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Theorem cdlemk35 36739
Description: Part of proof of Lemma K of [Crawley] p. 118. cdlemk29-3 36738 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
Dummy variables  d 
e  f  i  j are mutually distinct and distinct from all other variables.
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 2457 . . . 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 2457 . . . 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 2457 . . . 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 36737 . . 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 6306 . . . . . . . . . . 11  |-  ( Z 
.\/  ( R `  ( G  o.  `' b ) ) )  =  ( ( ( P  .\/  ( R `
 b ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( b  o.  `' F ) ) ) )  .\/  ( R `  ( G  o.  `' b ) ) )
1716oveq2i 6307 . . . . . . . . . 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 2486 . . . . . . . . 9  |-  Y  =  ( ( P  .\/  ( R `  G ) )  ./\  ( (
( P  .\/  ( R `  b )
)  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) ) 
.\/  ( R `  ( G  o.  `' b ) ) ) )
1918eqeq2i 2475 . . . . . . . 8  |-  ( ( z `  P )  =  Y  <->  ( z `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( (
( P  .\/  ( R `  b )
)  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) ) 
.\/  ( R `  ( G  o.  `' b ) ) ) ) )
2019imbi2i 312 . . . . . . 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 2888 . . . . . 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 11 . . . . 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 6275 . . . 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 2486 . . 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 2516 . 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 36738 . 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 2546 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   class class class wbr 4456    |-> cmpt 4515    _I cid 4799   `'ccnv 5007    |` cres 5010    o. ccom 5012   ` cfv 5594   iota_crio 6257  (class class class)co 6296    |-> cmpt2 6298   Basecbs 14643   lecple 14718   joincjn 15699   meetcmee 15700   Atomscatm 35089   HLchlt 35176   LHypclh 35809   LTrncltrn 35926   trLctrl 35984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34785
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-undef 7020  df-map 7440  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-p1 15796  df-lat 15802  df-clat 15864  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-llines 35323  df-lplanes 35324  df-lvols 35325  df-lines 35326  df-psubsp 35328  df-pmap 35329  df-padd 35621  df-lhyp 35813  df-laut 35814  df-ldil 35929  df-ltrn 35930  df-trl 35985
This theorem is referenced by:  cdlemk36  36740  cdlemk39  36743  cdlemk35s  36764
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