Metamath Proof Explorer

Theorem dihmeet2

Description: Reverse isomorphism H of a closed subspace intersection. (Contributed by NM, 15-Jan-2015)

Ref Expression
Hypotheses dihmeet2.m 
|- ./\ = ( meet ` K )
dihmeet2.h 
|- H = ( LHyp ` K )
dihmeet2.i 
|- I = ( ( DIsoH ` K ) ` W )
dihmeet2.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dihmeet2.x 
|- ( ph -> X e. ran I )
dihmeet2.y 
|- ( ph -> Y e. ran I )
Assertion dihmeet2 
|- ( ph -> ( `' I ` ( X i^i Y ) ) = ( ( `' I ` X ) ./\ ( `' I ` Y ) ) )

Proof

Step Hyp Ref Expression
1 dihmeet2.m 
 |-  ./\ = ( meet ` K )
2 dihmeet2.h 
 |-  H = ( LHyp ` K )
3 dihmeet2.i 
 |-  I = ( ( DIsoH ` K ) ` W )
4 dihmeet2.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
5 dihmeet2.x 
 |-  ( ph -> X e. ran I )
6 dihmeet2.y 
 |-  ( ph -> Y e. ran I )
7 2 3 dihcnvid2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( I ` ( `' I ` X ) ) = X )
8 4 5 7 syl2anc 
 |-  ( ph -> ( I ` ( `' I ` X ) ) = X )
9 2 3 dihcnvid2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ Y e. ran I ) -> ( I ` ( `' I ` Y ) ) = Y )
10 4 6 9 syl2anc 
 |-  ( ph -> ( I ` ( `' I ` Y ) ) = Y )
11 8 10 ineq12d 
 |-  ( ph -> ( ( I ` ( `' I ` X ) ) i^i ( I ` ( `' I ` Y ) ) ) = ( X i^i Y ) )
12 eqid 
 |-  ( Base ` K ) = ( Base ` K )
13 12 2 3 dihcnvcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( `' I ` X ) e. ( Base ` K ) )
14 4 5 13 syl2anc 
 |-  ( ph -> ( `' I ` X ) e. ( Base ` K ) )
15 12 2 3 dihcnvcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ Y e. ran I ) -> ( `' I ` Y ) e. ( Base ` K ) )
16 4 6 15 syl2anc 
 |-  ( ph -> ( `' I ` Y ) e. ( Base ` K ) )
17 12 1 2 3 dihmeet 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( `' I ` X ) e. ( Base ` K ) /\ ( `' I ` Y ) e. ( Base ` K ) ) -> ( I ` ( ( `' I ` X ) ./\ ( `' I ` Y ) ) ) = ( ( I ` ( `' I ` X ) ) i^i ( I ` ( `' I ` Y ) ) ) )
18 4 14 16 17 syl3anc 
 |-  ( ph -> ( I ` ( ( `' I ` X ) ./\ ( `' I ` Y ) ) ) = ( ( I ` ( `' I ` X ) ) i^i ( I ` ( `' I ` Y ) ) ) )
19 2 3 dihmeetcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. ran I /\ Y e. ran I ) ) -> ( X i^i Y ) e. ran I )
20 4 5 6 19 syl12anc 
 |-  ( ph -> ( X i^i Y ) e. ran I )
21 2 3 dihcnvid2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X i^i Y ) e. ran I ) -> ( I ` ( `' I ` ( X i^i Y ) ) ) = ( X i^i Y ) )
22 4 20 21 syl2anc 
 |-  ( ph -> ( I ` ( `' I ` ( X i^i Y ) ) ) = ( X i^i Y ) )
23 11 18 22 3eqtr4rd 
 |-  ( ph -> ( I ` ( `' I ` ( X i^i Y ) ) ) = ( I ` ( ( `' I ` X ) ./\ ( `' I ` Y ) ) ) )
24 12 2 3 dihcnvcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X i^i Y ) e. ran I ) -> ( `' I ` ( X i^i Y ) ) e. ( Base ` K ) )
25 4 20 24 syl2anc 
 |-  ( ph -> ( `' I ` ( X i^i Y ) ) e. ( Base ` K ) )
26 4 simpld 
 |-  ( ph -> K e. HL )
27 26 hllatd 
 |-  ( ph -> K e. Lat )
28 12 1 latmcl 
 |-  ( ( K e. Lat /\ ( `' I ` X ) e. ( Base ` K ) /\ ( `' I ` Y ) e. ( Base ` K ) ) -> ( ( `' I ` X ) ./\ ( `' I ` Y ) ) e. ( Base ` K ) )
29 27 14 16 28 syl3anc 
 |-  ( ph -> ( ( `' I ` X ) ./\ ( `' I ` Y ) ) e. ( Base ` K ) )
30 12 2 3 dih11 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( `' I ` ( X i^i Y ) ) e. ( Base ` K ) /\ ( ( `' I ` X ) ./\ ( `' I ` Y ) ) e. ( Base ` K ) ) -> ( ( I ` ( `' I ` ( X i^i Y ) ) ) = ( I ` ( ( `' I ` X ) ./\ ( `' I ` Y ) ) ) <-> ( `' I ` ( X i^i Y ) ) = ( ( `' I ` X ) ./\ ( `' I ` Y ) ) ) )
31 4 25 29 30 syl3anc 
 |-  ( ph -> ( ( I ` ( `' I ` ( X i^i Y ) ) ) = ( I ` ( ( `' I ` X ) ./\ ( `' I ` Y ) ) ) <-> ( `' I ` ( X i^i Y ) ) = ( ( `' I ` X ) ./\ ( `' I ` Y ) ) ) )
32 23 31 mpbid 
 |-  ( ph -> ( `' I ` ( X i^i Y ) ) = ( ( `' I ` X ) ./\ ( `' I ` Y ) ) )