Metamath Proof Explorer

Theorem cdleme29b

Description: Transform cdleme28 . (Compare cdleme25b .) TODO: FIX COMMENT. (Contributed by NM, 7-Feb-2013)

Ref Expression
Hypotheses cdleme26.b 
|- B = ( Base ` K )
cdleme26.l 
|- .<_ = ( le ` K )
cdleme26.j 
|- .\/ = ( join ` K )
cdleme26.m 
|- ./\ = ( meet ` K )
cdleme26.a 
|- A = ( Atoms ` K )
cdleme26.h 
|- H = ( LHyp ` K )
cdleme27.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme27.f 
|- F = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme27.z 
|- Z = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) )
cdleme27.n 
|- N = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) )
cdleme27.d 
|- D = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) )
cdleme27.c 
|- C = if ( s .<_ ( P .\/ Q ) , D , F )
Assertion cdleme29b 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( X e. B /\ -. X .<_ W ) ) -> E. v e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) -> v = ( C .\/ ( X ./\ W ) ) ) )

Proof

Step Hyp Ref Expression
1 cdleme26.b 
 |-  B = ( Base ` K )
2 cdleme26.l 
 |-  .<_ = ( le ` K )
3 cdleme26.j 
 |-  .\/ = ( join ` K )
4 cdleme26.m 
 |-  ./\ = ( meet ` K )
5 cdleme26.a 
 |-  A = ( Atoms ` K )
6 cdleme26.h 
 |-  H = ( LHyp ` K )
7 cdleme27.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdleme27.f 
 |-  F = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9 cdleme27.z 
 |-  Z = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) )
10 cdleme27.n 
 |-  N = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) )
11 cdleme27.d 
 |-  D = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) )
12 cdleme27.c 
 |-  C = if ( s .<_ ( P .\/ Q ) , D , F )
13 1 2 3 4 5 6 7 8 9 10 11 12 cdleme29ex 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( X e. B /\ -. X .<_ W ) ) -> E. s e. A ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) /\ ( C .\/ ( X ./\ W ) ) e. B ) )
14 eqid 
 |-  ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
15 eqid 
 |-  ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) )
16 eqid 
 |-  ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) ) ) = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) ) )
17 eqid 
 |-  if ( t .<_ ( P .\/ Q ) , ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) ) ) , ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) ) = if ( t .<_ ( P .\/ Q ) , ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) ) ) , ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) )
18 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 17 cdleme28 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( X e. B /\ -. X .<_ W ) ) -> A. s e. A A. t e. A ( ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) /\ ( -. t .<_ W /\ ( t .\/ ( X ./\ W ) ) = X ) ) -> ( C .\/ ( X ./\ W ) ) = ( if ( t .<_ ( P .\/ Q ) , ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) ) ) , ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) ) .\/ ( X ./\ W ) ) ) )
19 breq1 
 |-  ( s = t -> ( s .<_ W <-> t .<_ W ) )
20 19 notbid 
 |-  ( s = t -> ( -. s .<_ W <-> -. t .<_ W ) )
21 oveq1 
 |-  ( s = t -> ( s .\/ ( X ./\ W ) ) = ( t .\/ ( X ./\ W ) ) )
22 21 eqeq1d 
 |-  ( s = t -> ( ( s .\/ ( X ./\ W ) ) = X <-> ( t .\/ ( X ./\ W ) ) = X ) )
23 20 22 anbi12d 
 |-  ( s = t -> ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) <-> ( -. t .<_ W /\ ( t .\/ ( X ./\ W ) ) = X ) ) )
24 12 oveq1i 
 |-  ( C .\/ ( X ./\ W ) ) = ( if ( s .<_ ( P .\/ Q ) , D , F ) .\/ ( X ./\ W ) )
25 breq1 
 |-  ( s = t -> ( s .<_ ( P .\/ Q ) <-> t .<_ ( P .\/ Q ) ) )
26 oveq1 
 |-  ( s = t -> ( s .\/ z ) = ( t .\/ z ) )
