Metamath Proof Explorer


Theorem cdleme50trn123

Description: Part of proof that F is a translation. Combine all cases. 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 cdleme50trn123
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( 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 1 2 3 4 5 6 7 8 9 10 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 )
12 11 anass1rs
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) ) /\ P = Q ) -> ( ( R .\/ ( F ` R ) ) ./\ W ) = U )
13 1 2 3 4 5 6 7 8 9 10 cdleme50trn12
 |-  ( ( ( ( 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 )
14 13 anass1rs
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) ) /\ P =/= Q ) -> ( ( R .\/ ( F ` R ) ) ./\ W ) = U )
15 12 14 pm2.61dane
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( ( R .\/ ( F ` R ) ) ./\ W ) = U )