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Theorem cdlemkuv2 34403
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 16 on p. 119 for i = 1, where sigma1 (p) is  U, f1 is  D, and k1 is  O. (Contributed by NM, 2-Jul-2013.)
Hypotheses
Ref Expression
cdlemk1.b  |-  B  =  ( Base `  K
)
cdlemk1.l  |-  .<_  =  ( le `  K )
cdlemk1.j  |-  .\/  =  ( join `  K )
cdlemk1.m  |-  ./\  =  ( meet `  K )
cdlemk1.a  |-  A  =  ( Atoms `  K )
cdlemk1.h  |-  H  =  ( LHyp `  K
)
cdlemk1.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk1.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk1.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk1.o  |-  O  =  ( S `  D
)
cdlemk1.u  |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
Assertion
Ref Expression
cdlemkuv2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( U `  G ) `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) )
Distinct variable groups:    f, i,  ./\    .<_ , i    .\/ , f, i    A, i    D, f, i    f, F, i    i, H    i, K    f, N, i    P, f, i    R, f, i    T, f, i    f, W, i    ./\ , e    .\/ , e    D, e, j    e, G, j   
e, O    P, e    R, e    T, e    e, W    ./\ , j    .<_ , j    .\/ , j    A, j    D, j    j, F   
j, H    j, K    j, N    j, O    P, j    R, j    T, j   
j, W
Allowed substitution hints:    A( e, f)    B( e, f, i, j)    S( e, f, i, j)    U( e, f, i, j)    F( e)    G( f, i)    H( e, f)    K( e, f)    .<_ ( e, f)    N( e)    O( f, i)

Proof of Theorem cdlemkuv2
StepHypRef Expression
1 simp13 1037 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  G  e.  T )
2 cdlemk1.b . . . . 5  |-  B  =  ( Base `  K
)
3 cdlemk1.l . . . . 5  |-  .<_  =  ( le `  K )
4 cdlemk1.j . . . . 5  |-  .\/  =  ( join `  K )
5 cdlemk1.a . . . . 5  |-  A  =  ( Atoms `  K )
6 cdlemk1.h . . . . 5  |-  H  =  ( LHyp `  K
)
7 cdlemk1.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemk1.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
9 cdlemk1.m . . . . 5  |-  ./\  =  ( meet `  K )
10 cdlemk1.u . . . . 5  |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
112, 3, 4, 5, 6, 7, 8, 9, 10cdlemksv 34380 . . . 4  |-  ( G  e.  T  ->  ( U `  G )  =  ( iota_ j  e.  T  ( j `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) ) )
121, 11syl 17 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( U `  G )  =  (
iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 G ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( G  o.  `' D ) ) ) ) ) )
1312eqcomd 2430 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( iota_ j  e.  T  ( j `
 P )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) )  =  ( U `
 G ) )
14 cdlemk1.s . . . 4  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
15 cdlemk1.o . . . 4  |-  O  =  ( S `  D
)
162, 3, 4, 9, 5, 6, 7, 8, 14, 15, 10cdlemkuel 34401 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( U `  G )  e.  T
)
17 simp11l 1116 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  K  e.  HL )
18 simp11r 1117 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  W  e.  H )
19 simp33 1043 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
202, 3, 4, 9, 5, 6, 7, 8, 14, 15cdlemk16a 34392 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( (
( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) )  e.  A  /\  -.  ( ( P  .\/  ( R `  G ) )  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) 
.<_  W ) )
213, 5, 6, 7cdleme 34096 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  (
( ( P  .\/  ( R `  G ) )  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) )  e.  A  /\  -.  ( ( P  .\/  ( R `  G ) )  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) 
.<_  W ) )  ->  E! j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 G ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( G  o.  `' D ) ) ) ) )
2217, 18, 19, 20, 21syl211anc 1270 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  E! j  e.  T  ( j `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) )
23 nfcv 2580 . . . . . . 7  |-  F/_ j T
24 nfriota1 6274 . . . . . . 7  |-  F/_ j
( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) )
2523, 24nfmpt 4512 . . . . . 6  |-  F/_ j
( e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
2610, 25nfcxfr 2578 . . . . 5  |-  F/_ j U
27 nfcv 2580 . . . . 5  |-  F/_ j G
2826, 27nffv 5888 . . . 4  |-  F/_ j
( U `  G
)
29 nfcv 2580 . . . . . 6  |-  F/_ j P
3028, 29nffv 5888 . . . . 5  |-  F/_ j
( ( U `  G ) `  P
)
3130nfeq1 2595 . . . 4  |-  F/ j ( ( U `  G ) `  P
)  =  ( ( P  .\/  ( R `
 G ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( G  o.  `' D ) ) ) )
32 fveq1 5880 . . . . 5  |-  ( j  =  ( U `  G )  ->  (
j `  P )  =  ( ( U `
 G ) `  P ) )
3332eqeq1d 2424 . . . 4  |-  ( j  =  ( U `  G )  ->  (
( j `  P
)  =  ( ( P  .\/  ( R `
 G ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( G  o.  `' D ) ) ) )  <->  ( ( U `  G ) `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) ) )
3428, 31, 33riota2f 6288 . . 3  |-  ( ( ( U `  G
)  e.  T  /\  E! j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 G ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( G  o.  `' D ) ) ) ) )  ->  ( ( ( U `  G ) `
 P )  =  ( ( P  .\/  ( R `  G ) )  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) )  <-> 
( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 G ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( G  o.  `' D ) ) ) ) )  =  ( U `  G ) ) )
3516, 22, 34syl2anc 665 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( (
( U `  G
) `  P )  =  ( ( P 
.\/  ( R `  G ) )  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D ) ) ) )  <->  ( iota_ j  e.  T  ( j `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) )  =  ( U `
 G ) ) )
3613, 35mpbird 235 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( U `  G ) `  P )  =  ( ( P  .\/  ( R `  G )
)  ./\  ( ( O `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   E!wreu 2773   class class class wbr 4423    |-> cmpt 4482    _I cid 4763   `'ccnv 4852    |` cres 4855    o. ccom 4857   ` cfv 5601   iota_crio 6266  (class class class)co 6305   Basecbs 15120   lecple 15196   joincjn 16188   meetcmee 16189   Atomscatm 32798   HLchlt 32885   LHypclh 33518   LTrncltrn 33635   trLctrl 33693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-riotaBAD 32494
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-undef 7031  df-map 7485  df-preset 16172  df-poset 16190  df-plt 16203  df-lub 16219  df-glb 16220  df-join 16221  df-meet 16222  df-p0 16284  df-p1 16285  df-lat 16291  df-clat 16353  df-oposet 32711  df-ol 32713  df-oml 32714  df-covers 32801  df-ats 32802  df-atl 32833  df-cvlat 32857  df-hlat 32886  df-llines 33032  df-lplanes 33033  df-lvols 33034  df-lines 33035  df-psubsp 33037  df-pmap 33038  df-padd 33330  df-lhyp 33522  df-laut 33523  df-ldil 33638  df-ltrn 33639  df-trl 33694
This theorem is referenced by:  cdlemk18  34404  cdlemk7u  34406  cdlemk12u  34408  cdlemk21N  34409  cdlemk20  34410  cdlemkuv2-2  34421  cdlemk31  34432  cdlemkuv2-3N  34435
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