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Metamath Proof Explorer

Theorem cdleme32fva1

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 2-Mar-2013)

Ref Expression
Hypotheses cdleme32.b 
|- B = ( Base ` K )
cdleme32.l 
|- .<_ = ( le ` K )
cdleme32.j 
|- .\/ = ( join ` K )
cdleme32.m 
|- ./\ = ( meet ` K )
cdleme32.a 
|- A = ( Atoms ` K )
cdleme32.h 
|- H = ( LHyp ` K )
cdleme32.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme32.c 
|- C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme32.d 
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme32.e 
|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdleme32.i 
|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) )
cdleme32.n 
|- N = if ( s .<_ ( P .\/ Q ) , I , C )
cdleme32.o 
|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
cdleme32.f 
|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
Assertion cdleme32fva1 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) /\ P =/= Q ) -> ( F ` R ) = [_ R / s ]_ N )

Proof

Step Hyp Ref Expression
1 cdleme32.b 
 |-  B = ( Base ` K )
2 cdleme32.l 
 |-  .<_ = ( le ` K )
3 cdleme32.j 
 |-  .\/ = ( join ` K )
4 cdleme32.m 
 |-  ./\ = ( meet ` K )
5 cdleme32.a 
 |-  A = ( Atoms ` K )
6 cdleme32.h 
 |-  H = ( LHyp ` K )
7 cdleme32.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdleme32.c 
 |-  C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9 cdleme32.d 
 |-  D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
10 cdleme32.e 
 |-  E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
11 cdleme32.i 
 |-  I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) )
12 cdleme32.n 
 |-  N = if ( s .<_ ( P .\/ Q ) , I , C )
13 cdleme32.o 
 |-  O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
14 cdleme32.f 
 |-  F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
15 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) /\ P =/= Q ) -> R e. A )
16 1 5 atbase 
 |-  ( R e. A -> R e. B )
17 15 16 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) /\ P =/= Q ) -> R e. B )
18 simp3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) /\ P =/= Q ) -> P =/= Q )
19 simp2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) /\ P =/= Q ) -> -. R .<_ W )
20 13 14 cdleme31fv1s 
 |-  ( ( R e. B /\ ( P =/= Q /\ -. R .<_ W ) ) -> ( F ` R ) = [_ R / x ]_ O )
21 17 18 19 20 syl12anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) /\ P =/= Q ) -> ( F ` R ) = [_ R / x ]_ O )
22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 cdleme32fva 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) /\ P =/= Q ) -> [_ R / x ]_ O = [_ R / s ]_ N )
23 21 22 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) /\ P =/= Q ) -> ( F ` R ) = [_ R / s ]_ N )