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Theorem cdlemk35 30252
Description: Part of proof of Lemma K of [Crawley] p. 118. cdlemk29-3 30251 with shorter hypotheses. (Contributed by NM, 18-Jul-2013.)
Hypotheses
Ref Expression
cdlemk4.b  |-  B  =  ( Base `  K
)
cdlemk4.l  |-  .<_  =  ( le `  K )
cdlemk4.j  |-  .\/  =  ( join `  K )
cdlemk4.m  |-  ./\  =  ( meet `  K )
cdlemk4.a  |-  A  =  ( Atoms `  K )
cdlemk4.h  |-  H  =  ( LHyp `  K
)
cdlemk4.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk4.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk4.z  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
cdlemk4.y  |-  Y  =  ( ( P  .\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `  ( G  o.  `' b
) ) ) )
cdlemk4.x  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  Y ) )
Assertion
Ref Expression
cdlemk35  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  X  e.  T )
Distinct variable groups:    z, b,  ./\    .<_ , b, z    .\/ , b, z    A, b, z    B, b, z    F, b, z    G, b, z    H, b, z    K, b, z    N, b, z    P, b, z    R, b, z    T, b, z    W, b, z
Allowed substitution hints:    X( z, b)    Y( z, b)    Z( z, b)

Proof of Theorem cdlemk35
StepHypRef Expression
1 cdlemk4.b . . . 4  |-  B  =  ( Base `  K
)
2 cdlemk4.l . . . 4  |-  .<_  =  ( le `  K )
3 cdlemk4.j . . . 4  |-  .\/  =  ( join `  K )
4 cdlemk4.m . . . 4  |-  ./\  =  ( meet `  K )
5 cdlemk4.a . . . 4  |-  A  =  ( Atoms `  K )
6 cdlemk4.h . . . 4  |-  H  =  ( LHyp `  K
)
7 cdlemk4.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemk4.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
9 eqid 2256 . . . 4  |-  ( f  e.  T  |->  ( iota_ i  e.  T ( i `
 P )  =  ( ( P  .\/  ( R `  f ) )  ./\  ( ( N `  P )  .\/  ( R `  (
f  o.  `' F
) ) ) ) ) )  =  ( f  e.  T  |->  (
iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
10 eqid 2256 . . . 4  |-  ( d  e.  T ,  e  e.  T  |->  ( iota_ j  e.  T ( j `
 P )  =  ( ( P  .\/  ( R `  e ) )  ./\  ( (
( ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f )
)  ./\  ( ( N `  P )  .\/  ( R `  (
f  o.  `' F
) ) ) ) ) ) `  d
) `  P )  .\/  ( R `  (
e  o.  `' d ) ) ) ) ) )  =  ( d  e.  T , 
e  e.  T  |->  (
iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) ) `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) ) )
11 eqid 2256 . . . 4  |-  ( iota_ z  e.  T A. b  e.  T  ( (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  ->  z  =  ( b ( d  e.  T ,  e  e.  T  |->  ( iota_ j  e.  T ( j `
 P )  =  ( ( P  .\/  ( R `  e ) )  ./\  ( (
( ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f )
)  ./\  ( ( N `  P )  .\/  ( R `  (
f  o.  `' F
) ) ) ) ) ) `  d
) `  P )  .\/  ( R `  (
e  o.  `' d ) ) ) ) ) ) G ) ) )  =  (
iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  z  =  ( b ( d  e.  T ,  e  e.  T  |->  ( iota_ j  e.  T ( j `  P )  =  ( ( P  .\/  ( R `  e )
)  ./\  ( (
( ( f  e.  T  |->  ( iota_ i  e.  T ( i `  P )  =  ( ( P  .\/  ( R `  f )
)  ./\  ( ( N `  P )  .\/  ( R `  (
f  o.  `' F
) ) ) ) ) ) `  d
) `  P )  .\/  ( R `  (
e  o.  `' d ) ) ) ) ) ) G ) ) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11cdlemk34 30250 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  ( iota_ z  e.  T A. b  e.  T  (
( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) )  ->  z  =  ( b ( d  e.  T , 
e  e.  T  |->  (
iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) ) `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) ) ) G ) ) )  =  ( iota_ z  e.  T A. b  e.  T  ( (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  ->  ( z `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( (
( P  .\/  ( R `  b )
)  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) ) 
.\/  ( R `  ( G  o.  `' b ) ) ) ) ) ) )
13 cdlemk4.x . . . 4  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  Y ) )
14 cdlemk4.y . . . . . . . . . 10  |-  Y  =  ( ( P  .\/  ( R `  G ) )  ./\  ( Z  .\/  ( R `  ( G  o.  `' b
) ) ) )
15 cdlemk4.z . . . . . . . . . . . 12  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
1615oveq1i 5788 . . . . . . . . . . 11  |-  ( Z 
.\/  ( R `  ( G  o.  `' b ) ) )  =  ( ( ( P  .\/  ( R `
