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Metamath Proof Explorer

Theorem cdleme50ldil

Description: Part of proof of Lemma D in Crawley p. 113. F is a lattice dilation. TODO: fix comment. (Contributed by NM, 9-Apr-2013)

Ref Expression
Hypotheses cdlemef50.b 
|- B = ( Base ` K )
cdlemef50.l 
|- .<_ = ( le ` K )
cdlemef50.j 
|- .\/ = ( join ` K )
cdlemef50.m 
|- ./\ = ( meet ` K )
cdlemef50.a 
|- A = ( Atoms ` K )
cdlemef50.h 
|- H = ( LHyp ` K )
cdlemef50.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdlemef50.d 
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdlemefs50.e 
|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemef50.f 
|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )
cdleme50ldil.i 
|- C = ( ( LDil ` K ) ` W )
Assertion cdleme50ldil 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> F e. C )

Proof

Step Hyp Ref Expression
1 cdlemef50.b 
 |-  B = ( Base ` K )
2 cdlemef50.l 
 |-  .<_ = ( le ` K )
3 cdlemef50.j 
 |-  .\/ = ( join ` K )
4 cdlemef50.m 
 |-  ./\ = ( meet ` K )
5 cdlemef50.a 
 |-  A = ( Atoms ` K )
6 cdlemef50.h 
 |-  H = ( LHyp ` K )
7 cdlemef50.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdlemef50.d 
 |-  D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9 cdlemefs50.e 
 |-  E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
10 cdlemef50.f 
 |-  F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )
11 cdleme50ldil.i 
 |-  C = ( ( LDil ` K ) ` W )
12 eqid 
 |-  ( LAut ` K ) = ( LAut ` K )
13 1 2 3 4 5 6 7 8 9 10 12 cdleme50laut 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> F e. ( LAut ` K ) )
14 simpr 
 |-  ( ( P =/= Q /\ -. e .<_ W ) -> -. e .<_ W )
15 14 con2i 
 |-  ( e .<_ W -> -. ( P =/= Q /\ -. e .<_ W ) )
16 10 cdleme31fv2 
 |-  ( ( e e. B /\ -. ( P =/= Q /\ -. e .<_ W ) ) -> ( F ` e ) = e )
17 15 16 sylan2 
 |-  ( ( e e. B /\ e .<_ W ) -> ( F ` e ) = e )
18 17 ex 
 |-  ( e e. B -> ( e .<_ W -> ( F ` e ) = e ) )
19 18 rgen 
 |-  A. e e. B ( e .<_ W -> ( F ` e ) = e )
20 19 a1i 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> A. e e. B ( e .<_ W -> ( F ` e ) = e ) )
21 1 2 6 12 11 isldil 
 |-  ( ( K e. HL /\ W e. H ) -> ( F e. C <-> ( F e. ( LAut ` K ) /\ A. e e. B ( e .<_ W -> ( F ` e ) = e ) ) ) )
22 21 3ad2ant1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( F e. C <-> ( F e. ( LAut ` K ) /\ A. e e. B ( e .<_ W -> ( F ` e ) = e ) ) ) )
23 13 20 22 mpbir2and 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> F e. C )