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Theorem cdleme28 31171
Description: Quantified version of cdleme28c 31170. (Compare cdleme24 31150.) (Contributed by NM, 7-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
)
cdleme27.g  |-  G  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdleme27.o  |-  O  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) )
cdleme27.e  |-  E  =  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  O ) )
cdleme27.y  |-  Y  =  if ( t  .<_  ( P  .\/  Q ) ,  E ,  G
)
Assertion
Ref Expression
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 ) )  =  ( Y  .\/  ( X 
./\  W ) ) ) )
Distinct variable groups:    t, s, u, z, A    B, s,
t, u, z    u, F    u, G    H, s,
t, z    .\/ , s, t, u, z    K, s, t, z    .<_ , s, t, u, z    ./\ , s,
t, u, z    t, N, u    O, s, u    P, s, t, u, z    Q, s, t, u, z    U, s, t, u, z    W, s, t, u, z    X, s, z, t
Allowed substitution hints:    C( z, u, t, s)    D( z, u, t, s)    E( z, u, t, s)    F( z, t, s)    G( z, t, s)    H( u)    K( u)    N( z, s)    O( z, t)    X( u)    Y( z, u, t, s)    Z( z, u, t, s)

Proof of Theorem cdleme28
StepHypRef Expression
1 simp11 988 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( s  e.  A  /\  t  e.  A )  /\  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( -.  t  .<_  W  /\  ( t 
.\/  ( X  ./\  W ) )  =  X ) ) )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
2 simp12 989 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( s  e.  A  /\  t  e.  A )  /\  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( -.  t  .<_  W  /\  ( t 
.\/  ( X  ./\  W ) )  =  X ) ) )  ->  P  =/=  Q )
3 simp2l 984 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( s  e.  A  /\  t  e.  A )  /\  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( -.  t  .<_  W  /\  ( t 
.\/  ( X  ./\  W ) )  =  X ) ) )  -> 
s  e.  A )
4 simp3ll 1029 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( s  e.  A  /\  t  e.  A )  /\  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( -.  t  .<_  W  /\  ( t 
.\/  ( X  ./\  W ) )  =  X ) ) )  ->  -.  s  .<_  W )
53, 4jca 520 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( s  e.  A  /\  t  e.  A )  /\  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( -.  t  .<_  W  /\  ( t 
.\/  ( X  ./\  W ) )  =  X ) ) )  -> 
( s  e.  A  /\  -.  s  .<_  W ) )
6 simp2r 985 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( s  e.  A  /\  t  e.  A )  /\  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( -.  t  .<_  W  /\  ( t 
.\/  ( X  ./\  W ) )  =  X ) ) )  -> 
t  e.  A )
7 simp3rl 1031 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( s  e.  A  /\  t  e.  A )  /\  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( -.  t  .<_  W  /\  ( t 
.\/  ( X  ./\  W ) )  =  X ) ) )  ->  -.  t  .<_  W )
86, 7jca 520 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( s  e.  A  /\  t  e.  A )  /\  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( -.  t  .<_  W  /\  ( t 
.\/  ( X  ./\  W ) )  =  X ) ) )  -> 
( t  e.  A  /\  -.  t  .<_  W ) )
9 simp3lr 1030 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( s  e.  A  /\  t  e.  A )  /\  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( -.  t  .<_  W  /\  ( t 
.\/  ( X  ./\  W ) )  =  X ) ) )  -> 
( s  .\/  ( X  ./\  W ) )  =  X )
10 simp3rr 1032 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( s  e.  A  /\  t  e.  A )  /\  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( -.  t  .<_  W  /\  ( t 
.\/  ( X  ./\  W ) )  =  X ) ) )  -> 
( t  .\/  ( X  ./\  W ) )  =  X )
11 simp13 990 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( s  e.  A  /\  t  e.  A )  /\  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( -.  t  .<_  W  /\  ( t 
.\/  ( X  ./\  W ) )  =  X ) ) )  -> 
( X  e.  B  /\  -.  X  .<_  W ) )
12 cdleme26.b . . . . 5  |-  B  =  ( Base `  K
)
13 cdleme26.l . . . . 5  |-  .<_  =  ( le `  K )
14 cdleme26.j . . . . 5  |-  .\/  =  ( join `  K )
15 cdleme26.m . . . . 5  |-  ./\  =  ( meet `  K )
16 cdleme26.a . . . . 5  |-  A  =  ( Atoms `  K )
17 cdleme26.h . . . . 5  |-  H  =  ( LHyp `  K
)
18 cdleme27.u . . . . 5  |-  U  =  ( ( P  .\/  Q )  ./\  W )
19 cdleme27.f . . . . 5  |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
20 cdleme27.z . . . . 5  |-  Z  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )
21 cdleme27.n . . . . 5  |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W
) ) )
22 cdleme27.d . . . . 5  |-  D  =  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  N ) )
23 cdleme27.c . . . . 5  |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F
)
24 cdleme27.g . . . . 5  |-  G  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
25 cdleme27.o . . . . 5  |-  O  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) )
26 cdleme27.e . . . . 5  |-  E  =  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  O ) )
27 cdleme27.y . . . . 5  |-  Y  =  if ( t  .<_  ( P  .\/  Q ) ,  E ,  G
)
2812, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27cdleme28c 31170 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( C  .\/  ( X  ./\  W ) )  =  ( Y  .\/  ( X 
./\  W ) ) )
291, 2, 5, 8, 9, 10, 11, 28syl133anc 1208 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( s  e.  A  /\  t  e.  A )  /\  (
( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( -.  t  .<_  W  /\  ( t 
.\/  ( X  ./\  W ) )  =  X ) ) )  -> 
( C  .\/  ( X  ./\  W ) )  =  ( Y  .\/  ( X  ./\  W ) ) )
30293exp 1153 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  ( (
s  e.  A  /\  t  e.  A )  ->  ( ( ( -.  s  .<_  W  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( -.  t  .<_  W  /\  ( t  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( C  .\/  ( X  ./\  W
) )  =  ( Y  .\/  ( X 
./\  W ) ) ) ) )
3130ralrimivv 2798 1  |-  ( ( ( ( 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 ) )  =  ( Y  .\/  ( X 
./\  W ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   A.wral 2706   ifcif 3740   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   iota_crio 6543   Basecbs 13470   lecple 13537   joincjn 14402   meetcmee 14403   Atomscatm 30062   HLchlt 30149   LHypclh 30782
This theorem is referenced by:  cdleme29b  31173
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-poset 14404  df-plt 14416  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-p0 14469  df-p1 14470  df-lat 14476  df-clat 14538  df-oposet 29975  df-ol 29977  df-oml 29978  df-covers 30065  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150  df-llines 30296  df-lplanes 30297  df-lvols 30298  df-lines 30299  df-psubsp 30301  df-pmap 30302  df-padd 30594  df-lhyp 30786
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