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Metamath Proof Explorer

Theorem cdlemefr31fv1

Description: Value of ( FR ) when -. R .<_ ( P .\/ Q ) . TODO This may be useful for shortening others that now use riotasv 3d . TODO: FIX COMMENT. (Contributed by NM, 30-Mar-2013)

Ref Expression
Hypotheses cdlemefr27.b 
|- B = ( Base ` K )
cdlemefr27.l 
|- .<_ = ( le ` K )
cdlemefr27.j 
|- .\/ = ( join ` K )
cdlemefr27.m 
|- ./\ = ( meet ` K )
cdlemefr27.a 
|- A = ( Atoms ` K )
cdlemefr27.h 
|- H = ( LHyp ` K )
cdlemefr27.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdlemefr27.c 
|- C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdlemefr27.n 
|- N = if ( s .<_ ( P .\/ Q ) , I , C )
cdleme29fr.o 
|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
cdleme29fr.f 
|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
cdleme43frv.x 
|- X = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
Assertion cdlemefr31fv1 
|- ( ( ( ( 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 ) = X )

Proof

Step Hyp Ref Expression
1 cdlemefr27.b 
 |-  B = ( Base ` K )
2 cdlemefr27.l 
 |-  .<_ = ( le ` K )
3 cdlemefr27.j 
 |-  .\/ = ( join ` K )
4 cdlemefr27.m 
 |-  ./\ = ( meet ` K )
5 cdlemefr27.a 
 |-  A = ( Atoms ` K )
6 cdlemefr27.h 
 |-  H = ( LHyp ` K )
7 cdlemefr27.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdlemefr27.c 
 |-  C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9 cdlemefr27.n 
 |-  N = if ( s .<_ ( P .\/ Q ) , I , C )
10 cdleme29fr.o 
 |-  O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
11 cdleme29fr.f 
 |-  F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
12 cdleme43frv.x 
 |-  X = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
13 1 2 3 4 5 6 7 8 9 10 11 cdlemefr32fva1 
 |-  ( ( ( ( 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 / s ]_ N )
14 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 )
15 simp3 
 |-  ( ( ( ( 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 .<_ ( P .\/ Q ) )
16 8 9 12 cdleme31sn2 
 |-  ( ( R e. A /\ -. R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N = X )
17 14 15 16 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 .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N = X )
18 13 17 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 ) = X )