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Metamath Proof Explorer

Theorem dihcnvord

Description: Ordering property for converse of isomorphism H. (Contributed by NM, 17-Aug-2014)

Ref Expression
Hypotheses dihcnvord.l 
|- .<_ = ( le ` K )
dihcnvord.h 
|- H = ( LHyp ` K )
dihcnvord.i 
|- I = ( ( DIsoH ` K ) ` W )
dihcnvord.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dihcnvord.x 
|- ( ph -> X e. ran I )
dihcnvord.y 
|- ( ph -> Y e. ran I )
Assertion dihcnvord 
|- ( ph -> ( ( `' I ` X ) .<_ ( `' I ` Y ) <-> X C_ Y ) )

Proof

Step Hyp Ref Expression
1 dihcnvord.l 
 |-  .<_ = ( le ` K )
2 dihcnvord.h 
 |-  H = ( LHyp ` K )
3 dihcnvord.i 
 |-  I = ( ( DIsoH ` K ) ` W )
4 dihcnvord.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
5 dihcnvord.x 
 |-  ( ph -> X e. ran I )
6 dihcnvord.y 
 |-  ( ph -> Y e. ran I )
7 eqid 
 |-  ( Base ` K ) = ( Base ` K )
8 7 2 3 dihcnvcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( `' I ` X ) e. ( Base ` K ) )
9 4 5 8 syl2anc 
 |-  ( ph -> ( `' I ` X ) e. ( Base ` K ) )
10 7 2 3 dihcnvcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ Y e. ran I ) -> ( `' I ` Y ) e. ( Base ` K ) )
11 4 6 10 syl2anc 
 |-  ( ph -> ( `' I ` Y ) e. ( Base ` K ) )
12 7 1 2 3 dihord 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( `' I ` X ) e. ( Base ` K ) /\ ( `' I ` Y ) e. ( Base ` K ) ) -> ( ( I ` ( `' I ` X ) ) C_ ( I ` ( `' I ` Y ) ) <-> ( `' I ` X ) .<_ ( `' I ` Y ) ) )
13 4 9 11 12 syl3anc 
 |-  ( ph -> ( ( I ` ( `' I ` X ) ) C_ ( I ` ( `' I ` Y ) ) <-> ( `' I ` X ) .<_ ( `' I ` Y ) ) )
14 2 3 dihcnvid2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( I ` ( `' I ` X ) ) = X )
15 4 5 14 syl2anc 
 |-  ( ph -> ( I ` ( `' I ` X ) ) = X )
16 2 3 dihcnvid2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ Y e. ran I ) -> ( I ` ( `' I ` Y ) ) = Y )
17 4 6 16 syl2anc 
 |-  ( ph -> ( I ` ( `' I ` Y ) ) = Y )
18 15 17 sseq12d 
 |-  ( ph -> ( ( I ` ( `' I ` X ) ) C_ ( I ` ( `' I ` Y ) ) <-> X C_ Y ) )
19 13 18 bitr3d 
 |-  ( ph -> ( ( `' I ` X ) .<_ ( `' I ` Y ) <-> X C_ Y ) )