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Theorem cdlemk22-3 31712
Description: Part of proof of Lemma K of [Crawley] p. 118. Lines 26-27, p. 119 for i=1 and j=2. (Contributed by NM, 7-Jul-2013.)
Hypotheses
Ref Expression
cdlemk3.b  |-  B  =  ( Base `  K
)
cdlemk3.l  |-  .<_  =  ( le `  K )
cdlemk3.j  |-  .\/  =  ( join `  K )
cdlemk3.m  |-  ./\  =  ( meet `  K )
cdlemk3.a  |-  A  =  ( Atoms `  K )
cdlemk3.h  |-  H  =  ( LHyp `  K
)
cdlemk3.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk3.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk3.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk3.u1  |-  Y  =  ( d  e.  T ,  e  e.  T  |->  ( iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) ) )
Assertion
Ref Expression
cdlemk22-3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  C  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( C  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F ) )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  (
( D Y G ) `  P )  =  ( ( C Y G ) `  P ) )
Distinct variable groups:    e, d,
f, i,  ./\    .<_ , i    .\/ , d, e, f, i    A, i    j, d, D, e, f, i    f, F, i    G, d, e, j   
i, H    i, K    f, N, i    P, d, e, f, i    R, d, e, f, i    T, d, e, f, i    W, d, e, f, i    ./\ , j    .<_ , j    .\/ , j    A, j    j, F    j, H    j, K    j, N    P, j    R, j    S, d, e, j    T, j    j, W    F, d, e    .<_ , e    C, d, e, f, i, j   
f, G, i
Allowed substitution hints:    A( e, f, d)    B( e, f, i, j, d)    S( f, i)    H( e, f, d)    K( e, f, d)    .<_ ( f, d)    N( e, d)    Y( e, f, i, j, d)

Proof of Theorem cdlemk22-3
StepHypRef Expression
1 cdlemk3.b . . 3  |-  B  =  ( Base `  K
)
2 cdlemk3.l . . 3  |-  .<_  =  ( le `  K )
3 cdlemk3.j . . 3  |-  .\/  =  ( join `  K )
4 cdlemk3.m . . 3  |-  ./\  =  ( meet `  K )
5 cdlemk3.a . . 3  |-  A  =  ( Atoms `  K )
6 cdlemk3.h . . 3  |-  H  =  ( LHyp `  K
)
7 cdlemk3.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemk3.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
9 cdlemk3.s . . 3  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
10 eqid 2296 . . 3  |-  ( S `
 C )  =  ( S `  C
)
11 eqid 2296 . . 3  |-  ( e  e.  T  |->  ( iota_ j  e.  T ( j `
 P )  =  ( ( P  .\/  ( R `  e ) )  ./\  ( (
( S `  C
) `  P )  .\/  ( R `  (
e  o.  `' C
) ) ) ) ) )  =  ( e  e.  T  |->  (
iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  C ) `
 P )  .\/  ( R `  ( e  o.  `' C ) ) ) ) ) )
12 eqid 2296 . . 3  |-  ( S `
 D )  =  ( S `  D
)
13 eqid 2296 . . 3  |-  ( e  e.  T  |->  ( iota_ j  e.  T ( j `
 P )  =  ( ( P  .\/  ( R `  e ) )  ./\  ( (
( S `  D
) `  P )  .\/  ( R `  (
e  o.  `' D
) ) ) ) ) )  =  ( e  e.  T  |->  (
iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  D ) `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13cdlemk22 31704 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  C  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( C  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F ) )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  (
( ( e  e.  T  |->  ( iota_ j  e.  T ( j `  P )  =  ( ( P  .\/  ( R `  e )
)  ./\  ( (
( S `  D
) `  P )  .\/  ( R `  (
e  o.  `' D
) ) ) ) ) ) `  G
) `  P )  =  ( ( ( e  e.  T  |->  (
iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  C ) `
 P )  .\/  ( R `  ( e  o.  `' C ) ) ) ) ) ) `  G ) `
 P ) )
15 simp13 987 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  C  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( C  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F ) )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  D  e.  T )
16 simp212 1094 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  C  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( C  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F ) )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  G  e.  T )
17 cdlemk3.u1 . . . . 5  |-  Y  =  ( d  e.  T ,  e  e.  T  |->  ( iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) ) )
181, 2, 3, 4, 5, 6, 7, 8, 9, 17, 12, 13cdlemkuu 31706 . . . 4  |-  ( ( D  e.  T  /\  G  e.  T )  ->  ( D Y G )  =  ( ( e  e.  T  |->  (
iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  D ) `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) ) `  G ) )
1915, 16, 18syl2anc 642 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  C  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( C  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F ) )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  ( D Y G )  =  ( ( e  e.  T  |->  ( iota_ j  e.  T ( j `  P )  =  ( ( P  .\/  ( R `  e )
)  ./\  ( (
( S `  D
) `  P )  .\/  ( R `  (
e  o.  `' D
) ) ) ) ) ) `  G
) )
2019fveq1d 5543 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  C  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( C  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F ) )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  (
( D Y G ) `  P )  =  ( ( ( e  e.  T  |->  (
iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  D ) `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) ) `  G ) `
 P ) )
21 simp213 1095 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  C  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( C  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F ) )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  C  e.  T )
221, 2, 3, 4, 5, 6, 7, 8, 9, 17, 10, 11cdlemkuu 31706 . . . 4  |-  ( ( C  e.  T  /\  G  e.  T )  ->  ( C Y G )  =  ( ( e  e.  T  |->  (
iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  C ) `
 P )  .\/  ( R `  ( e  o.  `' C ) ) ) ) ) ) `  G ) )
2321, 16, 22syl2anc 642 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  C  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( C  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F ) )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  ( C Y G )  =  ( ( e  e.  T  |->  ( iota_ j  e.  T ( j `  P )  =  ( ( P  .\/  ( R `  e )
)  ./\  ( (
( S `  C
) `  P )  .\/  ( R `  (
e  o.  `' C
) ) ) ) ) ) `  G
) )
2423fveq1d 5543 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  C  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( C  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F ) )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  (
( C Y G ) `  P )  =  ( ( ( e  e.  T  |->  (
iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  C ) `
 P )  .\/  ( R `  ( e  o.  `' C ) ) ) ) ) ) `  G ) `
 P ) )
2514, 20, 243eqtr4d 2338 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  C  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( C  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  C )  /\  ( R `  C
)  =/=  ( R `
 F ) )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  C
)  =/=  ( R `
 D ) ) ) )  ->  (
( D Y G ) `  P )  =  ( ( C Y G ) `  P ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039    e. cmpt 4093    _I cid 4320   `'ccnv 4704    |` cres 4707    o. ccom 4709   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   iota_crio 6313   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Atomscatm 30075   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969
This theorem is referenced by:  cdlemk23-3  31713  cdlemk25-3  31715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970
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