Metamath Proof Explorer


Theorem cdlemefs32fva1

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT. (Contributed by NM, 29-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 ) )
Assertion 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 )

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 breq1
 |-  ( s = R -> ( s .<_ ( P .\/ Q ) <-> R .<_ ( P .\/ Q ) ) )
15 simp1
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
16 simp3l
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> s e. A )
17 simp3rl
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> -. s .<_ W )
18 16 17 jca
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> ( s e. A /\ -. s .<_ W ) )
19 simp3rr
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> s .<_ ( P .\/ Q ) )
20 simp2
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> P =/= Q )
21 1 2 3 4 5 6 7 8 9 10 11 cdlemefs27cl
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N e. B )
22 15 18 19 20 21 syl13anc
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> N e. B )
23 1 2 3 4 5 6 7 8 9 10 11 cdlemefs32snb
 |-  ( ( ( ( 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 e. B )
24 1 2 3 4 5 6 14 22 23 12 13 cdlemefrs32fva1
 |-  ( ( ( ( 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 )