Metamath Proof Explorer

Theorem cdlemefr44

Description: Value of f(r) when r is an atom not under pq, using more compact hypotheses. TODO: eliminate and use cdlemefr45 instead? TODO: FIX COMMENT. (Contributed by NM, 31-Mar-2013)

Ref Expression
Hypotheses cdlemef44.b 
|- B = ( Base ` K )
cdlemef44.l 
|- .<_ = ( le ` K )
cdlemef44.j 
|- .\/ = ( join ` K )
cdlemef44.m 
|- ./\ = ( meet ` K )
cdlemef44.a 
|- A = ( Atoms ` K )
cdlemef44.h 
|- H = ( LHyp ` K )
cdlemef44.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdlemef44.d 
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdlemef44.o 
|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , I , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) )
cdlemef44.f 
|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
Assertion cdlemefr44 
|- ( ( ( ( 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 )

Proof

Step Hyp Ref Expression
1 cdlemef44.b 
 |-  B = ( Base ` K )
2 cdlemef44.l 
 |-  .<_ = ( le ` K )
3 cdlemef44.j 
 |-  .\/ = ( join ` K )
4 cdlemef44.m 
 |-  ./\ = ( meet ` K )
5 cdlemef44.a 
 |-  A = ( Atoms ` K )
6 cdlemef44.h 
 |-  H = ( LHyp ` K )
7 cdlemef44.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdlemef44.d 
 |-  D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9 cdlemef44.o 
 |-  O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , I , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) )
10 cdlemef44.f 
 |-  F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
11 eqid 
 |-  ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
12 biid 
 |-  ( s .<_ ( P .\/ Q ) <-> s .<_ ( P .\/ Q ) )
13 vex 
 |-  s e. _V
14 8 11 cdleme31sc 
 |-  ( s e. _V -> [_ s / t ]_ D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) )
15 13 14 ax-mp 
 |-  [_ s / t ]_ D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
16 12 15 ifbieq2i 
 |-  if ( s .<_ ( P .\/ Q ) , I , [_ s / t ]_ D ) = if ( s .<_ ( P .\/ Q ) , I , ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) )
17 eqid 
 |-  ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
18 1 2 3 4 5 6 7 11 16 9 10 17 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 ) = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
19 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 )
20 8 17 cdleme31sc 
 |-  ( R e. A -> [_ R / t ]_ D = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
21 19 20 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 ) ) ) )
22 18 21 eqtr4d 
 |-  ( ( ( ( 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 )