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Metamath Proof Explorer

Theorem cdleme50trn2

Description: Part of proof that F is a translation. Remove S hypotheses no longer needed from cdleme50trn2a . TODO: fix comment. (Contributed by NM, 10-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 ) )
Assertion cdleme50trn2 
|- ( ( ( ( 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 .\/ ( F ` R ) ) ./\ W ) = U )

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 simp11 
 |-  ( ( ( ( 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 ) ) -> ( K e. HL /\ W e. H ) )
12 simp12 
 |-  ( ( ( ( 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 ) ) -> ( P e. A /\ -. P .<_ W ) )
13 simp13 
 |-  ( ( ( ( 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 ) ) -> ( Q e. A /\ -. Q .<_ W ) )
14 simp2l 
 |-  ( ( ( ( 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 ) ) -> P =/= Q )
15 2 3 5 6 cdlemb2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> E. e e. A ( -. e .<_ W /\ -. e .<_ ( P .\/ Q ) ) )
16 11 12 13 14 15 syl121anc 
 |-  ( ( ( ( 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 ) ) -> E. e e. A ( -. e .<_ W /\ -. e .<_ ( P .\/ Q ) ) )
17 simp1 
 |-  ( ( ( ( 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 ) /\ ( e e. A /\ ( -. e .<_ W /\ -. e .<_ ( P .\/ Q ) ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
18 simp2l 
 |-  ( ( ( ( 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 ) /\ ( e e. A /\ ( -. e .<_ W /\ -. e .<_ ( P .\/ Q ) ) ) ) ) -> P =/= Q )
19 simp2r 
 |-  ( ( ( ( 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 ) /\ ( e e. A /\ ( -. e .<_ W /\ -. e .<_ ( P .\/ Q ) ) ) ) ) -> ( R e. A /\ -. R .<_ W ) )
20 simp3rl 
 |-  ( ( ( ( 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 ) /\ ( e e. A /\ ( -. e .<_ W /\ -. e .<_ ( P .\/ Q ) ) ) ) ) -> e e. A )
21 simprrl 
 |-  ( ( R .<_ ( P .\/ Q ) /\ ( e e. A /\ ( -. e .<_ W /\ -. e .<_ ( P .\/ Q ) ) ) ) -> -. e .<_ W )
22 21 3ad2ant3 
 |-  ( ( ( ( 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 ) /\ ( e e. A /\ ( -. e .<_ W /\ -. e .<_ ( P .\/ Q ) ) ) ) ) -> -. e .<_ W )
23 20 22 jca 
 |-  ( ( ( ( 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 ) /\ ( e e. A /\ ( -. e .<_ W /\ -. e .<_ ( P .\/ Q ) ) ) ) ) -> ( e e. A /\ -. e .<_ W ) )
24 simp3l 
 |-  ( ( ( ( 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 ) /\ ( e e. A /\ ( -. e .<_ W /\ -. e .<_ ( P .\/ Q ) ) ) ) ) -> R .<_ ( P .\/ Q ) )
25 simprrr 
 |-  ( ( R .<_ ( P .\/ Q ) /\ ( e e. A /\ ( -. e .<_ W /\ -. e .<_ ( P .\/ Q ) ) ) ) -> -. e .<_ ( P .\/ Q ) )
26 25 3ad2ant3 
 |-  ( ( ( ( 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 ) /\ ( e e. A /\ ( -. e .<_ W /\ -. e .<_ ( P .\/ Q ) ) ) ) ) -> -. e .<_ ( P .\/ Q ) )
27 1 2 3 4 5 6 7 8 9 10 cdleme50trn2a 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( e e. A /\ -. e .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. e .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ ( F ` R ) ) ./\ W ) = U )
28 17 18 19 23 24 26 27 syl132anc 
 |-  ( ( ( ( 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 ) /\ ( e e. A /\ ( -. e .<_ W /\ -. e .<_ ( P .\/ Q ) ) ) ) ) -> ( ( R .\/ ( F ` R ) ) ./\ W ) = U )
29 28 3exp 
 |-  ( ( ( 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 ) /\ ( e e. A /\ ( -. e .<_ W /\ -. e .<_ ( P .\/ Q ) ) ) ) -> ( ( R .\/ ( F ` R ) ) ./\ W ) = U ) ) )
30 29 exp4a 
 |-  ( ( ( 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 ) -> ( ( e e. A /\ ( -. e .<_ W /\ -. e .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ ( F ` R ) ) ./\ W ) = U ) ) ) )
31 30 3imp 
 |-  ( ( ( ( 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 ) ) -> ( ( e e. A /\ ( -. e .<_ W /\ -. e .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ ( F ` R ) ) ./\ W ) = U ) )
32 31 expd 
 |-  ( ( ( ( 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 ) ) -> ( e e. A -> ( ( -. e .<_ W /\ -. e .<_ ( P .\/ Q ) ) -> ( ( R .\/ ( F ` R ) ) ./\ W ) = U ) ) )
33 32 rexlimdv 
 |-  ( ( ( ( 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 ) ) -> ( E. e e. A ( -. e .<_ W /\ -. e .<_ ( P .\/ Q ) ) -> ( ( R .\/ ( F ` R ) ) ./\ W ) = U ) )
34 16 33 mpd 
 |-  ( ( ( ( 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 .\/ ( F ` R ) ) ./\ W ) = U )