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Theorem cdleme29c 33708
Description: Transform cdleme28b 33703. (Compare cdleme25c 33687.) TODO: FIX COMMENT (Contributed by NM, 8-Feb-2013.)
Hypotheses
Ref Expression
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
Ref Expression
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 ) ) ) )
Distinct variable groups:    u, s,
z, A    B, s, u, z    u, F    H, s, z    .\/ , s, u, z    K, s, z    .<_ , s, u, z    ./\ , s, u, z    u, N    P, s, u, z    Q, s, u, z    U, s, u, z    W, s, u, z    X, s   
v, A    v, B    v, 
.\/    v,  .<_    v,  ./\    v, P    v, Q    v, U    v, W    v, C    v, s, Z, u    z, v, X
Allowed substitution hints:    C( z, u, s)    D( z, v, u, s)    F( z, v, s)    H( v, u)    K( v, u)    N( z, v, s)    X( u)    Z( z)

Proof of Theorem cdleme29c
StepHypRef Expression
1 cdleme26.b . . 3  |-  B  =  ( Base `  K
)
2 cdleme26.l . . 3  |-  .<_  =  ( le `  K )
3 cdleme26.j . . 3  |-  .\/  =  ( join `  K )
4 cdleme26.m . . 3  |-  ./\  =  ( meet `  K )
5 cdleme26.a . . 3  |-  A  =  ( Atoms `  K )
6 cdleme26.h . . 3  |-  H  =  ( LHyp `  K
)
7 cdleme27.u . . 3  |-  U  =  ( ( P  .\/  Q )  ./\  W )
8 cdleme27.f . . 3  |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
9 cdleme27.z . . 3  |-  Z  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )
10 cdleme27.n . . 3  |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W
) ) )
11 cdleme27.d . . 3  |-  D  =  ( iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  N ) )
12 cdleme27.c . . 3  |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F
)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cdleme29b 33707 . 2  |-  ( ( ( ( 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 1013 . . . 4  |-  ( ( ( ( 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 985 . . . 4  |-  ( ( ( ( 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 ) )
161, 2, 3, 4, 5, 6lhpmcvr2 33356 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  E. s  e.  A  ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X ) )
1714, 15, 16syl2anc 656 . . 3  |-  ( ( ( ( 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 4489 . . 3  |-  ( 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 ) ) ) ) )
1917, 18syl 16 . 2  |-  ( ( ( ( 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 ) ) ) ) )
2013, 19mpbird 232 1  |-  ( ( ( ( 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 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   E!wreu 2715   ifcif 3788   class class class wbr 4289   ` cfv 5415   iota_crio 6048  (class class class)co 6090   Basecbs 14170   lecple 14241   joincjn 15110   meetcmee 15111   Atomscatm 32596   HLchlt 32683   LHypclh 33316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-riotaBAD 32292
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-undef 6788  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-p1 15206  df-lat 15212  df-clat 15274  df-oposet 32509  df-ol 32511  df-oml 32512  df-covers 32599  df-ats 32600  df-atl 32631  df-cvlat 32655  df-hlat 32684  df-llines 32830  df-lplanes 32831  df-lvols 32832  df-lines 32833  df-psubsp 32835  df-pmap 32836  df-padd 33128  df-lhyp 33320
This theorem is referenced by:  cdleme29cl  33709
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