 b ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( b  o.  `' F ) ) ) )  .\/  ( R `  ( G  o.  `' b ) ) )
1716oveq2i 5789 . . . . . . . . . 10  |-  ( ( P  .\/  ( R `
 G ) ) 
./\  ( Z  .\/  ( R `  ( G  o.  `' b ) ) ) )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( (
( P  .\/  ( R `  b )
)  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) ) 
.\/  ( R `  ( G  o.  `' b ) ) ) )
1814, 17eqtri 2276 . . . . . . . . 9  |-  Y  =  ( ( P  .\/  ( R `  G ) )  ./\  ( (
( P  .\/  ( R `  b )
)  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) ) 
.\/  ( R `  ( G  o.  `' b ) ) ) )
1918eqeq2i 2266 . . . . . . . 8  |-  ( ( z `  P )  =  Y  <->  ( z `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( (
( P  .\/  ( R `  b )
)  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) ) 
.\/  ( R `  ( G  o.  `' b ) ) ) ) )
2019imbi2i 305 . . . . . . 7  |-  ( ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) )  ->  (
z `  P )  =  Y )  <->  ( (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  ->  ( z `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( (
( P  .\/  ( R `  b )
)  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) ) 
.\/  ( R `  ( G  o.  `' b ) ) ) ) ) )
2120ralbii 2540 . . . . . 6  |-  ( A. b  e.  T  (
( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) )  ->  (
z `  P )  =  Y )  <->  A. b  e.  T  ( (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  ->  ( z `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( (
( P  .\/  ( R `  b )
)  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) ) 
.\/  ( R `  ( G  o.  `' b ) ) ) ) ) )
2221a1i 12 . . . . 5  |-  ( z  e.  T  ->  ( A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  Y )  <->  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( (
( P  .\/  ( R `  b )
)  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) ) 
.\/  ( R `  ( G  o.  `' b ) ) ) ) ) ) )
2322riotabiia 6276 . . . 4  |-  ( iota_ z  e.  T A. b  e.  T  ( (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  ->  ( z `  P )  =  Y ) )  =  (
iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( (
( P  .\/  ( R `  b )
)  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) ) 
.\/  ( R `  ( G  o.  `' b ) ) ) ) ) )
2413, 23eqtri 2276 . . 3  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( (
( P  .\/  ( R `  b )
)  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) ) 
.\/  ( R `  ( G  o.  `' b ) ) ) ) ) )
2512, 24syl6eqr 2306 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  ( iota_ z  e.  T A. b  e.  T  (
( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) )  ->  z  =  ( b ( d  e.  T , 
e  e.  T  |->  (
iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) ) `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) ) ) G ) ) )  =  X )
261, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11cdlemk29-3 30251 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  ( iota_ z  e.  T A. b  e.  T  (
( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) )  ->  z  =  ( b ( d  e.  T , 
e  e.  T  |->  (
iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) ) `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) ) ) G ) ) )  e.  T )
2725, 26eqeltrrd 2331 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  X  e.  T )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   class class class wbr 3983    e. cmpt 4037    _I cid 4262   `'ccnv 4646    |` cres 4649    o. ccom 4651   ` cfv 4659  (class class class)co 5778    e. cmpt2 5780   iota_crio 6249   Basecbs 13096   lecple 13163   joincjn 14026   meetcmee 14027   Atomscatm 28604   HLchlt 28691   LHypclh 29324   LTrncltrn 29441   trLctrl 29498
This theorem is referenced by:  cdlemk36  30253  cdlemk39  30256  cdlemk35s  30277
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-map 6728  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-p1 14094  df-lat 14100  df-clat 14162  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-llines 28838  df-lplanes 28839  df-lvols 28840  df-lines 28841  df-psubsp 28843  df-pmap 28844  df-padd 29136  df-lhyp 29328  df-laut 29329  df-ldil 29444  df-ltrn 29445  df-trl 29499
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