Metamath Proof Explorer

Theorem dihjatc

Description: Isomorphism H of lattice join of an element under the fiducial hyperplane with atom not under it. (Contributed by NM, 26-Aug-2014)

Ref Expression
Hypotheses dihjatc.b 
|- B = ( Base ` K )
dihjatc.l 
|- .<_ = ( le ` K )
dihjatc.h 
|- H = ( LHyp ` K )
dihjatc.j 
|- .\/ = ( join ` K )
dihjatc.a 
|- A = ( Atoms ` K )
dihjatc.u 
|- U = ( ( DVecH ` K ) ` W )
dihjatc.s 
|- .(+) = ( LSSum ` U )
dihjatc.i 
|- I = ( ( DIsoH ` K ) ` W )
dihjatc.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dihjatc.x 
|- ( ph -> ( X e. B /\ X .<_ W ) )
dihjatc.p 
|- ( ph -> ( P e. A /\ -. P .<_ W ) )
Assertion dihjatc 
|- ( ph -> ( I ` ( X .\/ P ) ) = ( ( I ` X ) .(+) ( I ` P ) ) )

Proof

Step Hyp Ref Expression
1 dihjatc.b 
 |-  B = ( Base ` K )
2 dihjatc.l 
 |-  .<_ = ( le ` K )
3 dihjatc.h 
 |-  H = ( LHyp ` K )
4 dihjatc.j 
 |-  .\/ = ( join ` K )
5 dihjatc.a 
 |-  A = ( Atoms ` K )
6 dihjatc.u 
 |-  U = ( ( DVecH ` K ) ` W )
7 dihjatc.s 
 |-  .(+) = ( LSSum ` U )
8 dihjatc.i 
 |-  I = ( ( DIsoH ` K ) ` W )
9 dihjatc.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
10 dihjatc.x 
 |-  ( ph -> ( X e. B /\ X .<_ W ) )
11 dihjatc.p 
 |-  ( ph -> ( P e. A /\ -. P .<_ W ) )
12 9 simpld 
 |-  ( ph -> K e. HL )
13 hlop 
 |-  ( K e. HL -> K e. OP )
14 12 13 syl 
 |-  ( ph -> K e. OP )
15 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
16 1 15 op1cl 
 |-  ( K e. OP -> ( 1. ` K ) e. B )
17 14 16 syl 
 |-  ( ph -> ( 1. ` K ) e. B )
18 10 simpld 
 |-  ( ph -> X e. B )
19 11 simpld 
 |-  ( ph -> P e. A )
20 1 5 atbase 
 |-  ( P e. A -> P e. B )
21 19 20 syl 
 |-  ( ph -> P e. B )
22 1 2 15 ople1 
 |-  ( ( K e. OP /\ P e. B ) -> P .<_ ( 1. ` K ) )
23 14 21 22 syl2anc 
 |-  ( ph -> P .<_ ( 1. ` K ) )
24 hlol 
 |-  ( K e. HL -> K e. OL )
25 12 24 syl 
 |-  ( ph -> K e. OL )
26 eqid 
 |-  ( meet ` K ) = ( meet ` K )
27 1 26 15 olm12 
 |-  ( ( K e. OL /\ X e. B ) -> ( ( 1. ` K ) ( meet ` K ) X ) = X )
28 25 18 27 syl2anc 
 |-  ( ph -> ( ( 1. ` K ) ( meet ` K ) X ) = X )
29 10 simprd 
 |-  ( ph -> X .<_ W )
30 28 29 eqbrtrd 
 |-  ( ph -> ( ( 1. ` K ) ( meet ` K ) X ) .<_ W )
31 1 2 3 4 26 5 6 7 8 dihjatc3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( 1. ` K ) e. B /\ X e. B ) /\ ( P e. A /\ -. P .<_ W ) /\ ( P .<_ ( 1. ` K ) /\ ( ( 1. ` K ) ( meet ` K ) X ) .<_ W ) ) -> ( I ` ( ( ( 1. ` K ) ( meet ` K ) X ) .\/ P ) ) = ( ( I ` ( ( 1. ` K ) ( meet ` K ) X ) ) .(+) ( I ` P ) ) )
32 9 17 18 11 23 30 31 syl312anc 
 |-  ( ph -> ( I ` ( ( ( 1. ` K ) ( meet ` K ) X ) .\/ P ) ) = ( ( I ` ( ( 1. ` K ) ( meet ` K ) X ) ) .(+) ( I ` P ) ) )
33 28 fvoveq1d 
 |-  ( ph -> ( I ` ( ( ( 1. ` K ) ( meet ` K ) X ) .\/ P ) ) = ( I ` ( X .\/ P ) ) )
34 28 fveq2d 
 |-  ( ph -> ( I ` ( ( 1. ` K ) ( meet ` K ) X ) ) = ( I ` X ) )
35 34 oveq1d 
 |-  ( ph -> ( ( I ` ( ( 1. ` K ) ( meet ` K ) X ) ) .(+) ( I ` P ) ) = ( ( I ` X ) .(+) ( I ` P ) ) )
36 32 33 35 3eqtr3d 
 |-  ( ph -> ( I ` ( X .\/ P ) ) = ( ( I ` X ) .(+) ( I ` P ) ) )