Metamath Proof Explorer

Theorem dia11N

Description: The partial isomorphism A for a lattice K is one-to-one in the region under co-atom W . Part of Lemma M of Crawley p. 120 line 28. (Contributed by NM, 25-Nov-2013) (New usage is discouraged.)

Ref Expression
Hypotheses dia11.b 
|- B = ( Base ` K )
dia11.l 
|- .<_ = ( le ` K )
dia11.h 
|- H = ( LHyp ` K )
dia11.i 
|- I = ( ( DIsoA ` K ) ` W )
Assertion dia11N 
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( I ` X ) = ( I ` Y ) <-> X = Y ) )

Proof

Step Hyp Ref Expression
1 dia11.b 
 |-  B = ( Base ` K )
2 dia11.l 
 |-  .<_ = ( le ` K )
3 dia11.h 
 |-  H = ( LHyp ` K )
4 dia11.i 
 |-  I = ( ( DIsoA ` K ) ` W )
5 eqss 
 |-  ( ( I ` X ) = ( I ` Y ) <-> ( ( I ` X ) C_ ( I ` Y ) /\ ( I ` Y ) C_ ( I ` X ) ) )
6 1 2 3 4 diaord 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( I ` X ) C_ ( I ` Y ) <-> X .<_ Y ) )
7 1 2 3 4 diaord 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Y e. B /\ Y .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( I ` Y ) C_ ( I ` X ) <-> Y .<_ X ) )
8 7 3com23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( I ` Y ) C_ ( I ` X ) <-> Y .<_ X ) )
9 6 8 anbi12d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( ( I ` X ) C_ ( I ` Y ) /\ ( I ` Y ) C_ ( I ` X ) ) <-> ( X .<_ Y /\ Y .<_ X ) ) )
10 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> K e. HL )
11 10 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> K e. Lat )
12 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> X e. B )
13 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> Y e. B )
14 1 2 latasymb 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( ( X .<_ Y /\ Y .<_ X ) <-> X = Y ) )
15 11 12 13 14 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( X .<_ Y /\ Y .<_ X ) <-> X = Y ) )
16 9 15 bitrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( ( I ` X ) C_ ( I ` Y ) /\ ( I ` Y ) C_ ( I ` X ) ) <-> X = Y ) )
17 5 16 bitrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( ( I ` X ) = ( I ` Y ) <-> X = Y ) )