Metamath Proof Explorer


Theorem cdlemefr45e

Description: Explicit expansion of cdlemefr45 . TODO: use to shorten cdlemefr45 uses? TODO: FIX COMMENT. (Contributed by NM, 10-Apr-2013)

Ref Expression
Hypotheses cdlemef45.b
|- B = ( Base ` K )
cdlemef45.l
|- .<_ = ( le ` K )
cdlemef45.j
|- .\/ = ( join ` K )
cdlemef45.m
|- ./\ = ( meet ` K )
cdlemef45.a
|- A = ( Atoms ` K )
cdlemef45.h
|- H = ( LHyp ` K )
cdlemef45.u
|- U = ( ( P .\/ Q ) ./\ W )
cdlemef45.d
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdlemef45.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 cdlemefr45e
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( F ` R ) = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemef45.b
 |-  B = ( Base ` K )
2 cdlemef45.l
 |-  .<_ = ( le ` K )
3 cdlemef45.j
 |-  .\/ = ( join ` K )
4 cdlemef45.m
 |-  ./\ = ( meet ` K )
5 cdlemef45.a
 |-  A = ( Atoms ` K )
6 cdlemef45.h
 |-  H = ( LHyp ` K )
7 cdlemef45.u
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdlemef45.d
 |-  D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9 cdlemef45.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 ) )
10 1 2 3 4 5 6 7 8 9 cdlemefr45
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( F ` R ) = [_ R / t ]_ D )
11 simp2rl
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> R e. A )
12 eqid
 |-  ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
13 8 12 cdleme31sc
 |-  ( R e. A -> [_ R / t ]_ D = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
14 11 13 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 .<_ ( P .\/ Q ) ) -> [_ R / t ]_ D = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
15 10 14 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 .<_ ( P .\/ Q ) ) -> ( F ` R ) = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )