Metamath Proof Explorer

Theorem dihjatcclem3

Description: Lemma for dihjatcc . (Contributed by NM, 28-Sep-2014)

Ref Expression
Hypotheses dihjatcclem.b 
|- B = ( Base ` K )
dihjatcclem.l 
|- .<_ = ( le ` K )
dihjatcclem.h 
|- H = ( LHyp ` K )
dihjatcclem.j 
|- .\/ = ( join ` K )
dihjatcclem.m 
|- ./\ = ( meet ` K )
dihjatcclem.a 
|- A = ( Atoms ` K )
dihjatcclem.u 
|- U = ( ( DVecH ` K ) ` W )
dihjatcclem.s 
|- .(+) = ( LSSum ` U )
dihjatcclem.i 
|- I = ( ( DIsoH ` K ) ` W )
dihjatcclem.v 
|- V = ( ( P .\/ Q ) ./\ W )
dihjatcclem.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dihjatcclem.p 
|- ( ph -> ( P e. A /\ -. P .<_ W ) )
dihjatcclem.q 
|- ( ph -> ( Q e. A /\ -. Q .<_ W ) )
dihjatcc.w 
|- C = ( ( oc ` K ) ` W )
dihjatcc.t 
|- T = ( ( LTrn ` K ) ` W )
dihjatcc.r 
|- R = ( ( trL ` K ) ` W )
dihjatcc.e 
|- E = ( ( TEndo ` K ) ` W )
dihjatcc.g 
|- G = ( iota_ d e. T ( d ` C ) = P )
dihjatcc.dd 
|- D = ( iota_ d e. T ( d ` C ) = Q )
Assertion dihjatcclem3 
|- ( ph -> ( R ` ( G o. `' D ) ) = V )

Proof

Step Hyp Ref Expression
1 dihjatcclem.b 
 |-  B = ( Base ` K )
2 dihjatcclem.l 
 |-  .<_ = ( le ` K )
3 dihjatcclem.h 
 |-  H = ( LHyp ` K )
4 dihjatcclem.j 
 |-  .\/ = ( join ` K )
5 dihjatcclem.m 
 |-  ./\ = ( meet ` K )
6 dihjatcclem.a 
 |-  A = ( Atoms ` K )
7 dihjatcclem.u 
 |-  U = ( ( DVecH ` K ) ` W )
8 dihjatcclem.s 
 |-  .(+) = ( LSSum ` U )
9 dihjatcclem.i 
 |-  I = ( ( DIsoH ` K ) ` W )
10 dihjatcclem.v 
 |-  V = ( ( P .\/ Q ) ./\ W )
11 dihjatcclem.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
12 dihjatcclem.p 
 |-  ( ph -> ( P e. A /\ -. P .<_ W ) )
13 dihjatcclem.q 
 |-  ( ph -> ( Q e. A /\ -. Q .<_ W ) )
14 dihjatcc.w 
 |-  C = ( ( oc ` K ) ` W )
15 dihjatcc.t 
 |-  T = ( ( LTrn ` K ) ` W )
16 dihjatcc.r 
 |-  R = ( ( trL ` K ) ` W )
17 dihjatcc.e 
 |-  E = ( ( TEndo ` K ) ` W )
18 dihjatcc.g 
 |-  G = ( iota_ d e. T ( d ` C ) = P )
19 dihjatcc.dd 
 |-  D = ( iota_ d e. T ( d ` C ) = Q )
20 2 6 3 14 lhpocnel2 
 |-  ( ( K e. HL /\ W e. H ) -> ( C e. A /\ -. C .<_ W ) )
21 11 20 syl 
 |-  ( ph -> ( C e. A /\ -. C .<_ W ) )
22 2 6 3 15 18 ltrniotacl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( C e. A /\ -. C .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) -> G e. T )
23 11 21 12 22 syl3anc 
 |-  ( ph -> G e. T )
24 2 6 3 15 19 ltrniotacl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( C e. A /\ -. C .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> D e. T )
25 11 21 13 24 syl3anc 
 |-  ( ph -> D e. T )
26 3 15 ltrncnv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ D e. T ) -> `' D e. T )
27 11 25 26 syl2anc 
 |-  ( ph -> `' D e. T )
28 3 15 ltrnco 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ `' D e. T ) -> ( G o. `' D ) e. T )
29 11 23 27 28 syl3anc 
 |-  ( ph -> ( G o. `' D ) e. T )
30 2 4 5 6 3 15 16 trlval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( G o. `' D ) e. T /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( R ` ( G o. `' D ) ) = ( ( Q .\/ ( ( G o. `' D ) ` Q ) ) ./\ W ) )
31 11 29 13 30 syl3anc 
 |-  ( ph -> ( R ` ( G o. `' D ) ) = ( ( Q .\/ ( ( G o. `' D ) ` Q ) ) ./\ W ) )
32 13 simpld 
 |-  ( ph -> Q e. A )
33 2 6 3 15 ltrncoval 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ `' D e. T ) /\ Q e. A ) -> ( ( G o. `' D ) ` Q ) = ( G ` ( `' D ` Q ) ) )
34 11 23 27 32 33 syl121anc 
 |-  ( ph -> ( ( G o. `' D ) ` Q ) = ( G ` ( `' D ` Q ) ) )
35 2 6 3 15 19 ltrniotacnvval 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( C e. A /\ -. C .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( `' D ` Q ) = C )
36 11 21 13 35 syl3anc 
 |-  ( ph -> ( `' D ` Q ) = C )
37 36 fveq2d 
 |-  ( ph -> ( G ` ( `' D ` Q ) ) = ( G ` C ) )
38 2 6 3 15 18 ltrniotaval 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( C e. A /\ -. C .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( G ` C ) = P )
39 11 21 12 38 syl3anc 
 |-  ( ph -> ( G ` C ) = P )
40 37 39 eqtrd 
 |-  ( ph -> ( G ` ( `' D ` Q ) ) = P )
41 34 40 eqtrd 
 |-  ( ph -> ( ( G o. `' D ) ` Q ) = P )
42 41 oveq2d 
 |-  ( ph -> ( Q .\/ ( ( G o. `' D ) ` Q ) ) = ( Q .\/ P ) )
43 11 simpld 
 |-  ( ph -> K e. HL )
44 12 simpld 
 |-  ( ph -> P e. A )
45 4 6 hlatjcom 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) )
46 43 44 32 45 syl3anc 
 |-  ( ph -> ( P .\/ Q ) = ( Q .\/ P ) )
47 42 46 eqtr4d 
 |-  ( ph -> ( Q .\/ ( ( G o. `' D ) ` Q ) ) = ( P .\/ Q ) )
48 47 oveq1d 
 |-  ( ph -> ( ( Q .\/ ( ( G o. `' D ) ` Q ) ) ./\ W ) = ( ( P .\/ Q ) ./\ W ) )
49 48 10 eqtr4di 
 |-  ( ph -> ( ( Q .\/ ( ( G o. `' D ) ` Q ) ) ./\ W ) = V )
50 31 49 eqtrd 
 |-  ( ph -> ( R ` ( G o. `' D ) ) = V )