27 26 oveq1d 
 |-  ( s = t -> ( ( s .\/ z ) ./\ W ) = ( ( t .\/ z ) ./\ W ) )
28 27 oveq2d 
 |-  ( s = t -> ( Z .\/ ( ( s .\/ z ) ./\ W ) ) = ( Z .\/ ( ( t .\/ z ) ./\ W ) ) )
29 28 oveq2d 
 |-  ( s = t -> ( ( P .\/ Q ) ./\ ( Z .\/ ( ( s .\/ z ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) )
30 10 29 syl5eq 
 |-  ( s = t -> N = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) )
31 30 eqeq2d 
 |-  ( s = t -> ( u = N <-> u = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) ) )
32 31 imbi2d 
 |-  ( s = t -> ( ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) <-> ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) ) ) )
33 32 ralbidv 
 |-  ( s = t -> ( A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) <-> A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) ) ) )
34 33 riotabidv 
 |-  ( s = t -> ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = N ) ) = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) ) ) )
35 11 34 syl5eq 
 |-  ( s = t -> D = ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) ) ) )
36 oveq1 
 |-  ( s = t -> ( s .\/ U ) = ( t .\/ U ) )
37 oveq2 
 |-  ( s = t -> ( P .\/ s ) = ( P .\/ t ) )
38 37 oveq1d 
 |-  ( s = t -> ( ( P .\/ s ) ./\ W ) = ( ( P .\/ t ) ./\ W ) )
39 38 oveq2d 
 |-  ( s = t -> ( Q .\/ ( ( P .\/ s ) ./\ W ) ) = ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
40 36 39 oveq12d 
 |-  ( s = t -> ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) )
41 8 40 syl5eq 
 |-  ( s = t -> F = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) )
42 25 35 41 ifbieq12d 
 |-  ( s = t -> if ( s .<_ ( P .\/ Q ) , D , F ) = if ( t .<_ ( P .\/ Q ) , ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) ) ) , ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) ) )
43 42 oveq1d 
 |-  ( s = t -> ( if ( s .<_ ( P .\/ Q ) , D , F ) .\/ ( X ./\ W ) ) = ( if ( t .<_ ( P .\/ Q ) , ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) ) ) , ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) ) .\/ ( X ./\ W ) ) )
44 24 43 syl5eq 
 |-  ( s = t -> ( C .\/ ( X ./\ W ) ) = ( if ( t .<_ ( P .\/ Q ) , ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) ) ) , ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) ) .\/ ( X ./\ W ) ) )
45 23 44 reusv3 
 |-  ( E. s e. A ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) /\ ( C .\/ ( X ./\ W ) ) e. B ) -> ( A. s e. A A. t e. A ( ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) /\ ( -. t .<_ W /\ ( t .\/ ( X ./\ W ) ) = X ) ) -> ( C .\/ ( X ./\ W ) ) = ( if ( t .<_ ( P .\/ Q ) , ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) ) ) , ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) ) .\/ ( X ./\ W ) ) ) <-> E. v e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) -> v = ( C .\/ ( X ./\ W ) ) ) ) )
46 45 biimpd 
 |-  ( E. s e. A ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) /\ ( C .\/ ( X ./\ W ) ) e. B ) -> ( A. s e. A A. t e. A ( ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) /\ ( -. t .<_ W /\ ( t .\/ ( X ./\ W ) ) = X ) ) -> ( C .\/ ( X ./\ W ) ) = ( if ( t .<_ ( P .\/ Q ) , ( iota_ u e. B A. z e. A ( ( -. z .<_ W /\ -. z .<_ ( P .\/ Q ) ) -> u = ( ( P .\/ Q ) ./\ ( Z .\/ ( ( t .\/ z ) ./\ W ) ) ) ) ) , ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) ) .\/ ( X ./\ W ) ) ) -> E. v e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) -> v = ( C .\/ ( X ./\ W ) ) ) ) )
47 13 18 46 sylc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( X e. B /\ -. X .<_ W ) ) -> E. v e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) -> v = ( C .\/ ( X ./\ W ) ) ) )