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Metamath Proof Explorer

Theorem cdleme29c

Description: Transform cdleme28b . (Compare cdleme25c .) TODO: FIX COMMENT. (Contributed by NM, 8-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 cdleme29c 
|- ( ( ( ( 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 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 ) ) ) )
14 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( X e. B /\ -. X .<_ W ) ) -> ( K e. HL /\ W e. H ) )
15 simp3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( X e. B /\ -. X .<_ W ) ) -> ( X e. B /\ -. X .<_ W ) )
16 1 2 3 4 5 6 lhpmcvr2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> E. s e. A ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) )
17 14 15 16 syl2anc 
 |-  ( ( ( ( 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 ) )
18 reusv1 
 |-  ( E. s e. A ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) -> ( E! v e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) -> v = ( C .\/ ( X ./\ W ) ) ) <-> E. v e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) -> v = ( C .\/ ( X ./\ W ) ) ) ) )
19 17 18 syl 
 |-  ( ( ( ( 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 ) ) ) <-> E. v e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( X ./\ W ) ) = X ) -> v = ( C .\/ ( X ./\ W ) ) ) ) )
20 13 19 mpbird 
 |-  ( ( ( ( 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 ) ) ) )