Metamath Proof Explorer

Theorem cdleme50trn3

Description: Part of proof that F is a translation. P = Q case. TODO: fix comment. (Contributed by NM, 10-Apr-2013)

Ref Expression
Hypotheses cdlemef50.b 
|- B = ( Base ` K )
cdlemef50.l 
|- .<_ = ( le ` K )
cdlemef50.j 
|- .\/ = ( join ` K )
cdlemef50.m 
|- ./\ = ( meet ` K )
cdlemef50.a 
|- A = ( Atoms ` K )
cdlemef50.h 
|- H = ( LHyp ` K )
cdlemef50.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdlemef50.d 
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdlemefs50.e 
|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemef50.f 
|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )
Assertion cdleme50trn3 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( R .\/ ( F ` R ) ) ./\ W ) = U )

Proof

Step Hyp Ref Expression
1 cdlemef50.b 
 |-  B = ( Base ` K )
2 cdlemef50.l 
 |-  .<_ = ( le ` K )
3 cdlemef50.j 
 |-  .\/ = ( join ` K )
4 cdlemef50.m 
 |-  ./\ = ( meet ` K )
5 cdlemef50.a 
 |-  A = ( Atoms ` K )
6 cdlemef50.h 
 |-  H = ( LHyp ` K )
7 cdlemef50.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdlemef50.d 
 |-  D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9 cdlemefs50.e 
 |-  E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
10 cdlemef50.f 
 |-  F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )
11 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
12 simprr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R e. A /\ -. R .<_ W ) )
13 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
14 2 4 13 5 6 lhpmat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( R ./\ W ) = ( 0. ` K ) )
15 11 12 14 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R ./\ W ) = ( 0. ` K ) )
16 simprrl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> R e. A )
17 1 5 atbase 
 |-  ( R e. A -> R e. B )
18 16 17 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> R e. B )
19 simprl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> P = Q )
20 10 cdleme31id 
 |-  ( ( R e. B /\ P = Q ) -> ( F ` R ) = R )
21 18 19 20 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( F ` R ) = R )
22 21 oveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ ( F ` R ) ) = ( R .\/ R ) )
23 simpl1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. HL )
24 3 5 hlatjidm 
 |-  ( ( K e. HL /\ R e. A ) -> ( R .\/ R ) = R )
25 23 16 24 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ R ) = R )
26 22 25 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ ( F ` R ) ) = R )
27 26 oveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( R .\/ ( F ` R ) ) ./\ W ) = ( R ./\ W ) )
28 simpl2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
29 2 4 13 5 6 lhpmat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P ./\ W ) = ( 0. ` K ) )
30 11 28 29 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( P ./\ W ) = ( 0. ` K ) )
31 15 27 30 3eqtr4d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( R .\/ ( F ` R ) ) ./\ W ) = ( P ./\ W ) )
32 simpl2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> P e. A )
33 3 5 hlatjidm 
 |-  ( ( K e. HL /\ P e. A ) -> ( P .\/ P ) = P )
34 23 32 33 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( P .\/ P ) = P )
35 19 oveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( P .\/ P ) = ( P .\/ Q ) )
36 34 35 eqtr3d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> P = ( P .\/ Q ) )
37 36 oveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( P ./\ W ) = ( ( P .\/ Q ) ./\ W ) )
38 31 37 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( R .\/ ( F ` R ) ) ./\ W ) = ( ( P .\/ Q ) ./\ W ) )
39 38 7 eqtr4di 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P = Q /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( R .\/ ( F ` R ) ) ./\ W ) = U )