Oct-2011)

Ref Expression
Hypotheses pmap11.b 
|- B = ( Base ` K )
pmap11.m 
|- M = ( pmap ` K )
Assertion pmap11 
|- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( ( M ` X ) = ( M ` Y ) <-> X = Y ) )

Proof

Step Hyp Ref Expression
1 pmap11.b 
 |-  B = ( Base ` K )
2 pmap11.m 
 |-  M = ( pmap ` K )
3 eqss 
 |-  ( ( M ` X ) = ( M ` Y ) <-> ( ( M ` X ) C_ ( M ` Y ) /\ ( M ` Y ) C_ ( M ` X ) ) )
4 hllat 
 |-  ( K e. HL -> K e. Lat )
5 eqid 
 |-  ( le ` K ) = ( le ` K )
6 1 5 latasymb 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( ( X ( le ` K ) Y /\ Y ( le ` K ) X ) <-> X = Y ) )
7 4 6 syl3an1 
 |-  ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( ( X ( le ` K ) Y /\ Y ( le ` K ) X ) <-> X = Y ) )
8 1 5 2 pmaple 
 |-  ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( X ( le ` K ) Y <-> ( M ` X ) C_ ( M ` Y ) ) )
9 1 5 2 pmaple 
 |-  ( ( K e. HL /\ Y e. B /\ X e. B ) -> ( Y ( le ` K ) X <-> ( M ` Y ) C_ ( M ` X ) ) )
10 9 3com23 
 |-  ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( Y ( le ` K ) X <-> ( M ` Y ) C_ ( M ` X ) ) )
11 8 10 anbi12d 
 |-  ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( ( X ( le ` K ) Y /\ Y ( le ` K ) X ) <-> ( ( M ` X ) C_ ( M ` Y ) /\ ( M ` Y ) C_ ( M ` X ) ) ) )
12 7 11 bitr3d 
 |-  ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( X = Y <-> ( ( M ` X ) C_ ( M ` Y ) /\ ( M ` Y ) C_ ( M ` X ) ) ) )
13 3 12 bitr4id 
 |-  ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( ( M ` X ) = ( M ` Y ) <-> X = Y ) )