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Metamath Proof Explorer

Theorem cdlemefs31fv1

Description: Value of ( FR ) when R .<_ ( P .\/ Q ) . TODO This may be useful for shortening others that now use riotasv 3d . TODO: FIX COMMENT. ***END OF VALUE AT ATOM STUFF TO REPLACE ONES BELOW*** 

       "cdleme3xsn1aw" decreased using "cdlemefs32sn1aw"
       "cdleme32sn1aw" decreased from 3302 to 36 using "cdlemefs32sn1aw".
       "cdleme32sn2aw" decreased from 1687 to 26 using "cdlemefr32sn2aw".
       "cdleme32snaw" decreased from 376 to 375 using "cdlemefs32sn1aw".
       "cdleme32snaw" decreased from 375 to 368 using "cdlemefr32sn2aw".
       "cdleme35sn3a" decreased from 547 to 523 using "cdleme43frv1sn".
(Contributed by NM, 27-Mar-2013)

Ref Expression
Hypotheses cdlemefs32.b 
|- B = ( Base ` K )
cdlemefs32.l 
|- .<_ = ( le ` K )
cdlemefs32.j 
|- .\/ = ( join ` K )
cdlemefs32.m 
|- ./\ = ( meet ` K )
cdlemefs32.a 
|- A = ( Atoms ` K )
cdlemefs32.h 
|- H = ( LHyp ` K )
cdlemefs32.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdlemefs32.d 
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdlemefs32.e 
|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemefs32.i 
|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) )
cdlemefs32.n 
|- N = if ( s .<_ ( P .\/ Q ) , I , C )
cdleme29fs.o 
|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
cdleme29fs.f 
|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
cdleme43fsv.y 
|- Y = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme43fsv.z 
|- Z = ( ( P .\/ Q ) ./\ ( Y .\/ ( ( R .\/ S ) ./\ W ) ) )
Assertion cdlemefs31fv1 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( F ` R ) = Z )

Proof

Step Hyp Ref Expression
1 cdlemefs32.b 
 |-  B = ( Base ` K )
2 cdlemefs32.l 
 |-  .<_ = ( le ` K )
3 cdlemefs32.j 
 |-  .\/ = ( join ` K )
4 cdlemefs32.m 
 |-  ./\ = ( meet ` K )
5 cdlemefs32.a 
 |-  A = ( Atoms ` K )
6 cdlemefs32.h 
 |-  H = ( LHyp ` K )
7 cdlemefs32.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdlemefs32.d 
 |-  D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9 cdlemefs32.e 
 |-  E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
10 cdlemefs32.i 
 |-  I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) )
11 cdlemefs32.n 
 |-  N = if ( s .<_ ( P .\/ Q ) , I , C )
12 cdleme29fs.o 
 |-  O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
13 cdleme29fs.f 
 |-  F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
14 cdleme43fsv.y 
 |-  Y = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
15 cdleme43fsv.z 
 |-  Z = ( ( P .\/ Q ) ./\ ( Y .\/ ( ( R .\/ S ) ./\ W ) ) )
16 simp1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
17 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> P =/= Q )
18 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( R e. A /\ -. R .<_ W ) )
19 simp3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> R .<_ ( P .\/ Q ) )
20 1 2 3 4 5 6 7 8 9 10 11 12 13 cdlemefs32fva1 
 |-  ( ( ( ( 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 )
21 16 17 18 19 20 syl121anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( F ` R ) = [_ R / s ]_ N )
22 1 2 3 4 5 6 7 8 9 10 11 14 15 cdleme43fsv1sn 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> [_ R / s ]_ N = Z )
23 21 22 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( F ` R ) = Z )