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Theorem cdlemk35 30368
Description: Part of proof of Lemma K of [Crawley] p. 118. cdlemk29-3 30367 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 2284 . . . 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 2284 . . . 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 2284 . . . 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 30366 . . 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 5829 . . . . . . . . . . 11  |-  ( Z 
.\/  ( R `  ( G  o.  `' b ) ) )  =  ( ( ( P  .\/  ( R `
 b ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( b  o.  `' F ) ) ) )  .\/  ( R `  ( G  o.  `' b ) ) )
1716oveq2i 5830 . . . . . . . . . 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 2304 . . . . . . . . 9  |-  Y  =  ( ( P  .\/  ( R `  G ) )  ./\  ( (
( P  .\/  ( R `  b )
)  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) ) 
.\/  ( R `  ( G  o.  `' b ) ) ) )
1918eqeq2i 2294 . . . . . . . 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 2568 . . . . . 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 6317 . . . 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 2304 . . 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 2334 . 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 30367 . 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 2359 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 1628    e. wcel 1688    =/= wne 2447   A.wral 2544   class class class wbr 4024    e. cmpt 4078    _I cid 4303   `'ccnv 4687    |` cres 4690    o. ccom 4692   ` cfv 5221  (class class class)co 5819    e. cmpt2 5821   iota_crio 6290   Basecbs 13142   lecple 13209   joincjn 14072   meetcmee 14073   Atomscatm 28720   HLchlt 28807   LHypclh 29440   LTrncltrn 29557   trLctrl 29614
This theorem is referenced by:  cdlemk36  30369  cdlemk39  30372  cdlemk35s  30393
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-map 6769  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-llines 28954  df-lplanes 28955  df-lvols 28956  df-lines 28957  df-psubsp 28959  df-pmap 28960  df-padd 29252  df-lhyp 29444  df-laut 29445  df-ldil 29560  df-ltrn 29561  df-trl 29615
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